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% Project Gutenberg's A Textbook of General Astronomy, by Charles A. Young%
%                                                                         %
% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
% re-use it under the terms of the Project Gutenberg License included     %
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%                                                                         %
% Title: A Textbook of General Astronomy                                  %
%        For Colleges and Scientific Schools                              %
%                                                                         %
% Author: Charles A. Young                                                %
%                                                                         %
% Release Date: August 30, 2011 [EBook #37275]                            %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: ISO-8859-1                                      %
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% *** START OF THIS PROJECT GUTENBERG EBOOK A TEXTBOOK OF GENERAL ASTRONOMY ***
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\begin{verbatim}
Project Gutenberg's A Textbook of General Astronomy, by Charles A. Young

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.net


Title: A Textbook of General Astronomy
       For Colleges and Scientific Schools

Author: Charles A. Young

Release Date: August 30, 2011 [EBook #37275]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK A TEXTBOOK OF GENERAL ASTRONOMY ***
\end{verbatim}

\clearpage
\begin{center}
\pdfbookmark[0]{A Text-book of General Astronomy}{A Text-book of General Astronomy}
\small
Produced by Susan Skinner, Nigel Blower, Brenda Lewis and
the Online Distributed Proofreading Team at
http://www.pgdp.net (This file was produced from images
generously made available by the Bibliothčque nationale
de France (BnF/Gallica) at http://gallica.bnf.fr,
illustrations generously made available by The Internet
Archive/Canadian Libraries)
\end{center}
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  A small number of minor typographical errors and inconsistencies
  have been corrected. See the {\ttfamily\footnotesize DPtypo} command
  in the \LaTeX\ source for more information.
}
\clearpage

\frontmatter
%% -----File: 001.png---
%% -----File: 002.png---
%% -----File: 003.png---
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\nbVS

\includegraphicsmid{frontispiece}{FRONTISPIECE.

\textsc{The Great Telescope of the Lick Observatory,\\
        Mt.\ Hamilton, Cal.}

Object-Glass made by A. Clark \& Sons:\\
Aperture, 36~in.; Focal Length, 56~ft.\ 2~in.\\
Mounting by Warner \& Swasey.}

\nbVS

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%% -----File: 004.png---
\begin{center}
\nbVS
\so{\Large A~{}~TEXT-BOOK}

\nbVS
{\small OF}

\nbVS
{\Huge GENERAL ASTRONOMY}

\nbVS
{\small FOR}

\nbVS
{\large COLLEGES AND SCIENTIFIC SCHOOLS}

\nbVS[2]
{\small BY} \\[2ex]
{\large CHARLES A. YOUNG, \textsc{Ph}.D., LL.D.,}  \\[1ex]
\textsc{\small Professor of Astronomy in the College of New Jersey (Princeton)}.

\nbVS[2]
\nbrule

\nbVS[2]
\textsc{Boston, U.S.A., and London}:  \\
GINN \& COMPANY, PUBLISHERS.  \\
1889.
\end{center}
\clearpage

%% -----File: 005.png
\begin{center}
\nbVS

\textsc{Entered at Stationers' Hall}.

\nbrule

\textsc{Copyright, 1888, by}  \\
CHARLES A. YOUNG,

\nbrule

\textsc{All Rights Reserved}.

\nbVS

\textsc{Typography by J.~S. Cushing \& Co., Boston, U.S.A.}  \\
\nbrule[4cm] \\
\textsc{Presswork by Ginn \& Co., Boston, U.S.A.}
\end{center}
%% -----File: 006.png---Folio v-------

\clearpage
\pdfbookmark[1]{Preface}{Preface}
\chapter*{PREFACE.}
\markboth{PREFACE.}{}
\pagestyle{fancy} % for remainder of document
\thispagestyle{empty}% just for first page of Preface, ToC, Introduction
\nbrule\bigskip

\textsc{The} present work is designed as a text-book of Astronomy
suited to the \textit{general} course in our colleges and schools of
science, and is meant to supply that amount of information
upon the subject which may fairly be expected of every
``liberally educated'' person. While it assumes the previous
discipline and mental maturity usually corresponding to the
latter years of the college course, it does not demand the
peculiar mathematical training and aptitude necessary as the
basis of a \textit{special} course in the science---only the most elementary
knowledge of Algebra, Geometry, and Trigonometry
is required for its reading. Its aim is to give a clear, accurate,
and justly proportioned presentation of astronomical
facts, principles, and methods in such a form that they can
be easily apprehended by the average college student with a
reasonable amount of effort.

The limitations of time are such in our college course that
probably it will not be possible in most cases for a class to
take thoroughly everything in the book. The fine print is to
be regarded rather as collateral reading, important to anything
like a complete view of the subject, but not essential to
the course. Some of the chapters can even be omitted in
cases where it is found necessary to abridge the course as
much as possible; \textit{e.g.}, the chapters on Instruments and on
Perturbations.

While the work is no mere compilation, it makes no claims
to special originality: information and help have been drawn
%% -----File: 007.png---Folio vi-------
from all available sources. The author is under great obligations
to the astronomical histories of Grant and Wolf, and
especially to Miss Clerke's admirable ``History of Astronomy in
the Nineteenth Century.'' Many data also have been drawn
from Houzeau's valuable ``Vade Mecum de l'Astronomie.''

It has been intended to bring the book well down to date,
and to indicate to the student the sources of information on
subjects which are necessarily here treated inadequately on
account of the limitations of time and space.

Special acknowledgments are due to Professor Langley and
to his publishers, Messrs.\ Ticknor \&~Co., for the use of a
number of illustrations from his beautiful book, ``The New
Astronomy''; and also to D.~Appleton \&~Co.\ for the use of
several cuts from the author's little book on the Sun. Professor
Trowbridge of Cambridge kindly provided the original
negative from which was made the cut illustrating the comparison
of the spectrum of iron with that of the sun. Warner
\& Swasey of Cleveland and Fauth \&~Co.\ of Washington have
also furnished the engravings of a number of astronomical
instruments.

Professors Todd, Emerson, Upton, and McNeill have given
most valuable assistance and suggestions in the revision of the
proof; as indeed, in hardly a less degree, have several others.

The author will consider it a great favor if those who may
use the book will kindly communicate to him, either directly
or through the publishers, any errata, in order that they
may be promptly corrected.

{\footnotesize\textsc{Princeton, N.~J.}, August, 1888.}

\nbrule[4cm]

{\footnotesize\textsc{Note}. In this issue of the book a number of errors which appeared in the
first impression have been corrected.

\indent\indent April, 1889.}
%% -----File: 008.png---Folio vii-------

\clearpage
\pdfbookmark[1]{Table of Contents}{Table of Contents}
\chapter*{TABLE OF CONTENTS.}
\markboth{TABLE OF CONTENTS.}{}
\thispagestyle{empty}% just for first page of Preface, ToC, Introduction
\nbrule\bigskip

{\small
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\nbtocpagesheading

\nbtocentry{INTRODUCTION}{INTRODUCTION}

\nbtocentry{CHAPTERI}{CHAPTER I.---\textsc{The Doctrine of the Sphere}; Definitions and General
Considerations}

\nbtocentry{CHAPTERII}{\stretchyspace CHAPTER II.---\textsc{Astronomical Instruments}: the Telescope; Time-Keepers
and Chronograph; the Transit Instrument and Accessories;
the Meridian Circle and Reading Microscope; the Altitude and Azimuth
Instrument; the Equatorial Instrument and Micrometer; the
Sextant}

\nbtocentry{CHAPTERIII}{CHAPTER III.---\textsc{Corrections to Astronomical Observations}: the Dip
of the Horizon; Parallax; Semi-Diameter; Refraction; and Twilight}

\nbtocentry{CHAPTERIV}{CHAPTER IV.---\textsc{Problems of Practical Astronomy}: the Determination
of Latitude, of Time, of Longitude, of a Ship's Place at Sea,
of Azimuth, and of the Apparent Right Ascension and Declination
of a Heavenly Body; the Time of Sunrise or Sunset}

\nbtocentry{CHAPTERV}{CHAPTER V.---\textsc{The Earth}: the Approximate Determination of its
Dimensions and Form; Proofs of its Rotation; Accurate Determination
of its Dimensions by Geodetic Surveys and Pendulum Observations;
Determination of its Mass and Density}

\nbtocentry{CHAPTERVI}{CHAPTER VI.---\textsc{The Earth's Orbital Motion}: the Motion of the
Sun among the Stars; the Equation of Time; Precession; Nutation;
Aberration; the Calendar}

\nbtocentry{CHAPTERVII}{CHAPTER VII.---\textsc{The Moon}: her Orbital Motion; Distance and Dimensions;
Mass, Density, and Superficial Gravity; Rotation and
Librations; Phases; Light and Heat; Physical Condition; Influence
exerted on the Earth; Surface Structure; and Possible Changes}

\clearpage
\nbtocpagesheading

\nbtocentry{CHAPTERVIII}{CHAPTER VIII.---\textsc{The Sun}: Distance and Dimensions; Mass and Density;
Rotation; Solar Eye-Pieces, and Study of the Sun's Surface;
General Views as to Constitution; Sun Spots: their Appearance,
Nature, Distribution, and Periodicity; the Spectroscope and the Solar
Spectrum; Chemical Composition of the Sun; the Chromosphere and
Prominences; the Corona}

\nbtocentry{CHAPTERIX}{CHAPTER IX.---\textsc{The Sun's Light and Heat}: Comparison of Sunlight
with Artificial Lights; the Measurement of the Sun's Heat, and Determination
of the Solar Constant; the Pyrheliometer, Actinometer, and
Bolometer; the Sun's Temperature; Maintenance of the Sun's Radiation;
and Conclusions as to its Age and Future Endurance}
%% -----File: 009.png---Folio viii-------

\nbtocentry{CHAPTERX}{CHAPTER X.---\textsc{Eclipses}; Form and Dimensions of Shadows; Lunar
Eclipses; Solar Eclipses, Total, Annular, and Partial; Ecliptic Limits,
and Number of Eclipses in a Year; the Saros; Occultations}

\nbtocentry{CHAPTERXI}{CHAPTER XI.---\textsc{Central Forces}; Equable Description of Areas;
Areal, Linear, and Angular Velocities; Kepler's Laws and Inferences
from them; Gravitation demonstrated by the Moon's Motion; Conic
Sections as Orbits; the Problem of \textit{Two} Bodies; the ``Velocity from
Infinity'' and its Relation to the Species of Orbit described by a
Body moving under Gravitation; Intensity of Gravitation}

\nbtocentry{CHAPTERXII}{CHAPTER XII.---\textsc{The Problem of Three Bodies}; Disturbing Forces;
Lunar Perturbations and the Tides}

\nbtocentry{CHAPTERXIII}{CHAPTER XIII.---\textsc{The Planets: their Motions, Apparent and
Real}; the Ptolemaic, Tychonic, and Copernican Systems; the Orbits
and their Elements; Planetary Perturbations}

\nbtocentry{CHAPTERXIV}{CHAPTER XIV.---\textsc{The Planets themselves}; Methods of determining
their Diameters, Masses, Densities, Times of Rotation, etc.; the ``Terrestrial
Planets,''---Mercury, Venus, and Mars; the Asteroids; Intra-Mercurial
Planets and the Zodiacal Light}

\nbtocentry{CHAPTERXV}{CHAPTER XV.---\textsc{The Major Planets},---Jupiter, Saturn, Uranus, and
Neptune}

\clearpage
\nbtocpagesheading

\nbtocentry{CHAPTERXVI}{CHAPTER XVI.---\textsc{The Determination of the Sun's Horizontal Parallax
and Distance}; Oppositions of Mars and Transits of Venus;
Gravitational Methods; Determination by Means of the Velocity of
Light}

\nbtocentry{CHAPTERXVII}{CHAPTER XVII.---\textsc{Comets}: their Number, Motions, and Orbits; their
Constituent Parts and Appearance; their Spectra, Physical Constitution,
and Probable Origin}

\nbtocentry{CHAPTERXVIII}{CHAPTER XVIII.---\textsc{Meteors}: Aerolites, their Fall and Physical Characteristics;
Shooting Stars and Meteoric Showers; Connection between
Meteors and Comets}

\nbtocentry{CHAPTERXIX}{CHAPTER XIX.---\textsc{The Stars}: their Nature and Number; the Constellations;
Star-Catalogues; Designation of Stars; their Proper Motions,
and the Motion of the Sun in Space; Stellar Parallax}

\nbtocentry{CHAPTERXX}{CHAPTER XX.---\textsc{The Light of the Stars}; Star Magnitudes and Photometry;
Variable Stars; Stellar Spectra; Scintillation of the Stars}

\nbtocentry{CHAPTERXXI}{CHAPTER XXI.---\textsc{Aggregations of Stars}: Double and Multiple Stars;
Clusters; Nebulę; the Milky Way, and Distribution of Stars in Space;
Constitution of the Stellar Universe; Cosmogony and the Nebular
Hypothesis}

\nbtocentry{APPENDIX}{APPENDIX.---\textsc{Tables of Astronomical Data}}

\nbtocentry{INDEX}{INDEX}

\end{changemargin}
}%end small

%% -----File: 010.png---Folio ix-------
%% -----File: 011.png---Folio x-------

%% -----File: 012.png---Folio 1-------
\mainmatter

\clearpage
\pdfbookmark[1]{Introduction}{Introduction}
\chapter*{INTRODUCTION.}
\chslabel{INTRODUCTION}
\markboth{INTRODUCTION.}{}
\thispagestyle{empty}
\nbrule\bigskip

\nbarticle{1.} \nbparatext{Astronomy} (\mytextgreek{>'astron nómos})
is the science which treats of the
heavenly bodies. As such bodies we reckon the sun and moon, the
planets (of which the earth is one) and their satellites, comets and
meteors, and finally the stars and nebulę.

We have to consider in Astronomy:---
\begin{asparaenum}[(\itshape a\/\normalfont)]
\item The motions of these bodies, both real and apparent, and the
laws which govern these motions.
\item Their forms, dimensions, and masses.
\item Their nature and constitution.
\item The effects they produce upon each other by their attractions,
radiations, or by any other ascertainable influence.
\end{asparaenum}

It was an early, and has been a most persistent, belief that the
heavenly bodies have a powerful influence upon human affairs, so
that from a knowledge of their positions and ``aspects'' at critical
moments (as for instance at the time of a person's birth) one could
draw up a ``horoscope'' which would indicate the probable future.

The \textit{pseudo-science} which was founded on this belief was named
Astrology,---the elder sister of Alchemy,---and for centuries Astronomy
was its handmaid; \textit{i.e.}, astronomical observations and calculations
were made mainly in order to supply astrological data.

At present the end and object of astronomical study is chiefly
knowledge pure and simple; so far as now appears, its development
has less direct bearing upon the material interests of mankind than
that of any other of the natural sciences. It is not likely that great
inventions and new arts will grow out of its laws and principles, such
as are continually arising from physical, chemical, and biological
discoveries, though of course it would be rash to say that such outgrowths
are impossible. But the student of Astronomy must expect
his chief profit to be intellectual, in the widening of the range of
thought and conception, in the pleasure attending the discovery of
simple law working out the most complicated results, in the delight
%% -----File: 013.png---Folio 2-------
over the beauty and order revealed by the telescope in systems otherwise
invisible, in the recognition of the essential unity of the material
universe, and of the kinship between his own mind and the infinite
Reason that formed all things and is immanent in them.

At the same time it should be said at once that, even from the
lowest point of view, Astronomy is far from a useless science. The
\textit{art of navigation} depends for its very possibility upon astronomical
prediction. Take away from mankind their almanacs, sextants, and
chronometers, and commerce by sea would practically stop. The
science also has important applications in the survey of extended
regions of country, and the establishment of boundaries, to say
nothing of the accurate determination of time and the arrangement
of the calendar.

It need hardly be said that Astronomy is not separated from kindred
sciences by sharp boundaries. It would be impossible, for instance,
to draw a line between Astronomy on one side and Geology
and Physical Geography on the other. Many problems relating to
the formation and constitution of the earth belong alike to all three.

\nbarticle{2.} Astronomy is divided into many branches, some of which, as
ordinarily recognized, are the following:---

{1.} \nbparatext{Descriptive Astronomy.}---This, as its name implies, is merely
an orderly statement of astronomical facts and principles.

\subparagraph{2.} \nbparatext{Practical Astronomy.}---This is quite as much an art as a
science, and treats of the instruments, the methods of observation,
and the processes of calculation by which astronomical facts are
ascertained.

\subparagraph{3.} \nbparatext{Theoretical Astronomy}, which treats of the calculations of orbits
and ephemerides, including the effects of so-called ``perturbations.''

\subparagraph{4.} \nbparatext{Mechanical Astronomy}, which is simply the application of mechanical
principles to explain astronomical facts (chiefly the planetary
and lunar motions). It is sometimes called \textit{Gravitational} Astronomy,
because, with few exceptions, gravitation is the only force sensibly
concerned in the motions of the heavenly bodies. Until within thirty
years this branch of the science was generally designated as \textit{Physical
Astronomy}, but the term is now objectionable because of late it has
been used by many writers to denote a very different and comparatively
new branch of the science; viz.,---
%% -----File: 014.png---Folio 3-------

\subparagraph{5.} \nbparatext{Astronomical Physics, or Astro-physics.}---This treats of the
physical characteristics of the heavenly bodies, their brightness and
spectroscopic peculiarities, their temperature and radiation, the nature
and condition of their atmospheres and surfaces, and all phenomena
which indicate or depend on their physical condition.

\subparagraph{6.} \nbparatext{Spherical Astronomy.}---This, discarding all consideration of
absolute dimensions and distances, treats the heavenly bodies simply
as objects moving on the ``surface of the celestial sphere'': it has to
do only with angles and directions, and, strictly regarded, is in fact
merely Spherical Trigonometry applied to Astronomy.

\nbarticle{3.} The above-named branches are not distinct and separate, but
they overlap in all directions. Spherical Astronomy, for instance,
finds the demonstration of many of its formulę in Gravitational
Astronomy, and their application appears in Theoretical and Practical
Astronomy. But valuable works exist bearing all the different
titles indicated above, and it is important for the student to know
what subjects he may expect to find discussed in each; for this
reason it has seemed worth while to name and define the several
branches, although they do not distribute the science between them
in any strictly logical and mutually exclusive manner.

In the present text-book little regard will be paid to these subdivisions,
since the object of the work is not to present a complete
and profound discussion of the subject such as would be demanded
by a professional astronomer, but only to give so much knowledge of
the facts and such an understanding of the principles of the science
as may fairly be considered essential to a liberal education. If this
result is gained in the reader's case, it may easily happen that he will
wish for more than he can find in these pages, and then he must have
recourse to works of a higher order and far more difficult, dealing
with the subject more in detail and more thoroughly.

\sloppy
To master the present book no further preparation is necessary
than a very elementary knowledge of Algebra, Geometry, and Trigonometry,
and a similar acquaintance with Mechanics and Physics,
especially Optics. While nothing short of high mathematical attainments
will enable one to become eminent in the science, yet a perfect
comprehension of all its fundamental methods and principles, and a
very satisfactory acquaintance with its main results, is quite within
the reach of every person of ordinary intelligence, without any more
extensive training than may be had in our common schools. At the
%% -----File: 015.png---Folio 4-------
same time the necessary statements and demonstrations are so much
facilitated by the use of trigonometrical terms and processes that it
would be unwise to dispense with them entirely in a work to be used
by pupils who have already become acquainted with them.

\fussy
In discussing the different subjects which present themselves, the
writer will adopt whatever plan appears best fitted to convey to the
student clear and definite ideas, and to impress them upon the mind.
Usually it will be best to proceed in the Euclidean order, by first
stating the fact or principle in question, and then explaining its
demonstration. But in some cases the inverse process is preferable,
and the conclusion to be reached will appear gradually unfolding
itself as the result of the observations upon which it depends, just as
its discovery came about.

The frequent references to ``Physics'' refer to the ``Elementary
Text-Book of Physics,'' by Anthony \&~Brackett; 3d edition, 1887.
Wiley \&~Sons, N.Y.
\chelabel{INTRODUCTION}
%% -----File: 016.png---Folio 5-------

\Chapter{I}{The Doctrine of the Sphere}
\nbchaptercenter{THE ``DOCTRINE OF THE SPHERE,'' DEFINITIONS, AND GENERAL
CONSIDERATIONS.}

\textsc{Astronomy}, like all the other sciences, has a terminology of its
own, and uses technical terms in the description of its facts and
phenomena. In a popular essay it would of course be proper to
avoid such terms as far as possible, even at the expense of circumlocutions
and occasional ambiguity; but in a text-book it is desirable
that the reader should be introduced to the most important of them
at the very outset, and made sufficiently familiar with them to use
them intelligently and accurately.

%       *       *       *       *       *

\nbarticle{4.} \nbparatext{The Celestial Sphere.}---To an observer looking up to the
heavens at night it seems as if the stars were glittering points attached
to the inner surface of a dome; since we have no direct perception of
their distance there is no reason to imagine some nearer than others,
and so we involuntarily think of the surface as \textit{spherical} with ourselves
in its centre. Or if we sometimes feel that the stars and
other objects in the sky really differ in distance, we still instinctively
imagine an immense sphere surrounding and enclosing all. Upon
this sphere we imagine lines and circles traced, resembling more or
less the meridians and parallels upon the surface of the earth, and
by reference to these circles we are able to describe intelligently the
apparent positions and motions of the heavenly bodies.

This celestial sphere may be regarded in either of two different
ways, both of which are correct and lead to identical results.

\begin{asparaenum}[(\itshape a\/\normalfont)]
\item We may imagine it, in the first place, as transparent, and of
merely finite (though undetermined) dimensions, \textit{but in some way
so attached to, and connected with, the observer that his eye always
remains at its centre wherever he goes}. Each observer, in this way
of viewing it, carries his own sky with him, and is the centre of his
own heavens.

\item Or, in the second place,---and this is generally the more convenient
way of regarding the matter,---we may consider the celestial
%% -----File: 017.png---Folio 6-------
sphere as mathematically \textit{infinite} in its dimensions; then, let the
observer go where he will, he cannot sensibly get away from its
centre. Its radius being ``greater than any assignable quantity,''
the size of continents, the diameter of the earth, the distance of the
sun, the orbits of planets and comets, even the spaces between the
stars, are all insignificant, and the whole visible universe shrinks
\textit{relatively} to a mere point at its centre. In what follows we shall use
this conception of the celestial sphere.\footnote
  {To most persons the sky appears, not a true hemisphere, but a \textit{flattened} vault,
  as if the horizon were more remote than the zenith. This is a subjective effect
  due mainly to the intervening objects between us and the horizon. The sun and
  moon when rising or setting look much larger than when they are higher up, for
the same reason.}
\end{asparaenum}

\includegraphicsouter{illo001}{\textsc{Fig.~1.}}

The apparent place of any celestial body will then be the point
on the celestial sphere where the line drawn from the eye of the
observer in the direction in which he sees the object, and produced
indefinitely, pierces the sphere. Thus, in \figref{illo001}{Figure~1}, $A$, $B$, $C$ are
the apparent places of $a$, $b$, and $c$,
the observer being at $O$. The apparent
place of a heavenly body evidently
depends solely upon its \textit{direction}, and
is wholly independent of its \textit{distance}
from the observer.

\nbarticle{5.} \nbparatext{Linear and Angular Dimensions.}---Linear
dimensions are such as may
be expressed in \textit{linear} units; \textit{i.e.}, in
miles, feet, or inches; in metres or
millimetres. Angular dimensions are
expressed in \textit{angular} units; \textit{i.e.}, in
right angles, in radians,\footnote
  {A \textit{radian} is the angle which is measured by an arc equal in
  length to radius.
  Since a circle whose radius is unity has a circumference of $2 \pi$, and contains $360°$,
  or $21,600'$, or $1,296,000''$, it follows that a radian contains $\left(\dfrac{360}{2\pi}\right)°$
  or $\left(\dfrac{21600}{2\pi}\right)'$, or
  $\left(\dfrac{1296000}{2\pi}\right)''$; \textit{i.e.}\ (approximately), a radian
  $= 57.3°= 3437.7'= 206264.8''$. Hence,
  to reduce to seconds of arc an angle expressed in radians, we must multiply
  it by the number $206264.8$; a relation of which we shall have to make frequent
  use.

  See Halsted's Mensuration, p.~25.}
or (more commonly in astronomy) in degrees,
minutes, and seconds. Thus, for instance, the \textit{linear} semi-diameter
%% -----File: 018.png---Folio 7-------
of the sun is about 697,000 kilometers (433,000 miles),
while its \textit{angular} \DPtypo{semidiameter}{semi-diameter} is about $16'$, or a little more than
a quarter of a degree. Obviously, angular units alone can properly
be used in describing apparent distances and dimensions in the sky.
For instance, one cannot say correctly that the two stars which are
known as ``the pointers'' are two or five or ten \textit{feet} apart: their
distance is about five \textit{degrees}.

It is sometimes convenient to speak of ``\textit{angular area},'' the unit
of which is a ``square degree'' or a ``square minute''; \textit{i.e.}, a small
square in the sky of which each side is $1°$ or $1'$. Thus we may compare
the angular area of the constellation Orion with that of Taurus,
in \textit{square degrees}, just as we might compare Pennsylvania and New
Jersey in square miles.

\nbarticle{6.} \nbparatext{Relation between the Distance and Apparent Size of Object.}---Suppose
a globe having a radius $BC$ equal to $r$. As seen from
the point $A$ (\figref{illo002}{Fig.~2}) its apparent (\textit{i.e.}, \textit{angular}) \DPtypo{semidiameter}{semi-diameter} will
be $BAC$ or $s$, its distance being $AC$ or $R$.

\includegraphicsmid{illo002}{\textsc{Fig.~2.}}

We have immediately from Trigonometry, since $B$ is a right angle,
\[
  \sin s = \frac{r}{R}.
\]

If, as is usual in Astronomy, the diameter of the object is small
as compared with its distance, we may write
\[
s = \frac{r}{R},
\]
which gives $s$ in \textit{radians} (not in degrees or seconds). If we wish it
in the ordinary angular units,
\[
s° = 57.3 \frac{r}{R}, \quad \text{or} \quad s'' = 206264.8\frac{r}{R}.
\]
In either form of the equation we see that the apparent diameter
\textit{varies directly as the linear diameter, and inversely as the distance}.
%% -----File: 019.png---Folio 8-------

In the case of the moon, $R =$ about 239,000 miles; and $r$, 1081
miles. Hence $s = \frac{1081}{239000} = \frac{1}{221}$ of a radian, which is a little more
than $\frac{1}{4}$ of a degree.

\begin{fineprint}
It may be mentioned here as a rather curious fact that most persons say
that the moon appears about \textit{a foot in diameter}; at least, this seems to
be the average estimate. This implies that the surface of the sky appears
to them only about 110 feet away, since that is the distance at which a disc
one foot in diameter would have an angular diameter of $\frac{1}{110}$ of a radian, or $\frac{1}{2}°$.
\end{fineprint}

\nbarticle{7.} \nbparatext{Vanishing Point.}---Any system of parallel lines produced in
one direction will \textit{appear} to pierce the celestial sphere at a single
point. They actually pierce it at different points, separated on the
surface of the sphere by linear distances, equal to the actual distances
between the lines, but on the infinitely distant surface these linear
distances, being only finite, become invisible, subtending at the centre
angles less than anything assignable. The different points, therefore,
coalesce into a \textit{spot} of apparently infinitesimal size---the so-called
``vanishing point'' of perspective. Thus the axis of the earth and
\textit{all lines parallel to this axis} point to the celestial pole.

\nbthought

In order to describe intelligibly the apparent position of an object
in the sky, it is necessary to have certain points and lines from which
to reckon. We proceed to define some of these which are most
frequently used.

\nbarticle{8.} \nbparatext{The Zenith.}---The Zenith is the \textit{point vertically overhead}, \textit{i.e.},
the point where a plumb-line, produced upwards, would pierce the
sky: it is determined by the \textit{direction of gravity} where the observer
stands.

If the earth were exactly spherical, the zenith might also be defined
as the point where a line drawn \textit{from the centre of the earth upward
through the observer} meets the sky. But since, as we shall see
hereafter, the earth is not an exact globe, this second definition indicates
a point known as the \textit{Geocentric Zenith}, which is not identical
with the \textit{True} or \textit{Astronomical Zenith}, determined by the direction of
gravity.

\nbarticle{9.} \nbparatext{The Nadir.}---The Nadir is the point opposite the zenith---under
foot, of course.

Both zenith and nadir are derived from the Arabic, which language
has also given us many other astronomical terms.
%% -----File: 020.png---Folio 9-------

\nbarticle{10.} \nbparatext{Horizon.}---The Hor{\=\i}zon\footnote{%
  Beware of the common, but vulgar, pronunciation,
  \textit{Hór{\=\i}zon}.}
is a great circle of the celestial
sphere, having the zenith and nadir as its poles: it is therefore
half-way between them, and 90° from each.

A \textit{horizontal plane}, or the \textit{plane of the horizon}, is a plane perpendicular
to the direction of gravity, and the horizon may also be correctly
defined as the intersection of the celestial sphere by this plane.

Many writers make a distinction between the \textit{sensible} and \textit{rational}
horizons. The plane of the sensible horizon passes through the
observer; the plane of the rational horizon passes through the centre
of the earth, parallel to the plane of the sensible horizon: these two
planes, parallel to each other, and everywhere about 4000 miles
apart, trace out on the sky the two horizons, the sensible and the
rational. It is evident, however, that on the infinitely distant surface
of the celestial sphere, the two traces sensibly coalesce into one single
great circle, which is the horizon as first defined. In strictness,
therefore, while we can distinguish between the two horizontal \textit{planes},
we get but one \textit{horizon circle} in the sky.

\nbarticle{11.} \nbparatext{The Visible Horizon} is the line where sky and earth meet.
On land it is an irregular line, broken by hills and trees, and of no
astronomical value; but at sea it is a true circle, and of great importance
in observation. It is not, however, a \textit{great} circle, but,
technically speaking, only a \textit{small} circle; depressed below the true
horizon by an amount depending upon the observer's elevation above
the water. This depression is called the \textit{Dip of the Horizon}, and will
be discussed further on.

\nbarticle{12.} \nbparatext{Vertical Circles.}---These are great circles passing through
the zenith and nadir, and therefore necessarily perpendicular to the
horizon---\textit{secondaries} to it, to use the technical term.

\textbf{Parallels of Altitude,} or \textbf{Almucantars.}---These are small circles
parallel to the horizon: the term Almucantar is seldom used.

The points and circles thus far defined are determined entirely by
the \textit{direction of gravity} at the station occupied by the observer.

\nbthought

\nbarticle{13.} \nbparatext{The Diurnal Rotation of the Heavens.}---If one watches the
sky for a few hours some night, he will find that, while certain stars
rise in the east, others set in the west, and nearly all the constellations
change their places. Watching longer and more closely, it will
%% -----File: 021.png---Folio 10-------
appear that the stars move in circles, uniformly, in such a way as
not to disturb their relative configurations, but as if they were
attached to the inner surface of a revolving sphere, turning on its
axis once a day. The path thus daily described by a star is called its
``\textit{diurnal circle}.''

It is soon evident that in our latitude the visible ``pole'' of this
sphere---the point about which it turns---is in the north, not quite
half-way up from the horizon to the zenith, for in that region the stars
hardly move at all, but keep their places all night long.

\nbarticle{14.} \nbparatext{The Poles.}---The Poles may be defined as the two points in the
sky, one in the northern hemisphere and one in the southern, where a
\textit{star's diurnal circle reduces to zero}; \textit{i.e.}, points where, if a star were
placed, it would suffer no apparent change of place during the whole
twenty-four hours. The line joining these poles is, of course, the
\textit{axis} of the celestial sphere, about which it seems to rotate daily.

The exact place of the pole may be found by observing some star
very near the pole at two times 12 hours apart, and taking the middle
point between the two observed places of the star.

The definition of the pole just given is independent of any theory
as to the cause of the apparent rotation of the heavens. If, however,
%% -----File: 022.png---Folio 11-------
we admit that it is due to the earth's rotation on its axis, then
we may define the poles as the \textit{points where the earth's axis produced
pierces the celestial sphere}.

\nbarticle{15.} \nbparatext{The Pole-star (Polaris).}---The place of the northern pole is
very conveniently marked by the \textit{Pole-star}, a star of the second magnitude,
which is now only about $1\frac{1}{4}°$ from the pole: we say \textit{now}, because
on account of a slow change in the direction of the earth's
axis, called ``precession'' (to be discussed later), the distance between
the pole-star and the pole is constantly changing, and has been
for several centuries gradually decreasing.

The pole-star stands comparatively solitary in the sky, and may
easily be recognized by means of the so-called ``pointers,''---two
stars in the ``dipper'' (in the constellation of Ursa Major)---which
point very nearly to it, as shown in \figref{illo003}{Fig.~3}. The pole is very nearly
on the line joining Polaris with the star Mizar ($\zeta$~Urs.\ Maj., at the
bend in the handle of the dipper), and at a distance just about one-quarter
of the distance between the pointers, which are nearly $5°$
apart.

The southern pole, unfortunately, is not so marked by any conspicuous
star.

\includegraphicsmid{illo003}{\textsc{Fig.~3.}---The Pole Star and the Pointers.}

\nbarticle{16.} \nbparatext{The Celestial Equator}, or \nbparatext{Equinoctial Circle}.---This is a great
circle midway between the two poles, and of course $90°$ from each.
It may also be defined as the intersection of the plane of the earth's
equator with the celestial sphere. It derives its name from the fact
that, at the two dates in the year when the sun crosses this circle---about
March 20 and Sept.~22---the day and night are equal in length.

\nbthought

\nbarticle{17.} \nbparatext{The Vernal Equinox}, or \nbparatext{First of Aries}.---The Equinox, strictly
speaking, is the \textit{time when} the sun crosses the equator, but the term
has come by accommodation to denote also the \textit{point where} it crosses,
though in strictness it should be called the ``\textit{Equinoctial Point}.''
This crossing occurs twice a year, once in September and once in
March, and the \textit{Vernal Equinox is the point on the equator where
the sun crosses it in the spring}. It is sometimes called the \textit{Greenwich
of the Celestial Sphere}, because it is used as a reference point
in the sky, much as Greenwich is on the earth. Its position is not
marked by any conspicuous star.

Why this point is also called the ``First of Aries'' will appear
later, when we come to speak of the zodiac and its ``signs.''
%% -----File: 023.png---Folio 12-------

\nbarticle{18.} \nbparatext{Hour-Circles.}---Hour-circles are great circles of the celestial
sphere passing through its poles, and consequently perpendicular
to the celestial equator. They correspond exactly to the meridians
of the earth, and some writers call them ``Celestial Meridians''; but
the term is objectionable, as likely to lead to confusion with \textit{the}
Meridian, to be noted immediately.

\nbthought

\nbarticle{19.} \nbparatext{The Meridian and Prime Vertical.}---\textit{The Meridian is the great
circle passing through the pole and the zenith.} Since it is a great
circle, it must necessarily pass through \textit{both} poles, and through the
nadir as well as the zenith, and must be perpendicular both to the
equator and to the horizon.

It may also be correctly defined as the \textit{Vertical Circle} which passes
through the \textit{pole}; or, again, as the \textit{Hour-Circle} which passes through
the \textit{zenith}, since all vertical circles must pass through the zenith, and
all hour-circles through the pole.

\textit{The Prime Vertical} is the Vertical Circle (passing through the
zenith) at right angles to the meridian; hence lying \textit{east and west}
on the celestial sphere.

\nbarticle{20.} \nbparatext{The Cardinal Points.}---The North and South Points are the
points on the horizon where it is intersected by the meridian. The
East and West Points are where it is cut by the prime vertical, and
also by the equator. The North \textit{Point}, which is on the horizon, must
not be confounded with the North \textit{Pole}, which is not on the horizon,
but at an elevation equal (see \artref{Art.}{30}) to the latitude of the observer.

\nbthought

With these circles and points of reference we have now the means
to describe intelligibly the position of a heavenly body, in several
different ways.

We may give its \textit{altitude} and \textit{azimuth}, or its \textit{declination} and \textit{hour-angle};
or, if we know the time, its \textit{declination} and \textit{right ascension}.
Either of these pairs of co-ordinates, as they are called, will define
its place in the sky.

\nbarticle{21.} \nbparatext{Altitude and Zenith Distance}\ (\figref{illo004}{Fig.~4}).---The Altitude of a
heavenly body is \textit{its angular elevation above the horizon}, and is measured
by the arc of the vertical circle passing through the body, and
intercepted between it and the horizon.
%% -----File: 024.png---Folio 13-------

The Zenith Distance of a body is simply its angular distance from
the zenith, and is the complement of the altitude. Altitude + Zenith
Distance = $90°$.

\nbarticle{22.} \nbparatext{Azimuth and Amplitude} (\figref{illo004}{Fig.~4}).---The Azimuth of a body
is the \textit{angle at the zenith, between the meridian and the vertical circle,
which passes through the body}. It is measured also by the arc of the
horizon intercepted between the north or south point, and the foot
of this vertical. The word is of Arabic origin, and has the same
meaning as the \textit{true bearing} in surveying and navigation.

\includegraphicsmid{illo004}{\textsc{Fig.~4.}---The Horizon and Vertical Circles.\\[1ex]
\begin{tabular}{@{}l@{\;}|@{\;}l@{}}
  $O$, the place of the Observer. & $M$, some Star.\\
  $OZ$, the Observer's Vertical.  & $ZMH$, arc of the Star's Vertical Circle.\\
  $Z$, the Zenith; $P$, the Pole. & $TMR$, the Star's Almucantar.\\
  $SENW$, the Horizon.            & Angle $TZM$, or arc $SWNEH$, Star's \textit{Azimuth}.\\
  $SZPN$, the Meridian.           & Arc $HM$, Star's \textit{Altitude}.\\
  $EZW$, the Prime Vertical.      & Arc $ZM$, Star's \textit{Zenith Distance}.
\end{tabular}}

The \textit{Amplitude} of a body is the complement of the azimuth.
Azimuth + Amplitude = $90°$.

\begin{fineprint}
There are various ways of reckoning azimuth. Many writers express it
in the same manner as \textit{the bearing} is expressed in surveying; \textit{i.e.}, so many
degrees east or west of north or south; N.~$20°$ E., etc. The more usual
way at present is, however, to reckon it in degrees from the south point clear
round through the west to the point of beginning: thus an object in the
SW. would have an azimuth of $45°$; in the NW., $135°$; in the N., $180°$; in
the NE., $225°$; and in the SE., $315°$. For example, to find a star whose
azimuth is $260°$, and altitude $60°$, we must face N.~$80°$ E., and then look
up two-thirds of the way to the zenith. The object in this case has an
\textit{amplitude} of $10°$ N. of W., and a zenith distance of $30°$. Evidently both
the azimuth and altitude of a heavenly body are continually changing, except
in certain very special cases.
\end{fineprint}
%% -----File: 025.png---Folio 14-------

In \figref{illo004}{Fig.~4}, $SENW$ represents the horizon, $S$ being the south point,
and $Z$ the zenith. The angle $SZM$, which numerically equals the
arc $SH$, is the \textit{Azimuth} of the star $M$; while $EZM$, or $EH$ is its
\textit{Amplitude}. $MH$ is its \textit{Altitude}, and $ZM$ its \textit{Zenith Distance}.

\nbarticle{23.} \nbparatext{Declination and Polar Distance} (\figref{illo005}{Fig.~5}).---The Declination of
a heavenly body is its \textit{angular distance north or south of the celestial
equator}, and is measured by the arc of the hour-circle passing through
the object, intercepted between it and the equator. It is reckoned
positive ($+$) north of the celestial equator, and negative ($-$) south
of it. Evidently it is precisely analogous to the latitude of a place
on the earth. The \textit{north-polar distance} of a star is its angular distance
from the North Pole, and is simply the complement of the
declination. Declination $+$ North-Polar Distance = $90°$.

The declination of a star remains always the same; at least, the
slow changes that it undergoes need not be considered for our
present purpose. ``\textit{Parallels of Declination}'' are small circles parallel
to the celestial equator.

\nbarticle{24.} \nbparatext{The Hour-Angle} (\figref{illo005}{Fig.~5}).---The Hour-Angle of a star is the
\textit{angle at the pole between the meridian and the hour-circle passing
through the star}. It may be reckoned in degrees; but it also
may be, and most commonly is, reckoned in \textit{hours}, \textit{minutes}, and
\textit{seconds of time}; the hour being equivalent to fifteen degrees, and
the minute and second of time being equal to fifteen minutes and seconds
of arc respectively.

Of course the hour-angle of an object is continually changing,
being zero when the object is on the meridian, one hour, or fifteen
degrees, when it has moved that amount westward, and so on.

\nbarticle{25.} \nbparatext{Right Ascension} (\figref{illo005}{Fig.~5}).---The Right Ascension of a star
is \textit{the angle at the pole between the star's hour-circle and the hour-circle}
(\textit{called the Equinoctial Colure}), \textit{which passes through the vernal
equinox.}

It may be defined also as the arc of the equator, intercepted
between the vernal equinox and the foot of the star's hour-circle.

It is always reckoned from the equinox \textit{toward the east}; sometimes
in degrees, but usually in \textit{hours, minutes, and seconds of time.
The right ascension, like the declination, remains unchanged by the
diurnal motion.}
%% -----File: 026.png---Folio 15-------

\nbarticle{26.} \nbparatext{Sidereal Time} (\figref{illo005}{Fig.~5}).---For many astronomical purposes
it is convenient to reckon time, not by the sun's position in the sky,
but by that of the vernal equinox.

\includegraphicsmid{illo005}{\textsc{Fig.~5.}---Hour-Circles, etc.\\[1ex]
\begin{tabular}{@{}p{.48\textwidth}|p{.48\textwidth}@{}}
\hangindent=2em  $O$, place of the Observer; $Z$, his Zenith.

\hangindent=2em  $SENW$, the Horizon.

\hangindent=2em  $POP'$, line parallel to the Axis of the Earth.

\hangindent=2em  $P$ and $P'$, the two Poles of the Heavens.

\hangindent=2em  $EQWT$, the Celestial Equator, or Equinoctial.

\hangindent=2em  $X$, the Vernal Equinox, or ``First of Aries.''

&
\hangindent=2em  $PXP'$ the Equinoctial Colure, or Zero Hour-Circle.

\hangindent=2em  $m$, some Star.

\hangindent=2em  $Ym$, the Star's \textit{Declination}; $Pm$, its \textit{North-polar Distance.}

\hangindent=2em  Angle $mPR$ = arc $QY$, the Star's (eastern)
         \textit{Hour-Angle}; = $24^\text{h}$ \textit{minus} Star's (western)
         Hour-Angle.

\hangindent=2em  Angle $XPm$ = arc $XY$, Star's \textit{Right Ascension},
         Sidereal time at the moment = $24^\text{h}$ \textit{minus}
         angle $XPQ$.
\end{tabular}}

\textit{The Sidereal Time} at any moment may be defined as \textit{the hour-angle
of the vernal equinox}. It is sidereal \textit{noon}, when the equinoctial
point is on the meridian; 1~o'clock (sidereal) when its hour-angle
is $15°$; and 23~o'clock when its hour-angle is $345°$, \textit{i.e.}, when the vernal
equinox is an hour \textit{east} of the meridian; the time being reckoned
round through the whole 24~hours. On account of the annual motion
of the sun among the stars, the \textit{Solar Day}, by which time is reckoned
for ordinary purposes, is about 4~minutes longer than the sidereal
day. The exact difference is $3^{\text{m}}$ $56^{\text{s}}$.394 (sidereal), or just one day
in a year; there being $366\frac{1}{4}$ \textit{sidereal} days in the year, as against
$365\frac{1}{4}$ \textit{solar} days.

\nbarticle{27.} \nbparatext{Observatory Definition of Right Ascension.}---It is evident from
the above definition of sidereal time, that we may also define the
Right Ascension of a star as \textit{the sidereal time taken the star crosses
the meridian}. The Star and the Vernal Equinox are both of them
%% -----File: 027.png---Folio 16-------
fixed points in the sky, and do not change their relative position during
the sky's apparent daily revolution; a given star, therefore,
always comes to the meridian of any observer the same number of
hours after the vernal equinox has passed; and this number of hours
is the sidereal time at the moment of the star's transit, and measures
its right ascension. In the observatory, this definition of right ascension
is the most natural and convenient.

It is obvious that the right ascension of a star corresponds in the
sky exactly with the \textit{longitude} of a place on the earth; terrestrial
longitude being reckoned from Greenwich, just as right ascension
is reckoned from the vernal equinox.

N.B.\quad \textit{We shall find hereafter that the \emph{stars} have latitudes and
longitudes of their own; but unfortunately these \emph{celestial} latitudes and
longitudes do not correspond to the terrestrial, and great care is necessary
to prevent confusion}. (See \artref{Art.}{179}.)

\nbarticle{28.} An \textit{armillary sphere}, or some equivalent apparatus, is almost
essential to enable a beginner to get correct ideas of the points,
circles, and co-ordinates defined above, but the figures will perhaps
be of assistance.

The first of them (\figref{illo004}{Fig.~4}) represents the horizon, meridian, and
prime vertical, and shows how the position of a star is indicated by
its altitude and azimuth. This framework of circles, depending
upon the direction of gravity, of course always remains \textit{apparently}
unchanged in position, as if attached directly to the earth, while the
sky apparently turns around outside it.

The other figure (\figref{illo005}{Fig.~5}) represents the system of points and
circles which depend upon the earth's rotation, and are independent
of the direction of gravity. The vernal equinox and the hour-circles
apparently revolve with the stars while the pole remains fixed upon
the meridian, and the equator and parallels of declination, revolving
truly in their own planes, also appear to be at rest in the sky. But
the whole system of lines and points represented in the figure (horizon
and meridian alone excepted) may be considered as attached
to, or marked out upon, the inner surface of the celestial vault and
whirling with it.

It need hardly be said that the ``appearances are deceitful''---that
which is really carried around by the earth's rotation is the
observer, with his plumb-line and zenith, his horizon and meridian;
while the stars stand still---at least, their motions in a day are insensible
as seen from the earth.
%% -----File: 028.png---Folio 17-------

\begin{fineprint}
At the poles of the earth, which are, mathematically speaking, ``singular''
points, the definitions of the Meridian, of North and South, etc., break
down.

There the pole (celestial) and zenith coincide, and any number of circles
may be drawn through the two points, which have now become one. The
horizon and equator coalesce, and the only direction on the earth's surface
is due south (or north)---east and west have vanished.

A single step of the observer will, however, remedy the confusion: zenith
and pole will separate, and his meridian will again become determinate.
\end{fineprint}

\nbarticle{29.} To recapitulate: The \textit{direction of gravity} at the point where
the observer stands determines the Zenith and Nadir, the Horizon, and
the Almucantars (parallel to the Horizon), and all the vertical circles.
One of the verticals, the \textit{Meridian}, is singled out from the rest by
the circumstance that it passes through the \textit{pole} of the sky, marking
the North and South Points where it cuts the horizon.

Altitude and Azimuth (or their complements, Zenith Distance
and Amplitude) are the co-ordinates which designate the position
of a body by reference to the Zenith and the Meridian.

Similarly, the \textit{direction of the earth's axis} (which is independent
of the observer's place on the earth) determines the Poles, the
Equator, the Parallels of Declination, and the Hour-Circles. \textit{Two}
of these Hour-Circles are singled out as reference lines; one of them,
the Meridian, which passes through the Zenith, and is a purely
\textit{local} reference line; the other, the Equinoctial Colure, which passes
through the Vernal Equinox, a point chosen from its relation to the
sun's annual motion. Declination and \textit{Hour-Angle} are the co-ordinates
which refer the place of a star to the Pole and the Meridian;
while Declination and \textit{Right Ascension} refer it to the Pole and Equinoctial
Colure. The latter are the co-ordinates usually employed in
star-catalogues and ephemerides to define the positions of stars and
planets, and correspond exactly to Latitude and Longitude on the
earth, by means of which geographical positions are designated.

\nbarticle{30.} \nbparatext{Relation of the Apparent Diurnal Motion of the Sky to the
Observer's Latitude.}---Evidently the apparent motions of the stars
will be considerably influenced by the station of the observer, since
the place of the pole in the sky will depend upon it. The \textit{Altitude}
of the pole, or its \textit{height in degrees above} the horizon, is always equal
to the \textit{Latitude} of the observer. Indeed, the German word for latitude
(astronomical) is \textit{Polhöhe; i.e.}, simply ``Pole-height.''
%% -----File: 029.png---Folio 18-------

This will be clear from \figref{illo006}{Fig.~6}. The latitude of a place is
the \textit{angle between its plumb-line and the plane of the equator}; the
angle $ONQ$ in the figure. [If the earth were truly spherical, $N$
would coincide with $C$, the centre of the earth. The ordinary
definition of latitude given in the geographies disregards the slight
difference.]

\includegraphicsmid{illo006}{\textsc{Fig.~6.}---Relation of Latitude to the Elevation of the Pole.}

Now the angle $H'OP''$ is equal to $ONQ$, because their sides are
mutually perpendicular; and it is also the \textit{altitude of the pole}, because
the line $HH'$ is horizontal at $O$, and $OP''$ is parallel to $PP'$, the
earth's axis, and therefore points to the celestial pole.

This fundamental relation, \textit{that the altitude of the celestial pole is
the Latitude of the observer}, cannot be too strongly impressed on the
student's mind. The usual symbol for the latitude of a place is $\phi$.

\nbarticle{31.} \nbparatext{The Right Sphere.}---If the observer is situated at the
earth's equator, \textit{i.e.}, in latitude zero ($\phi = 0$), the pole will be in the
horizon, and the equator will pass vertically overhead through the
zenith.

The stars will rise and set vertically, and their diurnal circles will
all be bisected by the horizon, so that they will be 12 hours above
it and 12 below. This aspect of the heavens is called the \textit{Right
Sphere}.

\nbarticle{32.} \nbparatext{The Parallel Sphere.}---If the observer is at the pole of the
earth ($\phi = 90°$), then the celestial pole will be in the zenith, and
the equator will coincide with the horizon. If he is at the \textit{North}
Pole, all stars north of the celestial equator will remain permanently
%% -----File: 030.png---Folio 19-------
above the horizon, never rising or falling at all, but sailing around
on circles of altitude (or \textit{Almucantars}) parallel to the horizon.
Stars in the Southern Hemisphere, on the other hand, would never
rise to view. As the sun and the moon move in such a way that
during half the time they are alternately north and south of the
equator, they will be half the time above the horizon and half the
time below it. The moon would be visible for about a fortnight at a
time, and the sun for six months.

\includegraphicsmid{illo007}{\textsc{Fig.~7.}---The Oblique Sphere and Diurnal Circles.}

\nbarticle{33.} \nbparatext{The Oblique Sphere} (\figref{illo007}{Fig.~7}).---At any station between the
pole and equator the stars will move in circles oblique to the horizon,
$SENW$ in the figure. Those whose distance from the elevated pole
is less than the latitude of the place will, of course, never sink below
the horizon,---the radius of the ``\textit{Circle of Perpetual Apparition},''
as it is called (the shaded cap around $P$ in the figure), being just
equal to the height of the pole, and becoming larger as the latitude
increases. On the other hand, stars within the same distance of the
depressed pole will lie within the ``\textit{Circle of Perpetual Occultation},''
and will never rise above the horizon.

A star exactly on the celestial equator will have its diurnal circle
$EQWQ'$ bisected by the horizon, and will be above the horizon just
as long as below it. A star north of the equator (if the North Pole
is the elevated one) will have more than half of its diurnal circle
above the horizon, and will be visible more than half the time; as, for
instance, a star at $A$: and of course the reverse will be true of stars
%% -----File: 031.png---Folio 20-------
on the other side of the equator.\footnote
  {A Celestial Globe will be of great assistance in studying these diurnal circles.
  The north pole of the globe must be elevated to an angle equal to the latitude of
  the observer, which can be done by means of the degrees marked on the brass
  meridian. It will then at once be easily seen what stars never set, which ones
  never rise, and during what part of the 24 hours any heavenly body at a known
  distance from the equator is above or below the horizon.}
Whenever the sun is north of
the equator, the day will therefore be longer than the night for all
stations in northern latitude: how much longer will depend both on
the latitude of the place and the sun's distance from the celestial
equator.
\chelabel{CHAPTERI}

%% -----File: 032.png---Folio 21-------

\Chapter{II}{Astronomical Instruments}
\nbchaptercenter{ASTRONOMICAL INSTRUMENTS.}

\nbarticle{34.} \textsc{Astronomical} observations are of various kinds: sometimes
we desire to ascertain the apparent distance between two bodies at a
given time; sometimes the position which a body occupies at a given
time, or the moment it arrives at a given circle of the sky, usually
the meridian. Sometimes we wish merely to examine its surface, to
measure its light, or to investigate its spectrum; and for all these
purposes special instruments have been devised.

We propose in this chapter to describe very briefly a few of the
most important.

\nbarticle{35.} \nbparatext{Telescopes in General.}---Telescopes are of two kinds, refracting
and reflecting. The former were first invented, and are much
more used, but the largest instruments ever made are reflectors. In
both the fundamental principle is the same. The large lens, or mirror,
of the instrument forms at its focus a \textit{real image} of the object
looked at, and this image is then examined and magnified by the eye-piece,
which in principle is only a magnifying-glass.

\begin{fineprint}
In the form of telescope, however, introduced by Galileo,\footnote
  {In strictness, Galileo did not invent the telescope. Its \textit{first} invention
  seems to have been in 1608, by Lipperhey,
  a spectacle-maker of Middleburg,
  in Holland; though the honor has also been claimed for two or three other
  Dutch opticians. Galileo, in his ``Nuncius Sydereus,''
  published in March,
  1610, himself says that he had heard of the Dutch instruments in 1609, and
  by so hearing was led to construct his own, which, however, far excelled in
  power any that had been made previously; and he was the first to apply
  the telescope to Astronomy. See Grant's ``History of Astronomy,'' pp.~514
  and seqq.}
and still used
as the ``opera-glass,'' the rays from the object-glass are intercepted by a concave
lens which performs the office of an eye-piece \textit{before} they meet at the
focus to form the ``real image.'' But on account of the smallness of the
field of view, and other objections, this form of telescope is never used when
any considerable power is needed.
\end{fineprint}
%% -----File: 033.png---Folio 22-------

\nbarticle{36.} \nbparatext{Simple Refracting Telescope.}---This consists essentially as
shown in the figure (\figref{illo008}{Fig.~8}), of a tube containing two lenses: a single
convex lens, $A$, called the object-glass; and another, of smaller size
and short focus, $B$, called the eye-piece. Recalling the principles of
lenses the student will see that if the instrument be directed at a distant
object, the moon, for instance, all the rays, $a_0a_1a_2$, which fall
upon the object-glass from a point at the \textit{top} of the moon, will be
collected at $a$ in the focal plane, at the \textit{bottom} of the image. Similarly
rays from the \textit{bottom} of the moon will go to $b$ at the \textit{top} of the
image; moreover, since the rays that pass through the optical centre
of the lens, $o$, are undeviated,\footnote
  {In this explanation, we use the approximate theory of lenses (in which their
  thickness is neglected), as given in the elementary text-books. The more exact
  theory of Gauss and later writers would require some slight modifications in our
  statements, but none of any material importance. For a thorough discussion,
  see Jamin, ``\textit{Traité de Physique},'' or Encyc.\ Britannica,---Optics.}
the angle $a_0ob_0$ will equal $boa$; or, in
other words, if the focal length of the lens be five feet, for instance,
then the image of the moon, seen from a distance of five feet, will
appear just as large as the moon itself does in the sky,---it will
subtend the same angle. If we look at it from a smaller distance,
say from a distance of one foot, the image will look larger than the
moon; and in fact, without using an eye-piece at all, a person with
normal eyes can obtain considerable magnifying power from the
object-glass of a large telescope. With a lens of ten feet focal
length, such as is ordinarily used in an 8-inch telescope, one can
easily see the mountains on the moon and the satellites of Jupiter,
by taking out the eye-piece, and putting the eye in the line of vision
some eight or ten inches back of the eye-piece hole.

\includegraphicsmid{illo008}{\textsc{Fig.~8.}---Path of the Rays in the Astronomical Telescope.}

The image is a \textit{real} one; \textit{i.e.}, the rays that come from different
points in the object \textit{actually meet} at corresponding points in the image,
so that if a photographic plate were inserted at $ab$, and properly
exposed, a picture would be obtained.

If we look at the image with the naked eye, we cannot come nearer
%% -----File: 034.png---Folio 23-------
to the image (unless near-sighted) than eight or ten inches, and so
cannot get any great magnifying power; but if we use a magnifying-glass,
we can approach much closer.

\nbarticle{37.} \nbparatext{Magnifying Power.}---If the eye-piece $B$ is set at a distance
from the image equal to its principal focal distance, then any pencil of
rays from any point of the image will, after passing the lens, be converted
into a parallel beam, and will appear to the eye to come from
a point at an infinite distance, as if from an object in the sky. The
rays which came from the top of the moon, for instance, and are collected
at $a$ in the image, will reach the eye as a beam \textit{parallel to the
line $ac$, which connects $a$ with the optical centre of the eye-piece}. Similarly
with the rays which meet at $b$. The observer, therefore, will
see the \textit{top} of the moon's disc in the direction $ck$, and the \textit{bottom} in
the direction $cl$. It will appear to him \textit{inverted}, and greatly magnified;
its apparent diameter, as seen by the naked eye and measured
by the angle $aob$ (or its equal $b_0oa_0$); having been increased to $acb$.
Since both these angles are subtended by the same line $ab$, and are
\textit{small} (the \figref{illo008}{figure}, of course, is much out of proportion), they must
be inversely proportional to the distance $ob$ and $cb$; \textit{i.e.}, $boa : bca =
cb : ob$; or, putting this into words: The ratio between the natural
apparent diameter of the object, and its diameter as seen through the
telescope, \textit{is equal to the ratio between the focal lengths of the eye-lens
and object-glass}. This ratio is called the \textit{magnifying power}
of the telescope, and is therefore given by the simple formula
$M = \dfrac{F}{f}$, where $F$ is the focal length of the object-glass and $f$ that of
eye-piece,\footnote
  {A magnifying power of 1 is no magnifying power at all. Object and image
  subtend equal angles. A magnifying power denoted by a fraction, say $\frac{1}{4}$ would
  be a \textit{minifying} power, making the object look \textit{smaller}, as when we look at an object
  through the wrong end of a spy-glass.}
while $M$ is the magnifying power.

If, for example, the object-glass have a focal length of thirty feet,
and the eye-piece of one inch, the magnifying power will be 360; the
power may be changed at pleasure by substituting different eye-pieces,
of which every large telescope has an extensive stock.

\nbarticle{38.} \nbparatext{Brightness of Image.}---Since all the rays from a star which
fall upon the large object-glass are transmitted to the observer's eye
(neglecting the losses by absorption and reflection), he obviously
%% -----File: 035.png---Folio 24-------
receives a quantity of light much greater than he would naturally get:
as many times greater as the area of the object-lens is greater than
that of the pupil of the eye. If we estimate this latter as having a
diameter of one-fifth of an inch, then a 1-inch telescope would increase
the light twenty-five times, a 10-inch instrument 2500 times,
and the great Lick telescope, of thirty-six inches' aperture, 32,400
times, the amount being proportional to the \textit{square} of the diameter
of the lens.

It must not be supposed, however, that the apparent brightness of
an object like the moon, or a planet which shows a disc, is increased
in any such ratio, since the eye-piece spreads out the light to cover a
vastly more extensive angular area, according to its magnifying
power; in fact, it can be shown that no optical arrangement can
show an \textit{extended surface} brighter than it appears to the naked
eye. But the \textit{total quantity} of light utilized is greatly increased
by the telescope, and in consequence, multitudes of stars, far too
faint to be visible to the unassisted eye, are revealed; and, what is
practically very important, \textit{the brighter stars are easily seen by day}
with the telescope.

\nbarticle{39.} \nbparatext{Distinctness of Image.}---This depends upon the exactness
with which the lens gathers to a single \textit{point} in the focal image all
the rays which emanate from the corresponding point in the object.
A single lens, with spherical surfaces, cannot do this very perfectly,
the ``aberrations'' being of two kinds, the \textit{spherical} aberration and
the \textit{chromatic}. The former could be corrected, if it were worth while,
by slightly modifying the form of the lens-surfaces; but the latter,
which is far more troublesome, cannot be cured in any such way.
The violet rays are more refrangible than the red, and come to a
focus nearer the lens; so that the image of a star formed by such
a lens can never be a luminous point, but is a round patch of light
of different color at centre and edge.

\begin{fineprint}%comment as 032.png
\nbarticle{40.} \nbparatext{Long Telescopes.}---By making the diameter of the lens very
small as compared with its focal length, the aberration becomes less conspicuous;
and refractors were used, about 1660, having a length of more than
100 feet and a diameter of five or six inches. The object-glass was mounted
at the top of a high pole and the eye-piece was on a separate stand below.
With such an ``aerial telescope,'' of six inches aperture and 120 feet focus,
Huyghens discovered the rings of Saturn. His object-glass still exists, and
is preserved in the collection of the Royal Society in London.
\end{fineprint}
%% -----File: 036.png---Folio 25-------

\nbarticle{41.} \nbparatext{The Achromatic Telescope.}---The chromatic aberration of a
lens, as has been said, cannot be cured by any modification of the lens
itself; but it was discovered in England about 1760 that it can be
nearly corrected by making the object-glass of \textit{two} (or more) lenses,
of \textit{different kinds of glass}, one of the lenses being convex and the
other concave. The convex lens is usually made of \textit{crown} glass, the
concave of \textit{flint} glass. At the same time, by properly choosing the
curves, the \textit{spherical} aberration can also be destroyed, so that such a
compound object-glass comes reasonably near to fulfilling the condition,
that it should gather to a mathematical point in the image all
the rays that reach the object-glass from a single point in the object.

\begin{wrapfigure}{r}{0pt}
\footnotesize
\renewcommand\arraystretch{0.2}
\begin{tabular}[t]{c} 1\\ \includegfx{illo009a}\\ \textit{Clark}   \end{tabular}\:
\begin{tabular}[t]{c} 2\\ \includegfx{illo009b}\\ \textit{Gauss}   \end{tabular}\:
\begin{tabular}[t]{c} 3\\ \includegfx{illo009c}\\ \textit{Littrow} \end{tabular}
\caption*{\textsc{Fig.~9.}---Different Forms of the Achromatic Object-glass.}
\end{wrapfigure}

\begin{fineprint}
These object-glasses admit of considerable variety of forms. Formerly
they were generally made, as in \figref{illo009c}{Fig.~9}, No.~3, having the two lenses close
together, and the adjacent surfaces of the same, or nearly the same, curvature.
In small object-glasses the lenses are often cemented together with
Canada balsam or some other transparent medium. At present some of the
best makers separate the two lenses by a considerable distance, so as to
admit a free circulation of air between them; in the Pulkowa and Princeton
object-glasses, constructed by
Clark, the lenses are seven inches
apart, and in the Lick telescope six
and a half inches; as in No.~1. In
a form devised by Gauss (No.~2),
which has some advantages, but is
difficult of construction, the curves
are very deep, and both the lenses are of watch-glass form---concave on one
side and convex on the other. In all these forms the crown glass is outside;
Steinheil, Hastings, and others have constructed lenses with the \textit{flint-glass}
lens outside. Object-glasses are sometimes made with \textit{three} lenses instead
of two; a slightly better correction of aberrations can be obtained in this
way, but the gain is too small to pay for the extra expense and loss of light.
\end{fineprint}

\nbarticle{42.} \nbparatext{Secondary Spectrum.}---It is not, however, possible with the
kinds of glass at present available to secure a perfect correction of the
color. Our best achromatic lenses bring the yellowish green rays to
a focus \textit{nearer the lens} than they do the red and violet. In consequence,
the image of a bright star is surrounded by a purple halo,
which is not very noticeable in a good telescope of small size, but
is very conspicuous and troublesome in a large instrument.

\begin{fineprint}
This imperfection of achromatism makes it unsatisfactory to use an ordinary
lens (\textit{visually} corrected) for astronomical photography. To fit it to
make good photographs, it must either be specially corrected for the rays
%% -----File: 037.png---Folio 26-------
that are most effective in photography, the blue and violet (in which case it
will be almost worthless visually), or else a subsidiary lens, known as a ``photographic
corrector,'' may be provided, which can be put on in front of the
object-glass when needed. A new form of object-glass, devised independently
by Pickering in this country and Stokes in England, avoids the necessity
of a third lens by making the crown-glass lens of such a form that when
put close to the flint lens, with the \textit{flatter side out}, it makes a perfect object-glass
for visual purposes; but by simply reversing the crown lens, with the
more convex side outward, and separating the lenses an inch or two, it becomes
a photographic object-glass. A 13-inch object-glass of this construction
at Cambridge performs admirably.

Much is hoped from the new kind of glass now being made at Jena. In
combination with crown glass it produces lenses almost free from chromatic
aberration, and if it can be produced in homogeneous pieces of sufficient
size, it will revolutionize the art of telescope making.
\end{fineprint}

\nbarticle{43.} \nbparatext{Diffraction and Spurious Disc.}---Even if a lens were perfect
as regards the correction of aberrations, the ``wave'' nature of light
prevents the image of a luminous point from being also a point; the
image must \textit{necessarily} consist of a central \textit{disc}, brightest in the centre
and fading to darkness at the edge, and this is surrounded by a
series of bright rings, of which, however, only the smallest one is
generally easily seen. The size of this disc-and-ring system can be
calculated from the known wave-lengths of light and the dimensions
of the lens, and the results agree very precisely with observation.
The diameter of the ``spurious disc'' \textit{varies inversely} with the aperture
of the telescope. According to Dawes, it is about $4''.5$ for a
1-inch telescope; and consequently $1''$ for a $4\frac{1}{2}$-inch instrument, $0''.5$
for a 9-inch, and so on.

\begin{fineprint}
This circumstance has much to do with the superiority of large instruments
in showing minute details. No increase of magnifying power on a
small telescope can exhibit things as sharply as the same power on the larger
one; provided, of course, that the larger object-glass is equally perfect in
workmanship, and the \textit{air} in good optical condition.

If the telescope is a good one, and if the air is perfectly steady,---which
unfortunately is seldom the case,---the apparent disc of a star should be
perfectly round and well defined, without wings or tails of any kind, having
around it from one to three bright rings, separated by distances somewhat
greater than the diameter of the disc. If, however, the magnifying power
is more than about 50 to the inch of aperture, the edge of the disc will begin
to appear hazy. There is seldom any advantage in the use of a magnifying
power exceeding 75 to the inch, and for most purposes powers ranging from
20 to 40 to the inch are most satisfactory.
\end{fineprint}
%% -----File: 038.png---Folio 27-------

\nbarticle{44.} \nbparatext{Eye-Pieces.}---For many purposes, as for instance the examination
of close double stars, there is no better eye-piece than the simple
convex lens; but it performs well only when the object is exactly in
the centre of the field. Usually it is best to employ for the \textit{eye-piece}
a combination of two or more lenses which will give a more extensive
field of view.

Eye-pieces belong to two classes, the \textit{positive} and the \textit{negative}. The
former, which are much more generally useful, act as simple magnifying-glasses,
and can be used as hand magnifiers if desired. The focal
image formed by the object-glass lies \textit{outside} of the eye-piece.

In the \textit{negative} eye-pieces, on the other hand, the rays from the
object-glass are intercepted before they come to the focus, and the
image is formed between the lenses of the eye-piece. Such an eye-piece
cannot be used as a hand magnifier.

\begin{fineprint}
\nbarticle{45.} The simplest and most common forms of these eye-pieces are the
Ramsden (positive) and
Huyghenian (negative).
Each is composed of two
plano-convex lenses, but
the arrangement and
curves differ, as shown
in \figref{illo010a}{Fig.~10}. The former
gives a very flat field of
view, but is not achromatic;
the latter is more
nearly achromatic, and
possibly defines a little better just at the centre of the field; but the fact
that it is a \textit{negative} eye-piece greatly restricts its usefulness. In the Ramsden
eye-piece the focal lengths of the two component lenses, both of which
have their flat sides out, are about equal to each other, and their distance is
about one-third of the sum of the focal lengths. In the Huyghenian the
curved sides of the lenses are both turned towards the object-glass; the
focal distance of the field lens should be exactly \textit{three} times that of the lens
next the eye, and the distance between the lenses one-half the sum of the
focal lengths. The peculiarity of the Steinheil ``monocentric'' eye-piece
which is a triple achromatic positive lens, consisting of a central convex
lens of crown glass, with a concave meniscus of flint glass cemented to each
side, is that \textit{the curves are all struck from the same centre}, the thickness of the
lenses being so computed as to produce the needed corrections. It is free
from all internal reflections, which in other eye-pieces often produce ``ghosts,''
as they are called.

\begin{wrapfigure}{r}{0pt}
\footnotesize \itshape
\begin{tabular}[b]{c}Ramsden   \\(Positive)               \\\includegfx{illo010a}\end{tabular}\quad
\begin{tabular}[b]{c}Huyghenian\\(Negative)               \\\includegfx{illo010b}\end{tabular}\quad
\begin{tabular}[b]{c}Steinheil \\`Monocentric'\\(Positive)\\\includegfx{illo010c}\end{tabular}
\caption*{\textsc{Fig.~10.}---Various Forms of Telescope Eye-piece.}
\end{wrapfigure}

There are numerous other forms of eye-piece, each with its own advantages
and disadvantages. The \textit{erecting} eye-piece, used in spy-glasses, is
%% -----File: 039.png---Folio 28-------
essentially a compound microscope, and gives erect vision by again inverting
the already inverted image formed by the object-glass.

It is obvious that in a telescope of any size the object-glass is the most
important and expensive part of the instrument. Its cost varies from a few
hundred dollars to many thousands, while the eye-pieces generally cost only
from \$5 to \$20 apiece.
\end{fineprint}

\nbarticle{46.} \nbparatext{Reticle.}---When a telescope is used for \textit{pointing}, as in most
astronomical instruments, it must be provided with a \textit{reticle} of some
sort. This is usually a metallic frame with \textit{spider lines} stretched
across it, placed, not near the object-glass itself (as is often supposed),
but at the \textit{focus} of the object-glass, where the image is
formed, as at $a\:b$ in \figref{illo008}{Fig.~8}.

\begin{fineprint}
It is usually so arranged that it can be moved in or out a little to get it
exactly into the focal plane, and then, when the eye-piece (positive) is adjusted
for the observer's eye to give distinct vision of the object, the ``wires,''
as they are called, will also be equally distinct. As spider-threads are very
fragile, and likely to get broken or displaced, it is often better to substitute
a thin plate of glass with lines ruled upon it and blackened. Of course,
provision must be made for illuminating either the field of view or the
threads themselves, in order to make them visible in darkness.
\end{fineprint}

\nbarticle{47.} \nbparatext{The Reflecting Telescope.}---When the chromatic aberration
of lenses came to be understood through the optical discovery of
the dispersion of light by Newton, the reflecting telescope was
invented, and held its place as the instrument for star-gazing until
well into the present century, when large achromatics began to be
made. There are several varieties of reflecting telescope, all agreeing
in the substitution of a large concave mirror in place of the object-glass
of the refractor, but differing in the way in which they get at
the image formed by this mirror at its focus in order to examine it
with the eye-piece.

\begin{fineprint}
\nbarticle{48.} In the Herschellian form, which is the simplest, but only suited to
very large instruments, the mirror is \textit{tipped} a little, so as to throw the image
to the side of the tube, and the observer stands with his back to the object
and looks down into the tube. If the telescope is as much as two or three
feet in diameter, his head will not intercept enough light to do much harm,---not
nearly so much as would be lost by the second reflection necessary in
the other forms of the instrument. But the inclination of the mirror, and
the heat from the observer's person, are fatal to any very accurate definition,
and unfit this form of instrument for anything but the observation of nebulę
and objects which mainly require light-gathering power.
%% -----File: 040.png---Folio 29-------

\begin{wrapfigure}{r}{0pt}
\footnotesize
\begin{tabular}{@{}m{1em}@{}m{7.1cm}@{}}
1 & \includegfx{illo011a}\\
2 & \includegfx{illo011b}\\
3 & \includegfx{illo011c}\\
\end{tabular}
\caption*{\textsc{Fig.~11.}---Different Forms of Reflecting Telescope.\\[0.8ex]
1.~The~Herschellian;\; 2.~The~Newtonian;\; 3.~The~Gregorian.}
\end{wrapfigure}

\sloppy
In the Newtonian telescope, a small plane reflector standing at an angle
of 45° is placed in the centre of the tube, so as to intercept the rays reflected
by the large mirror a little before they come to their focus, and throw them
to the side of the tube, where the eye-piece is placed.


In the Gregorian form (which was the first invented), the large mirror is
pierced through its centre, and the rays from it are reflected through the
hole by a small \textit{concave} mirror, placed a little outside of the principal focus
at the mouth of the tube. With this instrument one looks directly at the
stars as with a refractor, and the image is erect.

\fussy
The Cassegrainian form is very similar, except that the small concave
mirror of the Gregorian is replaced by a \textit{convex} mirror, placed a little inside
the focus of the large mirror, which makes the instrument a little shorter,
and gives a flatter field
of view.

Formerly the great
mirror was always made
of a composition of copper
and tin (two parts
of copper to one of tin)
known as ``speculum
metal.'' At present it is
usually made of glass
\textit{silvered} on the front surface,
by a chemical process
which deposits the
metal in a thin, brilliant
film. These silver-on-glass
reflectors, when new,
reflect much more light
than the old specula, but
the film does not retain its polish so long. It is, however, a comparatively
simple matter to renew the film when necessary.

The largest telescopes ever made have been reflectors. At the head of the
list stands the enormous instrument of Lord Rosse, constructed in 1842, with
a mirror six feet in diameter and sixty feet focal length. Next in order are
a number of instruments of four feet aperture, first among which is the great
telescope of the elder Herschel, built in 1789, followed by the telescope
erected by Lassell at Malta in 1860, the Melbourne reflector by Grubb in 1870,
and the still more recent silver-on-glass reflector of the Paris observatory,
which, however, has proved a failure, owing to defective support of the mirror.

\nbarticle{49.} \nbparatext{Relative Advantages of Refractors and Reflectors.}---There has
been a good deal of discussion on this point, and each construction has its
partisans.

In favor of the reflectors we may mention,---

First. \textit{Ease of construction and consequent cheapness.} The concave mirror
%% -----File: 041.png---Folio 30-------
has but one surface to figure and polish, while an object-glass has four.
Moreover, as the light goes \textit{through} an object-glass, it is evident that the
glass employed must be perfectly clear and of uniform density through and
through; while in the case of the mirror, the light does not penetrate the
material at all. This makes it vastly easier to get the material for a large
mirror than for a large lens.

Second (and immediately connected with the preceding). \textit{The possibility
of making reflectors much larger than refractors.} Lord Rosse's great reflector
is six feet in diameter, while the Lick telescope, the largest of all refractors,
is only three.

Third. \textit{Perfect achromatism.} This is unquestionably a very great advantage,
especially in photographic and spectroscopic work.

But, on the whole, the advantages are generally considered to lie with the
refractors.

In their favor we mention:---

First. \textit{Great superiority in light.} No mirror (unless, perhaps, a \textit{freshly
polished} silver-on-glass film) reflects much
more than three-quarters of the incident
light; while a good (single) lens transmits
over 95 per cent. In a good refractor
about 82 per cent of the light
reaches the eye, after passing through
the four lenses of the object-glass and
eye-piece. In a Newtonian reflector, in
average condition, the percentage seldom
exceeds 50 per cent, and more
frequently is lower than higher.

Second. \textit{Better definition.}---Any slight
error at a point in the surface of a glass
lens, whether caused by faulty workmanship
or by distortion, affects the direction
of the ray passing through it only \textit{one-third} as much as the same error
on the surface of a mirror would do.

\includegraphicsouter{illo012}{\textsc{Fig.~12.}---Effect of Surface Errors in a Mirror and in a Lens.}

If, for instance, in \figref{illo012}{Fig.~12}, an element of the surface at $P$ is turned out
of its proper direction, $aa'$, by a small angle, so as to take the direction $bb'$,
then the \textit{reflected} ray will be sent to $f$, and its deviation will be \textit{twice} the
angle $aPb$. But since the index of refraction of glass is about 1.5 the
change in the direction of the \textit{refracted} ray from $R$ to $r$ will only be about
\textit{two-thirds} of $aPb$.

Moreover, so far as distortions are concerned, when a lens bends a little
by its own weight, \textit{both sides are affected in a nearly compensatory manner},
while in a mirror there is no such compensation. As a consequence, mirrors
very seldom indeed give any such definition as lenses do. The least fault
of workmanship, the least distortion by their own weight, the slightest
difference of temperature, between front and back, will absolutely ruin the
image, while a lens would be but slightly affected in its performance by
the same circumstances.
%% -----File: 042.png---Folio 31-------

Third. \textit{Permanence.} The lens, once made, and fairly taken care of,
suffers no deterioration from age; but the metallic speculum or the silver
film soon tarnishes, and must be repolished every few years. This alone
is decisive in most cases, and relegates the reflector mainly to the use of
these who are themselves able to construct their own instruments.

To these considerations we may add that a refractor, though more expensive
than a reflector of similar power, is not only more permanent, and less
likely to have its performance affected by accidental circumstances, but is
lighter and more convenient to use.
\end{fineprint}

\sloppy
\nbarticle{50.} \nbparatext{Time-Keepers and Time-Recorders.}---\textit{The Clock, Chronometer,
and Chronograph.}---Modern practical astronomy owes its development
as much to the clock and chronometer as to the telescope. The
ancients possessed no accurate instruments for the measurement of
time, and until within 200 years, the only reasonably precise method
of fixing the time of an important observation, as, for instance, of
an eclipse, was by noting the \textit{altitude} of the sun, or of some known
star at or very near the moment.

\fussy
It is true that the Arabian astronomer Ibn Jounis had made some
use of the pendulum about the year 1000 \textsc{a.d.}, more than 500 years
before Galileo introduced it to Europeans. But it was not until
nearly a century after Galileo's discovery that Huyghens applied it
to the construction of clocks (in 1657).

So far as the principles of construction are concerned, there is no
difference between an astronomical clock and any other. As a matter
of convenience, however, the astronomical clock is almost invariably
made to beat seconds (rarely half-seconds), and has a conspicuous
second-hand, while the hour-hand makes but \textit{one} revolution a day,
instead of two, as usual, and the face is marked for twenty-four hours
instead of twelve. Of course it is constructed with extreme care in
all respects.

\begin{fineprint}
\textit{The Escapement}, or ``\textit{Scapement},'' is often of the form known as the
``Graham Dead-beat''; but it is also frequently one of the numerous ``gravity'' escapements
which have been invented by ingenious mechanicians. The office of
the escapement is to be ``unlocked'' by the pendulum at each vibration, so
as to permit the wheel-work to advance one step, marking a second (or sometimes
two seconds), upon the clock-face; while, at the same time, the escapement
gives the pendulum a slight impulse, just equal to the resistance it has
suffered in performing the unlocking. The work done by the pendulum in
``unlocking'' the train, \textit{and the corresponding impulse, ought to be perfectly
constant}, in spite of all changes in the condition of the train of wheels; and
it is \textit{desirable}, though not \textit{essential}, that this work should be as
\textit{small} as possible.
\end{fineprint}
%% -----File: 043.png---Folio 32-------
\nbarticle{51.} The pendulum itself is usually suspended by a flat spring, and
great pains should be taken to have the support extremely firm: this
is often neglected, and the clock then cannot perform well.

\sloppy
\textit{Compensation for Temperature.}---In order to keep perfect time,
the pendulum must be a ``compensation pendulum''; \textit{i.e.}, constructed in such a way
that changes of temperature will
not change its length.

\fussy
An uncompensated pendulum, with steel rod,
changes its daily rate about one-third of a second
for each degree of temperature (centigrade).
A wooden pendulum rod is much less affected
by temperature, but is very apt to be disturbed
by changes of \textit{moisture}.

\begin{fineprint}
\includegraphicsouter{illo013}{\textsc{Fig.~13.}\\
Compensation Pendulums.\\
1.~Graham's Pendulum.\\
2.~Zinc-Steel Pendulum.}
Graham's mercurial pendulum (\figref{illo013}{Fig.~13}) is the
one most commonly used. It consists simply of a
jar (usually steel), three or four inches in diameter,
and about eight inches high, containing forty or fifty
pounds of mercury, and suspended at the end of a
steel rod. When the temperature rises, the rod
lengthens (which would make the clock go slower);
but, at the same time, the mercury expands, from
the bottom upwards, just enough to compensate.
This pendulum will perform well only when not
exposed to \textit{rapid changes} of temperature. Under
rapid changes the compensation \textit{lags}. If, for instance,
it grows warm quickly, the rod will expand
before the mercury does; so that, \textit{while the mercury is
growing warmer}, the clock will run slow, though after
it has become warm the rate may be all right.

A compensation pendulum, constructed on the
principle of the old gridiron pendulum of Harrison,
but of zinc and steel instead of brass and steel, is
now much used. The compensation is not so easily
adjusted as in the mercurial pendulum, but when properly made the mechanism
acts well, and bears rapid alterations of temperature much better than
the mercurial pendulum. The heavy pendulum-bob, a lead cylinder, is hung
at the end of a steel rod, which is suspended from the top of a zinc tube,
and hangs through the centre of it. This tube is itself supported \textit{at the bottom}
by three or four steel rods which hang from a piece attached to the pendulum
spring. The standard clock at Greenwich has a pendulum of this kind.
\end{fineprint}

\nbarticle{52.} \nbparatext{Effect of Atmospheric Pressure.}---In consequence of the
buoyancy of the air, and its resistance to motion, a pendulum swings
%% -----File: 044.png---Folio 33-------
a little more slowly than it would \textit{in vacuo}, and every change in the
density of the air affects its rate more or less. With mercurial
pendulums, of ordinary construction, the ``\textit{barometric coefficient},''
as it is called, is about one-third of a second for an inch of the
barometer; \textit{i.e.}, an increase of atmospheric density which would
raise the barometer one inch would make the clock \textit{lose} about one-third
of a second daily. It varies considerably, however, with different
pendulums.

\begin{fineprint}
It is not very usual to take any notice of this slight disturbance; but
when the extremest accuracy of time-keeping is aimed at, the clock is either
sealed in an air-tight case from which the air is partially exhausted (as at
Berlin), or else some special mechanism, controlled by a barometer, is devised
to compensate for the barometric changes, as at Greenwich. In the
Greenwich clock a magnet is raised or lowered by the rise or fall of the
mercury in a barometer attached to the clock-case. When the magnet rises,
it approaches a bit of iron two or three inches above it, fixed to the bottom
of the pendulum, and the increase of attraction accelerates the rate just
enough to balance the retardation due to the air's increased density and
viscosity. There are several other contrivances for the same purpose.
\end{fineprint}

\nbarticle{53.} \nbparatext{Error and Rate.}---The ``\textit{error},'' or ``\textit{correction}'' of a clock
is the amount that \textit{must be added} to the indication of the clock-face
at any moment in order to give the \textit{true time}; it is, therefore, \textit{plus}
($+$) when the clock is \textit{slow}, and \textit{minus} ($-$) when it is \textit{fast}. The
\textit{rate} of a clock is the amount of its \textit{daily gain or loss}; \textit{plus} ($+$) when
the clock is \textit{losing}. Sometimes the \textit{hourly rate} is used, but ``\textit{hourly}''
is then always specified.

A \textit{perfect} clock is one that has a \textit{constant rate}, whether that rate
be large or small. It is desirable, for convenience' sake, that both
error and rate should be small; but this is a mere matter of adjustment
by the user of the clock, who adjusts the error by setting the
hands, and the rate by raising or lowering the pendulum-bob.

\begin{fineprint}
The final adjustment of rate is often obtained by first setting the pendulum-bob
so that the clock will run slow a second or two daily, and then
putting on the top of the bob little weights of a gramme or two, which will
accelerate the motion. They can be dropped into place or knocked off without
stopping the clock or perceptibly disturbing it.

The very best clocks will run three or four years without being stopped
for cleaning, and will retain their rate without a change of more than one-fifth
of a second, one way or the other, during the whole time. But this is
%% -----File: 045.png---Folio 34-------
exceptional performance. In a run as long as that, most clocks would
be liable to change their rate as much as half a second or more, and to do
it somewhat irregularly.
\end{fineprint}

\sloppy
\nbarticle{54.} \nbparatext{The Chronometer.}---The pendulum-clock not being portable,
it is necessary to provide time-keepers that are. The chronometer is
merely a carefully made watch, with a balance wheel compensated
to run, as nearly as possible, at the same rate in different temperatures,
and with a peculiar escapement, which, though unsuited to
watches exposed to ordinary rough usage, gives better results than
any other when treated carefully.

\fussy
\begin{fineprint}
The \textit{box-chronometer} used on ship-board is usually about twice the diameter
of a common pocket watch, and is mounted on gimbals, so as to keep horizontal
at all times, notwithstanding the motion of the vessel. It usually
beats half-seconds. It is not possible to secure in the chronometer-balance
as perfect a temperature correction as in the pendulum. For this and other
%% -----File: 046.png---Folio 35-------
reasons the best chronometers cannot quite compete with the best clocks in
precision of time-keeping; but they are sufficiently accurate for most purposes,
and of course are vastly more convenient for field operations. They
are simply indispensable at sea. \textit{Never turn the hands of a chronometer
backward.}
\end{fineprint}

\includegraphicsmid{illo014}{\textsc{Fig.~14.}---A Chronograph by Warner and Swasey.}

\nbarticle{55.}  Before the invention of the telegraph it was customary to note
time merely ``by eye and ear.'' The observer, keeping his time-piece
near him, listened to the clock-beats, and estimated as closely
as he could, in seconds and \textit{tenths} of seconds, the moment when the
phenomenon he was watching occurred---the moment, for instance,
when a star passed across a wire in the reticle of his telescope.
At present the record is usually made by simply pressing a ``key''
in the hand of the observer, and this, by a telegraphic connection,
makes a mark upon a strip or sheet of paper, which is moved at a
uniform rate by clock-work, and graduated by seconds-signals from
the clock or chronometer.

\nbarticle{56.} \nbparatext{The Chronograph.}---This is the instrument which carries the
marking-pen and moves the paper on which the time-record is made.
The paper is wrapped upon a cylinder, six or seven inches in diameter,
and fifteen or sixteen inches long. This cylinder is made to revolve
once a minute, by clock-work, while the pen rests lightly upon
the paper and is slowly drawn along by a screw-motion, so that it
marks a continuous spiral. The pen is carried on the armature of
an electro-magnet, which every other second (or sometimes every
second) receives a momentary current from the clock, causing it to
make a mark like those which break the lines in the \figref{illo015}{figure} annexed.

\includegraphicsmid{illo015}{\textsc{Fig.~15.}---Part of a Chronograph Record.}

The beginning of a new minute (the 60th sec.) is indicated either
by a double mark as shown, or by the omission of a mark. When
the observer touches his key he also sends a current through the
magnet, and thus interpolates a mark of his own on the record, as
at $X$ in the \figref{illo015}{figure}: the \textit{beginning} of the mark is the instant noted---in
this case 54.9$^{\text{s}}$. Of course the minutes when the chronograph was
started and stopped are noted by the observer on the sheet, and so
enable him to identify the minutes and seconds all through the record.
%% -----File: 047.png---Folio 36-------

\begin{fineprint}
Many European observatories use chronographs in which the record is
made upon a long fillet of paper, instead of a sheet on a cylinder. The
instrument is lighter and cheaper than the American form, but much less
convenient.

The regulator of the clock-work must be a ``continuous'' regulator, working
continuously, and not by beats like a clock-escapement. There are
various forms, most of which are centrifugal governors, acting either by
friction (like the one in the \figref{illo014}{figure}) or by the resistance of the air; or else
``spring-governors,'' in which the motion of a train, with a pretty heavy
fly-wheel, is slightly checked at regular intervals by a pendulum.

\sloppy
\nbarticle{57.} \nbparatext{Clock-Breaks.}---The arrangements by which the clock is made to
send regular electric signals are also various. One of the earliest and simplest
is a fine platinum wire attached to the pendulum, which swings through a
drop of mercury at each vibration. All of the arrangements, however, in
which the pendulum itself has to make the electric contact are objectionable,
and for clocks using the Graham dead-beat scapement no absolutely satisfactory
means of giving the signals has yet been devised. Clocks with the
gravity escapements have a decided advantage in this respect. Their wheel-work
has no direct action in driving the pendulum, and so may be made to
do any reasonable amount of outside work in the way of ``key-manipulation''
without affecting the clock-rate in the least. Usually a wheel on the
axis of the scape-wheel is made to give the electric signals by touching a
light spring with one of its teeth every other second.

\fussy
Chronometers are now also fitted up in the same way, to be used with the
chronograph.

The signals sent are sometimes ``breaks'' in a continuous current, and
sometimes ``makes'' in an open circuit. Usage varies in this respect, and
each method has its advantages. The break-circuit system is a little simpler
in its connections, and possibly the signals are a little more sharp, but it
involves a much greater consumption of battery material, as the current is
always circulating, except during the momentary breaks.
\end{fineprint}

\nbarticle{58.} \nbparatext{Meridian Observations.}---A large proportion of all astronomical
observations are made at the time when the heavenly body
observed is crossing the meridian, or very near it. At that moment
the effects of refraction and parallax (to be discussed hereafter) are
a minimum, and as they act only in a vertical plane, they do not have
any influence on the \textit{time} at which the body crosses the meridian.

\nbarticle{59.} \nbparatext{The Transit Instrument} is the instrument used, in connection
with a clock or chronometer, and often with a chronograph also, to
observe the time of a star's ``transit'' across the meridian.

If the error of the (sidereal) clock is known at the moment, this
observation will determine the right ascension of the body, which, it
%% -----File: 048.png---Folio 37-------
will be remembered, is simply \textit{the sidereal time at which it crosses the
meridian}; \textit{i.e.}, the number of hours, minutes, and seconds by which it
follows the vernal equinox.

\textit{Vice versa}, if the right ascension is known, the error or correction
of the clock will be determined.

\sloppy
The instrument (\figref{illo016}{Fig.~16}) consists essentially of a telescope mounted
upon a stiff axis perpendicular to the telescope tube. This axis is
placed horizontal, east and west, and turns on \textit{pivots} at its extremities,
in Y-bearings upon the top of two fixed piers or pillars. A
small graduated circle is attached, to facilitate ``setting'' the telescope
at any designated altitude or declination.

\fussy
\includegraphicsmid{illo016}{\textsc{Fig.~16.}---The Transit Instrument (Schematic).}

The telescope carries at the eye-end, in the focal plane of the
object-glass, a reticle of some odd number of vertical wires,---five
or more,---one of which is always in the centre, and the others
are usually placed at equal distances on each side of it. One or
two wires also cross the field horizontally.

If the pivots are true, and the instrument accurately adjusted, it
is evident that the \textit{central vertical wire will always follow the meridian
as the instrument is turned}; and the instant when a star crosses this
wire will be the true moment of the star's meridian transit. The
%% -----File: 049.png---Folio 38-------
object in having a number of wires is, of course, simply to gain
accuracy by taking the mean of a number of observations instead of
depending upon a single one.

In order to ``\textit{level}'' the axis properly, a delicate spirit-level is an
essential adjunct; it is usual, also, (and important) to provide a convenient
``\textit{reversing apparatus},'' by which the instrument can be
turned half round, making the eastern and
western pivots change places.

\includegraphicsouter{illo017}{\textsc{Fig.~17.}---Reticle of the Transit Instrument.}

The instrument must be thoroughly stiff
and rigid, without loose joints or shaky
screws; and the two pivots must be \textit{accurately
round}, \textit{precisely in line with each other},
\textit{free from taper}, and precisely \textit{of the same
size}; all of which conditions may be summed
up by saying that they \textit{must be portions of
one and the same geometrical cylinder}.

\begin{fineprint}
The proper construction and grinding of
these pivots, which are usually of hard bell metal (sometimes of steel),
taxes the art of the most skilful mechanician. The level, also, is a delicate
instrument, and difficult to construct.

Provision is made, of course, for illuminating the field of view at night
so as to make the reticle wires visible. Usually one (or both) of the pivots
is pierced, and a lamp throws light through the opening upon a small mirror
in the centre of the tube, which reflects it down upon the reticle.

The Y's are used instead of round bearings, in order to prevent any
\textit{rolling} or \textit{shake} of the pivots as the instrument turns.

Fig.~18 shows a modern transit instrument (portable) as actually constructed
by Fauth \& Co.
\end{fineprint}

Another form of the instrument is much used, which is often
designated as the ``Broken Transit.'' A reflector in the central cube
throws the rays coming from the object-glass, out at right angles
through one end of the axis, where the eye-piece is placed; so that
the observer does not have to change his position at all for different
stars, but simply looks straight forward horizontally. It is very
convenient and rapid in actual work, but the observations require a
considerable correction for \textit{flexure of the axis}.

\nbarticle{60.} \begin{fineprint}\textit{Adjustments.}---(1) Focus and verticality of wires. (2)~Collimation.
(3)~Level. (4)~Azimuth.

First. The first thing to do after the instrument is set on its supports and
the axis roughly levelled, is \textit{to adjust the reticle}. The eye-piece is drawn out
%% -----File: 050.png---Folio 39-------
or pushed in until the wires appear perfectly sharp, and then the instrument
is directed to a star or to some \textit{distant} object (not less than a mile away),
and without disturbing the eye-piece, the sliding-tube, which carries the
reticle, is drawn out or pushed in until the object is also distinct at the
%% -----File: 051.png---Folio 40-------
same time with the wires. If this adjustment is correctly made, motion of
the eye in front of the eye-piece will not produce any apparent displacement
of the object in the field, with reference to the wires. To test the verticality
of the wires, the telescope is moved up and down a little, while looking at the
object; if the axis is level and the wires vertical, the wire will not move off
from the object sideways. There are screws provided to turn the reticle a
little, so as to effect this adjustment.

\includegraphicsmid{illo018}{\textsc{Fig.~18.}---A 3-inch Transit, with reversing apparatus. Fauth \& Co.}

When the wires have been thus adjusted for focus and verticality, the
reticle-slide should be tightly clamped and never disturbed again. The \textit{eye-piece}
can be moved in and out at pleasure, to secure distinct vision for different
eyes, but it is \textit{essential that the distance between the object-glass and the
reticle remain constant}.

Second. \textit{Collimation.} The line joining the optical centre of the object-glass
with the middle wire of the reticle is called the ``\textit{line of collimation},''
and this line must be made exactly perpendicular to the axis of rotation
by moving the reticle slightly to one side or the other by means of the
adjusting screws provided for the purpose. The simplest way of effecting
the adjustment is to point the instrument on some well-defined distant
object, like a nail-head or a joint in brickwork, and then carefully
to ``reverse'' the instrument without disturbing the stand. If the middle
wire, after reversal, points just as it did before, the ``collimation'' is correct;
if not, the middle wire must be moved \textit{half way} towards the object by the
screws.

\medskip

\textbf{Collimator.}---It is not always easy to find a distant object on which to
make this adjustment, and a ``\textit{collimator}'' may be substituted with advantage.
This is simply a telescope mounted horizontally on a pier in front of the
transit instrument, so that when the transit telescope is horizontal, it can
look straight into the collimator, which ought to be of about the same size
as the transit itself.

In the focus of the collimator object-glass are placed two wires forming
an X, and thus placed they can be seen by a telescope looking into the collimator
just as distinctly as if they were at an infinite distance and really celestial
objects. The instrument furnishes us a mark \textit{optically} celestial, but
\textit{mechanically} within reach of our finger-ends for illumination, adjustment,
etc. If the pier on which it is mounted is firm, the collimator cross is in all
respects as good as a star, and much more convenient.

Third. \textit{Level.} The adjustment for \textit{level} is made by setting a striding
level on the pivots of the axis, reading the level, then reversing the level
(not the transit) and reading it again. If the pivots are round and of the
same size, the difference between the level-readings direct and reversed will
indicate the amount by which one pivot is higher than the other. One of
the Y's is made so that it can be raised and lowered slightly by means of a
screw, and this gives the means of making the axis horizontal. If the
pivots are not of the same size (and they never are \textit{absolutely}), the astronomer
must determine and allow for the difference.
%% -----File: 052.png---Folio 41-------

Fourth. \textit{Azimuth.} In order that the instrument may indicate the meridian
truly, its axis must lie exactly east and west; \textit{i.e.}, its \textit{azimuth} must be $90°$.
This adjustment must be made by means of observations upon the stars, and
is an excellent example of the method of successive approximations, which
is so characteristic of astronomical investigation. (\textit{a}) After adjusting carefully
the focus and collimation of the instrument, we set it north and south
\textit{by guess}, and level it as precisely as possible. By looking at the pole star,
and remembering how the pole itself lies with reference to it, one can easily
set the instrument \textit{pretty nearly}; \textit{i.e.}, within half a degree or so. The middle
wire will now describe in the sky a vertical circle, which crosses the meridian
at the zenith, and lies very near the meridian for a considerable distance
each side of the zenith.

(\textit{b}) We must next get an ``approximate'' time; \textit{i.e.}, set our clock or
chronometer \textit{nearly} right. To do this, we select from the list of several
hundred stars in the Nautical Almanac (which is to be regarded in about
the same light with the clock and the spirit level, as an indispensable accessory
to the transit) a star which is about to cross the meridian \textit{near the zenith}.
The difference between the right ascension of the star as given in the
Almanac, and the time shown by the clock-face, will be \textit{very nearly} the
error of the clock at the time of the observation: not \textit{exactly}, unless the declination
of the star is such that it passes \textit{exactly} through the zenith, but
\textit{very nearly}, since the star crosses the meridian near the zenith. We now
have the time within a second or two.

(\textit{c}) Next turn down the telescope upon some Almanac star, which is
soon to cross the meridian within $10°$ of the pole. It will appear to move
very slowly. A little before the time it should reach the meridian, move the
whole frame of the instrument until the middle wire points upon it, and
then, by means of the ``Azimuth Screw,'' which gives a slight horizontal
motion to one of the Y's, \textit{follow the star} until the \textit{indicated moment of its transit};
\textit{i.e.}, until the clock (corrected for clock error) shows on its face the star's
right ascension. If the clock correction had been known with absolute exactness,
the instrument would now be \textit{truly} in the meridian; as the clock error,
however, is only approximate, the instrument will only be approximately in
the meridian; but---and this is the essential point---it will be \textit{very much
more nearly so} than at the beginning of the operation. The supposed incorrectness,
amounting perhaps to one or two seconds, in the time at which the
instrument was set on the circumpolar star will, on account of the slow motion
of the star, make almost no perceptible difference in the direction given
to the axis.

A repetition of the operation may possibly be needed to secure all the
desired precision. The accuracy of this azimuth adjustment can then be
verified by three successive ``culminations'' or transits of the pole star, or
any other circumpolar. The interval occupied in passing from the upper to
the lower culmination on the west side of the meridian ought, of course,
to be exactly equal to the time on the eastern side; \textit{i.e.}, twelve sidereal
hours.
\end{fineprint}
%% -----File: 053.png---Folio 42-------

\nbarticle{61.} The final test of \textit{all} the adjustments, and of the accurate going
of the clock, is obtained by observing a number of Almanac stars of
widely different declination. If they all indicate \textit{identically} the same
clock correction, the instrument is in adjustment; if not, and if the
differences are not very great, it is possible to deduce from the
observations themselves the true clock error, and the adjustment
errors of the instrument.

\begin{fineprint}
\includegraphicsouter{illo019}{\textsc{Fig.~19.}---The Meridian Circle (Schematic).}
It is to be added, in this connection, that the astronomer can never assume
that \textit{adjustments are perfect}: even if once perfect, they would not stay so, on
account of changes of temperature and other causes. Nor are observations
ever absolutely accurate. The problem is, from observations more or less
\textit{inaccurate} but \textit{honest}, with instruments more or less \textit{maladjusted} but \textit{firm}, to
find the result that would have been obtained by a perfect observation with
a perfect and perfectly adjusted instrument. It can be more nearly done
than one might suppose. But the discussion of the subject belongs to
Practical Astronomy, and cannot be entered into here.
\end{fineprint}

\nbarticle{62.} \nbparatext{Prime Vertical Instrument.}---For certain purposes, a Transit
Instrument, provided with an apparatus for rapid reversal, is turned
quarter-way round and mounted with the axis \textit{north} and \textit{south}, so
that the plane of rotation lies \textit{east} and \textit{west}, instead of in the meridian.
It is then called a Prime Vertical Transit.

\nbarticle{63.} \nbparatext{The Meridian Circle.}---In
order to determine the \textit{Declination}
or \textit{Polar Distance} of an
object, it is necessary to have
some instrument for measuring
angles; mere time-observations
will not suffice. The instrument
most used for this purpose is the
\textit{Meridian Circle}, or \textit{Transit Circle},
which is simply a transit instrument,
with a graduated circle
attached to its axis, and revolving
with the telescope. Sometimes
there are two circles, one
at each end of the axis.

\sloppy
Fig.~19 represents the instrument
``schematically,'' showing merely the essential parts. \figref{illo020}{Fig.~20}
is a meridian circle, with a 4-inch telescope, constructed by Fauth
\& Co.
%% -----File: 054.png---Folio 43-------

\includegraphicsmid{illo020}{\textsc{Fig.~20.}---Meridian Circle.\\[1.5ex]

\begin{tabular}{@{}p{.48\textwidth}|p{.48\textwidth}@{}}
\hangindent=2em  $A$, $B$, $C$, $D$, the Reading Microscopes.

\hangindent=2em  $K$, the Graduated Circle.

\hangindent=2em  $H$, the Roughly Graduated Setting Circle.

\hangindent=2em  $I$, the Index Microscope. This is usually, however,
     placed half way between $A$ and $D$.
&
\hangindent=2em  $F$, the Clamp.    $G$, the Tangent Screw.

\hangindent=2em  $LL$, the Level, only placed in position occasionally.

\hangindent=2em  $M$, the Right Ascension Micrometer.

\hangindent=2em  $WW$, Counterpoises, which take part of the weight
      of the instrument off from the Y's.
\end{tabular}}
%% -----File: 055.png---Folio 44-------

\fussy
\begin{fineprint}
In observatory instruments the circle is usually from two to four feet in
diameter; larger circles were once used, but it is found that their weight,
and the consequent strains and flexures, render them actually less accurate
than the smaller ones. The utmost resources of mechanical art are exhausted
in making the graduation as precise as possible and in providing for
its accurate reading, as well as in securing the maximum firmness and stability
of every part of the instrument. The actual divisions are usually
$5'$ apart (in very large instruments sometimes only $2'$), but the circle is
``read'' to seconds and tenths of seconds of arc by means of \textit{reading microscopes},
from two to six in number, fixed to the pier of the instrument. In a
circle of forty inches diameter, $1''$ is a little less than $\frac{1}{10000}$ of an inch,
($\frac{20}{206265}$ inch), so that the necessity of fine workmanship is obvious.
\end{fineprint}

\nbarticle{64.} \nbparatext{The Reading Microscope} (\figref{illo021}{Fig.~21}).---This consists essentially
of a compound microscope, which forms a magnified image of
the graduation at the focus of its object-glass, where this image is
viewed by a positive eye-piece. At the
place where the image is formed a pair of
parallel spider-lines or a cross is placed,
movable in the plane of the image by a
``micrometer screw''; \textit{i.e.}, a fine screw
with a graduated head, usually divided into
sixty parts. One revolution of the screw
carries the wire $1'$ of arc, which makes
one division of the screw-head $1''$, the
tenths of seconds being estimated.

\includegraphicsouter{illo021}{\textsc{Fig.~21.}---The Reading Microscope.}

\begin{fineprint}
The adjustment of the microscope for
``runs,'' as it is called (that is, to make one
revolution of the micrometer screw exactly
equal to $1'$), is effected as follows. By setting
the wires first on one of the graduation marks
visible in the field of view, and then on the
next mark, it is immediately evident whether
five revolutions of the screw ``run'' over or
fall short of $5'$ of the graduation. If they \textit{overrun}, it shows that the image
of the graduation formed by the microscope objective is too small to fit the
screw, and \textit{vice-versa}. Now, by simply increasing or decreasing the distance
$A\,B$ between the objective and the micrometer box, the size of the image
can be altered at will, and the objective is therefore so mounted that this
can be done. Of course, every change in the length of the microscope tube
will also require a readjustment of the distance between the ``limb,'' or
graduated surface, of the circle and the microscope, in order to secure distinct
vision; but by a few trials the adjustment is easily made sufficiently precise.
%% -----File: 056.png---Folio 45-------

The reading of the circle is as follows: An extra index-microscope,
with low power and large field of view, shows by inspection the degrees
and minutes. The reading-microscopes are only used to give
the odd seconds, which is done by turning the screw until the parallel
spider-lines are made to include one of the graduation lines half-way
between themselves; the head of the screw then shows directly the
seconds and tenths, to be added to the degrees and minutes shown
by the index. Thus in \figref{illo022}{Fig.~22}, the reading of the microscope is
$3'\,22''.1$, the $3'$ being given by the \textit{scale} in the field, the $22''.1$ by the
screw-head.
\end{fineprint}

\includegraphicsouter{illo022}{\textsc{Fig.~22.}---Field of View of Reading Microscope.}

\nbarticle{65.} \nbparatext{Method of observing a Star.}---A minute or two before the star
reaches the meridian the instrument is approximately pointed, so that
the star will come into the field of view. As soon as it makes its
appearance, the instrument
is moved by the slow-motion
tangent-screw until
the star is ``bisected'' by
the fixed horizontal wire
of the reticle, and the
star is kept bisected until
it reaches the middle vertical
wire which marks the
meridian. The microscopes are then read, and their mean result is
the star's ``circle-reading.''

\begin{fineprint}
Frequently the star is bisected, not by moving the whole instrument, but
by means of a ``micrometer wire,'' which moves up and down in the field of
view. The micrometer reading then has to be combined with the reading
of the microscope, to get the true circle-reading.
\end{fineprint}

\nbarticle{66.} \nbparatext{Zero Points.}---In determining the declination or meridian
altitude of a star by means of its circle-reading, it is necessary to
know the ``\textit{zero point}'' of the circle. For declinations, the ``zero
point'' is either the polar or the equatorial reading of the circle; \textit{i.e.},
the reading of the circle when the telescope is pointed at the pole
or at the equator.

\textit{The ``polar point''} may be found by observing some circumpolar
star above the pole, and again, twelve hours later, below it. When
the two circle-readings have been \textit{duly corrected for refraction and
instrumental errors}, their mean will be the polar point.
%% -----File: 057.png---Folio 46-------

\begin{fineprint}
Suppose, for instance, that $\delta$~Ursę Minoris, at the ``upper culmination,''
gives a corrected reading of $52°\, 18'\, 25''.3$, while at the lower culmination the
reading is $45°\, 31'\, 35''.7$, then the mean of these, $48°\, 55'\, 00''.5$, is the polar
point, and of course the equatorial reading is $138°\, 55'\, 00''.5$,---just $90°$
greater. The \textit{polar distance} of the star would be the \textit{half-difference} of the
two readings, or $3°\, 23'\, 24''.8$.
\end{fineprint}

\nbarticle{67.} \nbparatext{Nadir Point.}---The determination of the polar point requires
two observations of the same star at an interval of twelve hours. It is
often difficult to obtain such a pair; moreover, the \textit{refraction} complicates
the matter, and renders the result less trustworthy. Accordingly
it is now usual to use the nadir or the horizontal reading as the
zero, rather than the polar point.

\includegraphicsouter{illo023}{\textsc{Fig.~23.}\\The Collimating Eye-Piece.}

The \textit{nadir point} is determined by pointing the telescope downwards
to a basin of mercury, moving the telescope until the image
of the horizontal wire of the reticle, as seen by reflection, coincides
with the wire itself. Since the reticle is exactly in the principal
focus of the object-glass, rays of light emitted by any point in the
reticle will become a parallel beam after passing the lens, and if this
beam strikes a plane mirror perpendicularly and
is returned, the rays will come just as if from a
real object in the sky, and will form an image
at the focal plane. When, therefore, the image
of the central wire of the reticle, seen in the
mercury basin by reflection, coincides with the
wire itself, we know that the line of collimation
must be exactly perpendicular to the surface of
the mercury; \textit{i.e.}, vertical.

\begin{fineprint}
To make the image visible it is necessary to illuminate the reticle by light
thrown \textit{towards} the object-glass from behind the wires, instead of light
coming from the object-glass towards the eye as usual. This peculiar illumination
is commonly effected by means of Bohnenberger's ``collimating
eye-piece,'' shown in \figref{illo023}{Fig.~23}. In the simplest form it is merely a common
Ramsden eye-piece, with a hole in one side, and a thin glass plate inserted
at an angle of $45°$. A light from one side, entering through the hole, will be
(partially) reflected towards the wires, and will illuminate them sufficiently.

The \textit{horizontal point} of course differs just $90°$ from the nadir point. It
may also be found independently by noting the circle-readings of some star
observed one night directly, and the next night by reflection in mercury; or,
if the star is a close circumpolar, both observations may be made the same
evening, one a few minutes before its meridian passage, the other just as
long after. But the method of the collimating eye-piece is fully as accurate
and vastly more convenient.

\includegraphicsmid{illo024}{\textsc{Fig.~24.}---Altitude and Azimuth Instrument.}
%% -----File: 058.png---Folio 47-------

\nbarticle{68.} \nbparatext{Differential Use of the Instrument.}---We now know the places of
several hundred stars with so much precision that in many cases it is quite
sufficient to observe one or two of these ``\textit{standard stars}'' in connection with
the bodies whose places we wish to determine. The difference between the
declination of the known star and that of any star whose place is to be
determined, will, of course, be simply the difference of their circle-readings,
corrected for refraction, etc. The meridian circle is said to be used ``\textit{differentially}''
when thus treated.
\end{fineprint}

\nbarticle{69.} \nbparatext{Errors of Graduation, etc.}---If the circle is from a reputable
maker, and has four or six microscopes, and if the observations are
carefully made and all the microscopes read each time, results of
sufficient precision for most purposes may be obtained by merely
correcting the observations for ``runs'' and refraction. The outstanding
errors ought not to exceed a second or two. But when the
\textit{tenths} of a second are in question, the case is different. It will not
then do for the astronomer to assume the accuracy of the graduation
of his circle, but he must investigate the \textit{errors of its divisions}, the
errors of the \textit{micrometer screws} in the microscopes, the \textit{flexure} of the
telescope, and the effect of differences of temperature in shifting
the zero points of the circle, by slightly disturbing the position or
direction of the microscopes. Of course this is not the place to
enter into such details, but it is an opportunity to impress again upon
the student the fact that truth and accuracy are only attainable by
immense painstaking and labor.

\begin{fineprint}
\nbarticle{70.} \nbparatext{Mural Circle.}---This instrument is in principle the same as the
meridian circle, which has superseded it. It consists of a circle, carrying a
telescope mounted on the face of a \textit{wall} of masonry (as its name implies)
and free to revolve in the plane of the meridian. The wall furnishes a convenient
support for the microscopes.
\end{fineprint}

\includegraphicsouter{illo025}{\textsc{Fig.~25.}\\The Equatorial (Schematic).}

\nbarticle{71.} \nbparatext{Altitude and Azimuth Instrument.}---Since the transit instrument
and meridian circle are confined to the plane of the meridian,
their usefulness is obviously limited. Meridian observations, when
they are to be had, are better and more easily used than any others,
but are not always attainable. We must therefore have instruments
which will follow an object to any part of the heavens.

\sloppy
The \textit{altitude and azimuth instrument} is simply a surveyor's theodolite
on a large scale. It has a horizontal circle turning upon a \textit{vertical
axis}, and read by verniers or microscopes. Upon this circle, and
turning with it, are supports which carry the \textit{horizontal axis} of the
telescope with its vertical circle, also read by microscopes. Obviously
%% -----File: 059.png---Folio 48-------
the readings of these two circles, when the instrument is properly
adjusted and the zero points determined, will give the altitude
and azimuth of the body pointed on. \figref{illo024}{Fig.~24} represents a small instrument
of this kind.
%% -----File: 060.png---Folio 49-------

\includegraphicsmid{illo026}{\textsc{Fig.~26.}---The 23-inch Princeton Telescope.}

\fussy
\nbarticle{72.} \nbparatext{The Equatorial.}---The essential characteristic of this instrument
is that its principal axis, \textit{i.e.}, the axis which rests in \textit{fixed} bearings,
instead of being either horizontal or vertical, is inclined at an angle
equal to the latitude of the place, and directed towards the pole, thus
placing it parallel to the earth's axis of rotation. This axis of the
instrument is called its \textit{polar axis}; and the graduated circle which it
carries, and which is parallel to the celestial
equator, is called the \textit{hour-circle}, because
its reading gives the \textit{hour-angle} of
the body upon which the telescope happens
to be pointed. Sometimes, also, it is
called the Right Ascension Circle. Upon
this polar-axis are secured the bearings
of the \textit{declination axis}, which is perpendicular
to the polar axis, and carries the
telescope itself and the declination circle.

In the instruments before described, the
telescope is a mere \textit{pointer}, and wholly
subsidiary to the circles; in the equatorial
the telescope is usually the main thing,
and the circles are subordinate, serving
only to aid the observer in finding or
identifying the body upon which the telescope is directed.

Fig.~25 exhibits schematically the ordinary form of equatorial
mounting, of which there are numerous modifications. \figref{illo026}{Fig.~26} is the
23-inch Clark telescope at Princeton, and \figref{illo027}{Fig.~27} is the 4-foot
Melbourne reflector. The frontispiece is the great Lick telescope
of thirty-six inches diameter.

\includegraphicsmid{illo027}{\textsc{Fig.~27.}---The Melbourne Reflector.}

The advantages of the equatorial mounting for a large telescope
are very great as regards convenience. In the first place, when the
telescope is once pointed upon a star or planet, it is only necessary
to turn the polar axis with a uniform motion in order to ``follow'' the
star, which otherwise would be carried out of the field of view in a
few moments by the diurnal motion. This motion, since it is uniform,
can be, and in all large instruments usually is, given by clock-work,
with a continuous regulator of some kind, similar to that used
in the chronograph. The instrument once directed and clamped,
and the clock-work started, the object will continue apparently immovable
in the field of view as long as may be desired.

In the next place, it is very easy to find an object, even if invisible
to the naked eye, like a faint comet or nebula, or a star in the daytime,
%% -----File: 061.png---Folio 50-------
provided we know its declination and right ascension, and
have the sidereal time; for which reason a sidereal clock or chronometer
is an indispensable adjunct of the equatorial.

\begin{fineprint}
To find an object, the telescope is turned in declination until the reading
of the declination circle corresponds to the declination of the object, and
then the polar axis is turned until the hour-circle of the instrument (not to
be confounded with an hour-circle in the sky) reads the \textit{hour-angle} of the
object. This hour-angle, it will be remembered, is simply the difference between
%% -----File: 062.png---Folio 51-------
the sidereal time and the right ascension of the object. The hour-angle
is east if the right ascension exceeds the time; west, if it is less.
When the telescope is thus set, the object will be found (with a low magnifying
power) in the field of view, unless it is near the horizon, in which
case refraction must be taken into account.
\end{fineprint}

While the instrument cannot give very accurate determinations of
the positions of bodies by the direct readings of its circles, on account
of the irregular flexures of its axes, it may do so indirectly; that is,
it may be used to determine very accurately the \textit{difference} between
the right ascension and declination of a comet or planet, for instance,
and that of some neighboring star, whose place has been already
determined by the meridian circle; and this is one of the most important
uses of the instrument.
%% -----File: 063.png---Folio 52-------

\includegraphicsouter[14]{illo028}{\textsc{Fig.~28.}---The Filar Position-Micrometer.}

\nbarticle{73.} \nbparatext{The Micrometer.}---Micro\-meters of various sorts are employed
for the purpose. The most common and most generally useful is the
so-called ``\textit{filar position-micrometer},'' \figref{illo028}{Fig.~28}, which is an indispensable
auxiliary of every good
telescope.

It is a small instrument, much
like the upper part of the reading
microscope, but more complicated.
It usually contains a
reticle of fixed wires, two or
three parallel to each other, and
crossed at right angles by a
second set. Then there are two
or three wires parallel to the first
set, and movable by an accurately
made screw with a graduated
head and a counter, or
scale, for indicating the number
of entire revolutions made by
the screw. The box containing
these wires, and carrying the eye-piece and screw, can itself be
turned around in a plane perpendicular to the optical axis of the
telescope, and set in any desired position; for example, so that the
movable wires shall be parallel to the celestial equator, while the
other set run north and south. This ``position angle'' is read on a
graduated circle, which forms part of the instrument. Means of
illumination are provided, giving at pleasure either dark wires in a
bright field, or \textit{vice versa}.

\includegraphicsmid{illo029}{\textsc{Fig.~29.}---Construction of the Micrometer.}

With this instrument one can measure the distance (in seconds of
arc), and the direction between any two stars which are near enough
to be seen at once in the same field of view. This range in small
%% -----File: 064.png---Folio 53-------
telescopes may reach $30'$ of arc; while in the larger instruments,
which, with the same eye-pieces have much higher magnifying powers,
it is necessarily less,---not more than from $5'$ to $10'$.

\begin{fineprint}
\nbarticle{74.} A new form of equatorial, known as the
\textit{Equatorial Coudé}, or \textit{Elbowed} Equatorial, has been
recently introduced at the Paris Observatory. With large instruments
of the ordinary form a great deal of inconvenience is encountered by
the observer, in moving about to follow the eye-piece into the
various positions into which it is forced by the inconsiderateness
of the heavenly bodies. Moreover, the revolving dome, which is
usually erected to shelter a great telescope, is an exceedingly
cumbrous and expensive affair.

In the Equatorial Coudé, \figref{illo030}{Fig.~30}, these difficulties are overcome by
the use of mirrors. The observer sits always in one fixed position,
looking obliquely down through the polar axis, which is also the
telescope tube.

The Paris instrument has an object-glass about ten inches in
diameter, and performs very satisfactorily. The two reflections,
however, cause a considerable loss of light, and some injury to the
definition. The mirrors, and the consequent complications, also add
heavily to the cost of the instrument. \figref{illo030}{Fig.~30} is from a photograph
of this instrument.
\end{fineprint}

\includegraphicsmid{illo030}{\textsc{Fig.~30.}---The Equatorial Coudé.}
\nbthought

\nbarticle{75.} All the instruments so far described, except the chronometer,
are \textit{fixed} instruments; of use only when they can be set up
firmly and carefully adjusted to established positions. Not one of
them would be of the slightest use on \DPtypo{shipboard}{ship-board}.
%% -----File: 065.png---Folio 54-------

We have now to describe the instrument which, with the help of
the chronometer, is the main dependence of the mariner. It is an
instrument with which the observer measures the angular distance
between two objects; as, for instance, the sun and the visible horizon,
not by pointing first on one and then afterwards on the other, but by
\textit{sighting them both, simultaneously and in apparent coincidence}; which
can be done even when he has no fixed position or stable footing.


\nbarticle{76.} \nbparatext{The Sextant.}---The graduated limb of the sextant is carried
by a light framework, usually of metal, provided with a suitable handle
$X$. The arc is about one-sixth of a circle, as the name implies, and
is usually from five to eight inches radius. It bears a graduation of
half-degrees, \textit{numbered as whole degrees}, so that it can measure any
angle less than $120°$.

An ``\textit{index-arm},'' $MN$ in the \figref{illo031}{figure}, is pivoted at the centre of the
arc, and carries a vernier which slides along the limb, and can be
fixed at any point by a clamp and delicately moved by the attached
tangent screw, $T$. The reading of this vernier gives the angle
measured by the instrument. The best instruments read to $10''$.

Just over the centre of motion, the ``\textit{index-mirror}'' $M$, about
two inches by one and one-half in size, is fastened securely to the
index-arm, so as to be perpendicular to the plane of the limb. At
%% -----File: 066.png---Folio 55-------
$H$, the ``horizon-glass,'' about an inch wide and of the same height
as the index-glass, is secured firmly to the frame of the instrument,
in such position that, when the vernier of the index-arm reads \textit{zero},
the index-mirror and horizon-glass will be parallel to each other.
Only \textit{half} of the horizon-glass is silvered, the upper half being left
transparent. $E$ is a small telescope.

\includegraphicsmid{illo031}{\textsc{Fig.~31.}---The Sextant.}

If the vernier stands \textit{near}, but not \textit{at} zero, the observer looking
into the telescope will see together in the field of view \textit{two} separate
images of the object; and if, while still looking, he slides
the vernier a little, he will see that one of the images remains fixed,
while the other moves. The fixed image is due to the rays which
reach the object-glass of the telescope directly, coming through
the unsilvered half of the horizon-glass; the movable image, on the
other hand, is produced by rays which have suffered two reflections,---first,
from the index-mirror to the horizon-glass; and second, at
the lower half of the horizon-glass. When the two mirrors are
parallel, and the vernier reads zero, the two images coincide, provided
the object is at a considerable distance.

If now the vernier does not stand at or near zero, the observer,
looking at any object directly through the horizon-glass, will see,
not only that object, but also whatever other object is so situated
as to send its rays to the telescope by reflection upon the mirrors;
\textit{and the reading of the vernier will give the angle at the instrument
between the two objects whose images thus coincide}; the angle
between the planes of the two mirrors being just half that between
the objects, and the half-degrees on the limb being numbered as
whole ones.

\nbarticle{77.} The principal use of the instrument is in measuring the altitude
of the sun. At sea the observer, holding the instrument with his right
hand and keeping the plane of the arc vertical, looks \textit{directly} towards
the visible horizon at the point under the sun, through the horizon-glass
(whence its name); then by moving the vernier with his
left hand, he inclines the index-glass upwards until one edge of the
reflected image of the sun is brought just to touch the horizon-line,
noting the exact time by the chronometer, if necessary. The reading
of the vernier, after correcting for the semi-diameter of the sun, the
dip of the horizon, the refraction, and the parallax (and for the
``index-error'' of the sextant, if the vernier does not read strictly
zero when the mirrors are parallel) gives the sun's true altitude at the
moment.
%% -----File: 067.png---Folio 56-------

\begin{fineprint}
\includegraphicsouter{illo032}{\textsc{Fig.~32.}---Principle of the Sextant.}

\nbarticle{78.} On land the \textit{visible} horizon is of no use, and we have recourse to an
``\textit{artificial horizon},'' as it is called. This is merely a shallow basin of mercury,
covered, when necessary to protect it from the wind, with a roof made of
glass plates having their sides \textit{plane} and \textit{parallel}.

In this case we measure the angle between the sun's image reflected in the
mercury and the sun itself. The reading of the instrument, corrected for
index-error, gives \textit{twice the sun's apparent altitude}; which apparent altitude,
corrected as before for refraction and
parallax, but not for dip of the horizon,
gives the true altitude. The skilful use
of the sextant requires steadiness of
hand and considerable dexterity, and
from the small size of the telescope the
angles measured are of course less precise
than if determined by large fixed
instruments. But its portability and
applicability at sea render it absolutely
invaluable.

\nbarticle{79.} The principle that the true angle
between the objects whose images coincide
is twice the angle between the mirrors
(or between their normals) is easily
demonstrated as follows (\figref{illo032}{Fig.~32}):---

The ray $SM$ coming from an object, after reflection first at $M$ (the index-mirror),
and then at $H$ (the horizon-glass), is made to coincide with the
ray $OH$ coming from the horizon. We must prove that the angle $SEO$, between
the object and the horizon, as seen from the point $E$ in the instrument,
is double the angle $Q$, between $MQ$ and $HQ$, which are normals to the mirrors,
and therefore double $Q'$, which is the angle between the planes of the
mirrors.

First, from the law of reflection, we have,
\[
SMP = HMP, \text{ or } SMH = 2 × PMH.
\]
\begin{flalign*}
\text{\indent Similarly, }&& MHE &= 2 × MHQ. &&\phantom{\text{\indent Similarly, }}
\end{flalign*}

From the geometric principle that the exterior angle $SMH$ of the triangle
$HME$ is equal to the sum of the opposite interior angles at $H$ and $E$, we get
\[
HEM = SMH - MHE = 2\,PMH - 2\,MHQ = 2(PMH-MHQ).
\]

Similarly, from the triangle $HMQ$, we have
\[
HQM = PMH - MHQ,
\]
which is half the value just found for $HEM$, and proves the proposition.
\end{fineprint}
%% -----File: 068.png---Folio 57-------

Of course with the sextant, as with all other instruments, it is
necessary for the observer who aims at the utmost precision to investigate,
and take into account its errors of graduation, construction
and adjustment; but their discussion does not belong here.

\nbarticle{80.} Besides the instruments we have described, there are many
others designed for special work, some of which, as the zenith telescope,
and heliometer, will be mentioned hereafter as it becomes
necessary. There is also a whole class of physical instruments,
photometers, spectroscopes, heat-measuring appliances, and photographic
apparatus, which will have to be considered in due time.

\sloppy
But with clock, meridian circle, and equatorial and their usual
accessories, all the fundamental observations of theoretical and spherical
astronomy can be supplied. The chronometer and sextant are
practically the only astronomical instruments of any use at sea.
\chelabel{CHAPTERII}
%% -----File: 069.png---Folio 58-------

\fussy
\Chapter{III}{Corrections to Astronomical Observations}
\nbchapterhang{\stretchyspace CORRECTIONS TO ASTRONOMICAL OBSERVATIONS, DIP OF THE
HORIZON, PARALLAX, SEMI-DIAMETER, REFRACTION, AND
TWILIGHT.}

\nbarticle{81.} \nbparatext{Dip of the Horizon.}---In observations of the altitude of
a heavenly body at sea, where the measurement is made from
the \textit{sea-line}, a correction is needed on account of the fact that
this visible horizon does not coincide with the true astronomical
horizon (which is $90°$ from the zenith), but
falls sensibly below it by an amount known
as the \textit{Dip of the Horizon}. The amount of this dip
depends upon the size of the earth
and the height of the observer's eye above
the sea-level.

\includegraphicsouter{illo033}{\textsc{Fig.~33.}---Dip of the Horizon.}

In \figref{illo033}{Fig.~33}, $C$ is the centre of the earth,
$AB$ a portion of its level surface, and $O$ the
observer, at an elevation $h$ above $A$. The
line $OH$ is truly horizontal, while the tangent
line, $OB$, corresponds to the line drawn
from the eye to the visible horizon. The
angle $HOB$ is the \textit{dip}. This is obviously equal to the angle $OCB$
at the centre of the earth, if we regard the earth as spherical, as we
may do with quite sufficient accuracy for the purpose in hand.

From the right-angled triangle $OBC$ we have directly
\[
\cos OCB = \frac{BC}{CO}.
\]
Putting $R$ for the radius of the earth, and $\Delta$ for the dip, this becomes
\[
\cos \Delta  = \frac{R}{R + h}.
\]

\begin{fineprint}
This formula is exact, but inconvenient, because it gives the small angle
$\Delta$ by means of its cosine. Since, however, $1- \cos \Delta = 2 \sin^2 \frac{1}{2} \Delta$, we easily
obtain the following:---
\[
\sin \tfrac{1}{2} \Delta = \sqrt{\frac{h}{2(R + h)}}.
\]
%% -----File: 070.png---Folio 59-------

This gives the true depression of the sea horizon, as it would be if the
line of sight, drawn from the eye to the horizon line, were \textit{straight}. On
account of refraction it is not straight, however, and the amount of this
``terrestrial refraction'' is very variable and uncertain. It is usual to
diminish the dip computed from the formula by one-eighth its whole amount.
\end{fineprint}

An approximate formula\footnote%
  {\setlength{\abovedisplayskip}{0.5\abovedisplayskip}
   \setlength{\belowdisplayskip}{0.5\belowdisplayskip}
  This approximate formula may be obtained thus:---
  \begin{align*}
    &2 \sin^2 (\tfrac{1}{2} \Delta) = \frac{h}{R+h}
  = \left( \frac{h}{R} \right) + \left( 1 + \frac{h}{R} \right).\\[-2ex]
  \intertext{\indent But since $\dfrac{h}{R}$ is a very small fraction, it may be neglected in the divisor $\left( 1 + \dfrac{h}{R} \right)$,
  and the expression becomes simply,}\\[-6ex]
    &2 \sin^2 \tfrac{1}{2} \Delta = \frac{h}{R}; \text{ whence }
      \sin   \tfrac{1}{2} \Delta = \sqrt{\frac{h}{2R}}.
  \intertext{\indent Since $\Delta$ is a very small angle,}\\[-4ex]
  & \Delta = \sin \Delta = 2 \sin \tfrac{1}{2} \Delta, \quad \text{so that }\\
  & \Delta \text{ (in} \textit{ radians})
  = 2 \sqrt{ \frac{h}{2R} }
  =   \sqrt{ \frac{h}{\frac{1}{2}R} }.\\
  \intertext{\indent To reduce radians to minutes, we must multiply by 3438, the number of minutes
  in a radian. Accordingly, }\\[-6ex]
  & \Delta'\ (\text{in minutes of arc})
  = 3438 \sqrt{ \frac{h}{\frac{1}{2}R} }.\\[-2ex]
  \intertext{\indent If we express $h$ in feet, we must also use the same units for $R$. The mean
  radius of the earth is about 20,884,000 feet, one-half of which is 10,442,000, and
  the square root of this is 3231; so that the formula becomes }\\[-4ex]
  & \Delta' = \frac{3438}{3231} \sqrt{h\text{ (feet)}},
  \end{align*}
  which is near enough to that given in the text.

  In fact, the refraction makes so much difference that after taking the
  numerical factor, $\dfrac{3438}{3231}$, as unity, the formula still gives $\Delta' $ about $\frac{1}{20}$ part too large.

  The formula $\Delta' = \sqrt{ 3 h \ (\textit{metres})}$ is yet more nearly correct.
  } %endfootnote
for the dip is
\[
\Delta \text{ (in minutes of arc) } = \sqrt{h \text{ (feet)}};
\]
or, in words, \textit{the square root of the elevation of the eye $($in feet$)$ gives
the dip in minutes}. This gives a value about $\frac{1}{20}$  part too large.

Since the dip is applicable only to sextant observations made at
sea, where, from the nature of the instrument, and the rising and
falling of the observer with the vessel's motion, it is not possible to
measure altitudes more closely than within about $15''$, there is no
need of any extreme precision in its calculation.
%% -----File: 071.png---Folio 60-------

\nbarticle{82.} \nbparatext{Parallax.}---In the most general sense, ``parallax'' is the change
of a body's direction resulting from the observer's displacement. In
the restricted and technical sense in which we are to employ it now,
it may be defined as the difference \textit{between the direction of a body as
actually observed and the direction it would have if seen from the earth's
centre.} Thus in the figure, \figref{illo034}{Fig.~34}, where the observer is supposed
to be at $O$, the position of $P$ in the sky (as seen from $O$) would be
marked by the point where $OP$ produced would pierce the celestial sphere. Its position as seen from $C$ would be determined in the
same way by producing $CP$ to which $OX$ is drawn parallel. The
angle $POX$, therefore, or its equal, $OPC$, is \textit{the parallax of $P$ for
an observer at $O$}.

\includegraphicsouter{illo034}{\textsc{Fig.~34.}---Diurnal Parallax.}

Obviously, from the \figref{illo034}{figure}, we may also give the following definition
of the parallax. \textit{It is the angular
distance} (number of seconds of
arc) \textit{between the observer's station and
the centre of the earth's disc, as seen
from the body observed}. The moon's
parallax at any moment for me is my
angular distance from the earth's centre,
as seen by ``the man in the moon.''

When a body is in the zenith its
parallax is zero, and it is a maximum
at the horizon. In all cases it
\textit{depresses} a body, diminishing the
altitude \textit{without changing the azimuth.}

The ``law'' of the parallax is, that \textit{it varies as the sine of the zenith distance
directly, and inversely as the linear distance} (in miles) \textit{of the body.}

This follows easily from the triangle $COP$, where we have
$PC : OC =\sin COP : \sin CPO$.

Put $D$ for $PC$, the distance of the body from the earth; $R$ for
the earth's radius, $CO$; $CPO$, the parallax; $\zeta$ for $ZOP$, the apparent
zenith distance, and remember that the sine of $\zeta$ is equal to the
sine of its supplement, $COP$: we then have as the translation of
the above proportion,
\begin{flalign*}
&&D : R &= \sin \zeta : \sin p.&&\\
&\hspace{\parindent}\text{This gives us } &\sin p &= \frac{R}{D} \cdot \sin \zeta;&\phantom{This gives us }\\
\intertext{or, since $p$ is always a small angle,}
&&p'' &= 206265'' \frac{R}{D} \cdot \sin \zeta.&&
\end{flalign*}
%% -----File: 072.png---Folio 61-------

\nbarticle{83.} \nbparatext{Horizontal Parallax.}---When a body is at the horizon ($P_{h}$ in
the \figref{illo034}{figure}), then $\zeta$ becomes $90°$, and $\sin \zeta = 1$. In this case the parallax
reaches its maximum value, which is called \textit{the horizontal parallax}
of the body. Taking  $p_h$  as the symbol for this, we have
\[
\sin p_h = \frac{R}{D};\mbox{ or, nearly enough, }p_{h} = 206265''\,\frac{R}{D}.
\]
Comparing this with the formula above, we see that the parallax of
a body at any zenith distance equals the \textit{horizontal parallax multiplied
by the sine of the zenith distance}; i.e.,  $p = p_{h} \sin \zeta$.

\textbf{N.B.} A glance at the \figref{illo034}{figure} will show that we may define the
\textit{horizontal parallax}, $OPC$, of any body, as \textit{the angular semi-diameter
of the earth seen from that body}. To say, for instance, that the sun's
horizontal parallax is $8''.8$, amounts to saying that, \textit{seen from the sun},
the earth's apparent diameter is twice $8''.8$, or $17''.6$.

\nbarticle{84.} \nbparatext{Relation between Horizontal Parallax and Distance.}\allowbreak---Since we have
\[
\sin p_{h} =  \frac{R}{D},
\]
it follows of course that $D = R ÷ \sin p_h$;
\begin{flalign*}
&\text{or, (nearly) }& D &= \frac{206265''}{p_h''} × R. &&\phantom{\text{or, (nearly) }}
\end{flalign*}
If the sun's parallax equals $8''.8$,
\[
\mbox{its distance} =  \frac{206265}{8.8} × R = 23439\, R.
\]

\sloppy
\nbarticle{85.} \nbparatext{Equatorial Parallax.}---Owing to the ``ellipticity'' or ``oblateness''
of the earth the horizontal parallax of a body varies
slightly at different places, being a maximum at the equator, where
the distance of an observer from the earth's centre is greatest. It
is agreed to take as the standard the \textit{equatorial} horizontal parallax;
\textit{i.e.}, the earth's \textit{equatorial} semi-diameter as seen from the body.

\fussy
\nbarticle{86.} \nbparatext{Diurnal Parallax.}---The parallax we have been discussing is
sometimes called the \textit{diurnal parallax}, because it runs through all its
possible changes in one day.

\begin{fineprint}
When the sun, for instance, is rising, its parallax is a maximum, and by
throwing it down towards the east, increases its apparent right ascension.
At noon, when the sun is on the meridian, its parallax is a minimum, and
%% -----File: 073.png---Folio 62-------
affects only the declination. At sunset it is again a maximum, but
now throws the sun's apparent place down towards the west. Although
the sun is invisible while below the horizon, yet the parallax,
\textit{geometrically considered}, again becomes a minimum at
midnight, regaining its original value at the next sunrise.
\end{fineprint}

The qualifier, ``diurnal,'' is seldom used except when it is
necessary to distinguish between this kind of parallax and the
\textit{annual} parallax of the fixed stars, which is due to the
earth's orbital motion. The stars are so far away that they have no
sensible \textit{diurnal} parallax (the earth is an infinitesimal
point as seen from them); but some of them do have a slight and
measurable \textit{annual} parallax, by means of which we can
roughly determine their distances.\quad (\chapref{CHAPTERXIX}{Chap. XIX.})

\nbarticle{87.} \nbparatext{Smallness of Parallax.}---The horizontal
\textit{parallax} of even the nearest of the heavenly bodies is
always small. In the case of the moon the average value is about
$57'$, varying with her continually changing distance. Excepting now
and then a stray comet, no other heavenly body ever comes within a
distance a hundred times as great as hers. Venus and Mars approach
nearest, but the parallax of neither of them ever reaches $40''$.

\nbarticle{88.} \nbparatext{Semi-Diameter.}---In order to obtain the true
altitude of an object it is necessary, if the edge, or
\textit{``limb,''} as it is called, has been observed, to add or
deduct the apparent semi-diameter of the object. In most cases this
will be sensibly the same in all parts of the sky, but the moon is
so near that there is quite a perceptible difference between her
diameter when in the zenith and in the horizon.

\begin{fineprint}
A glance at \figref{illo034}{Fig.~34} shows that in the zenith the moon's distance is
less than at the horizon, by almost exactly the earth's radius---the
difference
between the lines $OZ$ and $OP_h$. Now this is very nearly one-sixtieth part
of the moon's distance, and consequently the moon, on a night when
its apparent diameter at rising is $30'$, will be $30''$
\textit{larger} when near the zenith. Since the semi-diameter given
in the almanac is what would be seen from the \textit{centre of the
earth}, every measure of the moon's distance from stars or from the
horizon will require us to take into account this ``augmentation of
the semi-diameter,'' as it is technically called.

The formula, easily deduced from the \figref{illo034}{figure} by remembering that the
angle $PCO = \zeta - p$ (zenith distance $-$ parallax), and that the
apparent and ``almanac'' diameters will be inversely proportional to
the two distances $OP$ and $CP$, is
\[
  \text{apparent semi-diameter } = \text{ almanac s.\ d.} × \frac {\sin \zeta}{\sin \left( \zeta - p \right)}.
\]
\end{fineprint}
%% -----File: 074.png---Folio 63-------

This measurable increase of the moon's angular diameter at high
altitudes has nothing to do with the purely subjective illusion which
makes the disc \textit{look} larger to us when \textit{near the horizon}. That it \textit{is} a
mere illusion may be made evident by simply looking through a dark
glass just dense enough to hide the horizon and intervening landscape.
The moon or sun then seems to shrink at once to normal
dimensions.

\includegraphicsouter{illo035}{\textsc{Fig.~35.}---Atmospheric Refraction.}

\sloppy
\nbarticle{89.} \nbparatext{Refraction.}---\\
Rays of light have their direction changed by
refraction in passing through the air, and as the \textit{direction in which we
see a body is that in which its light reaches the eye}, it follows that this
refraction apparently displaces
the stars and all
bodies seen through the
atmosphere. So far as
the action is regular, the
effect is to bend the rays
directly \textit{downwards}, and
thus to make the objects
appear \textit{higher} in the sky.
Refraction \textit{increases the
altitude} of a celestial object
\textit{without altering the
azimuth}. Like parallax,
it is zero at the zenith
and a maximum at the
horizon; but it follows a
different law. It is entirely independent of the distance of the
object, and its amount varies (nearly) as the \textit{tangent} of the zenith
distance---not as \textit{the sine}, as in the case of parallax.

\fussy
\begin{fineprint}
\nbarticle{90.}  This approximate law of the refraction is easily proved.

Suppose in \figref{illo035}{Fig.~35} that the observer at $O$ sees a star in the direction $OS$,
at the zenith distance $ZOS$ or $\zeta$. The light has reached him from $S'$ by a
path which was straight until the ray met the upper surface of the air at $A$,
but afterwards curved continually downwards as it passed from rarer to
denser regions.

We know that the atmosphere is very shallow as compared with the size
of the earth, and it is exceedingly rare in the upper portions, so that, as
far as concerns refraction, we may assume that the point $A$, where the first
perceptible bending of the ray occurs, is not more than fifty miles high,
and that the \textit{vertical $AZ'$ is sensibly parallel} to $OZ$; consequently, also,
%% -----File: 075.png---Folio 64-------
that \textit{all the successive ``strata of equal density'' are parallel to each other and
to the upper surface of the air}.

[This amounts to neglecting the earth's curvature between $O$ and $B$.]

The \textit{true} zenith distance (as it would be if there were no refraction) is
$ZDS'$, which equals $Z'AS'$; and since the retraction, $r$, may be defined as
the difference between the true and apparent zenith distances, this true
zenith distance will $= \zeta + r$.

Now from optical principles, when a ray of light passes through a
medium composed of parallel strata, the final direction of the ray is the
same as if the medium had throughout the density of the last stratum,
and therefore the final direction, $SO$, will be the same as if all the air, from
$A$ down, had the same density as at $O$, with the same index of refraction,
$n$. We may therefore apply the law of refraction directly at $A$, and write
$\sin Z'A S' = n \sin BAC \left( = ZOS \right)$, or $ \sin \left( \zeta +r \right) = n \sin \zeta$;
$AC$ being drawn
parallel to $OS$.

{\allowdisplaybreaks
Developing the first member, we have
\begin{flalign*}
&&&\sin \zeta \cos r + \cos \zeta \sin r = n \sin \zeta.&&&
\intertext{\indent But $r$ is always a small angle, never exceeding $40'$; we may therefore take
$ \cos r = 1$. Doing this and transposing the first term, we get}
&&&\cos \zeta \sin r
= n \sin \zeta - \sin \zeta = \left( n - 1 \right) \sin \zeta.&&&\\[1ex]
&\text{\indent Whence,}&&\sin r = (n- 1) \tan \zeta;&&&  \\[1ex]
&\text{or, }     &&r'' =(n-1) 206265 \tan \zeta \; \text{(nearly)}.&&&
\intertext{\indent The index of refraction for air, \textit{at zero centigrade} and \textit{a barometric
pressure of} $760^\text{mm}$, is $1.000294$; whence,}
&&&r'' = .000294 × 206265 × \tan \zeta = 60''.6 \tan \zeta.&&&
\end{flalign*}
}

This equation holds very nearly indeed down to a zenith distance
of $70°$, but fails as we approach the horizon. For rays coming nearly
horizontal, the points $A$ and $B$ are so far from $O$ that the normal
$AZ'$ is no longer practically parallel to $OZ$; and many of the other
fundamental assumptions on which the formula is based also break
down.

At the horizon, where $\zeta = 90°$ and $\tan \zeta = \textit{infinity}$, the formula
would give $\sin r = \textit{infinity}$ also; an absurdity, since no sine can
exceed unity. The refraction there is really about $37'$, under the
circumstances of temperature and pressure above indicated.
\end{fineprint}
%% -----File: 076.png---Folio 65-------

\nbarticle{91.} \nbparatext{Effect of Temperature and Barometric Pressure.}---The index
of refraction of air depends of course upon its temperature and pressure.
As the air grows \textit{warmer}, its refractive power \textit{decreases}; as it grows
\textit{denser}, the refraction \textit{increases}. Hence, in all precise observations of
the altitude (or zenith distance), it is necessary to note both the
thermometer and the barometer, in order to compute the refraction
with accuracy. For rough work, like ordinary sextant observations,
it will answer to use the ``mean refraction,'' corresponding to an
average state of things.

\begin{fineprint}
\textbf{Tables of Refraction.}---The computation of the refraction is best
effected by special tables made for the purpose; of these, Bessel's tables are
the most convenient, best known, and probably even yet the most accurate.
It must be always borne in mind, however, that from the action of wind and
other causes the condition of the air along the path of the ray is seldom perfectly
normal; in consequence, the actual refraction in any given case is liable
to differ from the computed by as much as one or even two per cent.
No amount of care in observation can evade this difficulty; the only remedy
is a sufficient repetition of observations under varying atmospheric conditions.
Observations at an altitude below $10°$ or $15°$ are never much to be
trusted.

\textbf{Lateral Refraction.}---When the air is much disturbed, sometimes objects
are displaced horizontally as well as vertically. Indeed, as a general
rule, when one looks at a star with a large telescope and high power, it will
seem to ``dance'' more or less---the effect of the varying refraction which
continually displaces the image.
\end{fineprint}

\nbarticle{92.} \nbparatext{Effect on the Time of Sunrise and Sunset.}---The horizontal refraction,
ranging as it does from $34'$ to $39'$, according to temperature,
is always somewhat greater than the diameter of either the sun or
the moon. At the moment, therefore, when the sun's lower limb
appears to be just rising, the whole disc is really below the plane
of the horizon; and the \textit{time} of sunrise in our latitudes is thus
accelerated from two to four minutes, according to the inclination of
the sun's diurnal circle to the horizon, which inclination varies with
the time of the year. Of course, sunset is delayed by the same
amount, and thus the day is lengthened by refraction from four
to eight minutes, at the expense of the night.

\nbarticle{93.} \nbparatext{Effect on the Form and Size of the Discs of the Sun and Moon.}---Near
the horizon the refraction changes very rapidly. While under
ordinary summer temperature it is about $35'$ \textit{at} the horizon, it is
%% -----File: 077.png---Folio 66-------
only $29'$ at an elevation of half a degree; so that, as the sun or moon
rises, the bottom of the disc is lifted $6'$ more than the top, and the
vertical diameter is thus made apparently about one-fifth part shorter
than the horizontal. This distorts the disc into the form of an oval,
flattened on the under side. In cold weather the effect is much more
marked. As the horizontal diameter is not at all increased by the
refraction, the apparent \textit{area} of the disc is notably diminished by it;
so that it is evident that refraction cannot be held in any way responsible
for the apparent enlargement of the rising luminary.

\nbarticle{94.} \nbparatext{Determination of the Refraction.}---1.~\textit{Physical Method.}
Theory furnishes the \textit{law} of astronomical refraction, though the
mathematical expression becomes rather complicated when we attempt
to make it exact. In order, therefore, to determine the astronomical
refraction under all possible circumstances, it is only necessary to
determine the index of refraction of air, and its variations with temperature
and pressure, by laboratory experiments, and to introduce the
constants thus obtained into the formulę. It is difficult, however, to
make these determinations with the necessary precision. In fact, at
present our knowledge of the constants of air rests mainly on astronomical
work.

2.~\textit{By Observations of Circumpolar Stars.} At an observatory whose
latitude exceeds $45°$ select some star which passes \textit{through the zenith}
at the upper culmination. (Its declination must equal the latitude of
the observatory.) It will not be affected by refraction at the zenith,
while at the lower culmination, twelve hours later, it will. With the
meridian circle observe its \textit{polar distance} in both positions, determining
the ``polar point'' of the circle as described on pp.~46--47. If the
polar point were not itself affected by refraction, the simple difference
between the two results for the star's polar distance, obtained
from the upper and lower observations, would be the refraction at
the lower point.

As a \textit{first approximation}, however, we may neglect the refraction
at the pole, and thus obtain a \textit{first} approximate lower refraction.
By means of this we may compute an \textit{approximate} polar refraction,
and so get a first ``corrected polar point.'' With this compute a
\textit{second} approximate lower refraction, which will be much more nearly
right than the first; this will give a \textit{second}  ``corrected polar point'';
this will in turn give us a \textit{third} approximation to the refraction; and
so on. But it would never be necessary to go beyond the third, as
the approximation is very rapid. If the star does not go exactly
%% -----File: 078.png---Folio 67-------
through the zenith, it is only necessary to compute each time approximate
refractions for its upper observation, as well as for the polar
point.

At present, however, the refraction is so well known that the
method actually used is to form ``equations of condition'' from the
observations of the altitude of known stars under varying circumstances,
and from these to deduce such corrections to the star places
and refraction constants as will best harmonize the whole mass of
material.

\begin{fineprint}
\nbarticle{95.} 3.~\textit{By Observations of the Altitudes of Equatorial Stars made at an Observatory
near the Equator}. For an observer so situated, stars that are on the
celestial equator $(\delta = 0)$ will come to the meridian at the zenith, and will rise
and fall \textit{vertically, with a motion strictly proportional to the time}; the \textit{true} zenith
distance of the star at any moment being just equal to its \textit{hour-angle} in
degrees. We have only, then, to observe the \textit{apparent} zenith distance of a
star with the corresponding time, and the refraction comes out directly.

If the station is not exactly on the equator, and if the star's declination is
not exactly zero, it is only necessary to know the latitude and declination
\textit{approximately} in order to get the refraction very accurately; a considerable
error in either latitude or declination will affect the result but slightly.

4.~The French astronomer Loewy has recently proposed a method which
promises well. He puts a pair of reflectors, inclined to each other at a convenient
angle of from $45°$ to $50°$ (a glass wedge with silvered sides), in front
of the object-glass of an equatorial. This will bring to the eye two rays
which make a strictly constant angle with each other, and there is no difficulty
in finding pairs of stars so situated that their images will come into
the field of view together. Now, were it not for refraction, these images
would always keep their relative position unchanged, notwithstanding the
diurnal motion; but on account of the changes in the refraction, as one star
rises and the other falls, they will shift in the field, and micrometric measures
will determine the shifting, and so the refraction, with great precision.
\end{fineprint}

\nbarticle{96.} \nbparatext{Twilight.}---{\footnotesize(Although this subject is outside the main purpose of this
chapter, which deals with corrections to be applied to astronomical observations, we
treat it here because, like refraction, it is a purely atmospheric phenomenon, and
finds no other more convenient place.)}

\smallskip
Twilight, the illumination of the sky which begins before sunrise,
and continues after sunset, is caused by the reflection of light to the
observer from the upper regions of the earth's atmosphere. It is not
yet certain whether this is due to reflection from foreign matter in the
air, such as minute crystals of ice and salt, particles of dust of
various kinds, and infinitesimal drops of water, or whether the pure
gases themselves have some power of reflecting light. There is no
%% -----File: 079.png---Folio 68-------
doubt, however, that air, under the ordinary conditions, possesses
considerable power of reflection; so that, as long as any air upon
which the sun is shining is visible to the observer, it will send him
more or less light, and appear illuminated.

\includegraphicsmid{illo036}{\textsc{Fig.~36.}---Twilight.}

Suppose the atmosphere to have the depth indicated in the \figref{illo036}{figure}.
Then, if the sun is at $S$, \figref{illo036}{Fig.~36}, it will just have set to an observer at $1$,
but all the air within his range of vision will still be illuminated. When,
by the earth's rotation, he has been transported to $2$, he will see the
``twilight bow'' rising in the east, a faintly reddish arc separating
the illuminated part of the sky from the darkened part below, which
lies in the shadow of the earth. When he reaches $3$, the western
half of the sky alone remains bright, but the arc of separation between
the light and darkness has become vague and indefinite; when
he reaches $4$, only a glow remains in the west; and when he comes
to $5$, night closes in on him. Nothing remains in sight on which
the sun is shining.

\begin{fineprint}
\nbarticle{97.} \textit{Duration of Twilight.}---This depends upon the height of the atmosphere,
and the angle at which the sun's diurnal circle cuts the horizon. It is
found as a matter of observation, not admitting, however, of much precision,
that twilight lasts until the sun has sunk about $18°$ below the horizon; that
is to say, the angle $1\,C\,5$ in the \figref{illo036}{figure} is about $18°$.

The time required to reach this point in latitude $40°$ varies from two
hours at the longest days in summer, to one hour thirty minutes about Oct.\
12 and March~1, when it is least. At the winter solstice it is about one
hour and thirty-five minutes.

In higher latitudes the twilight lasts longer, and the variation is more
considerable; the date of the minimum also shifts.

Near the equator the duration is shorter, hardly exceeding an hour at the
%% -----File: 080.png---Folio 69-------
sea-level; while at high elevations (where the amount of air above the
observer's level is less) it becomes very brief. At Quito and Lima it is
said not to last more than twenty minutes. Probably, also, in mountain
regions the clearness of the air, and its purity contribute to the effect.
\end{fineprint}

\nbarticle{98.} \nbparatext{Height of the Atmosphere.}---It is evident from the \figref{illo036}{figure} that
at the moment twilight ceases, the last visible portion of illuminated
air is at the top of the atmosphere, and just half-way between the observer
and the nearest point where the sun is setting. If the whole arc
$1$,~$5$ is $18°$, $1$,~$3$ is $9°$: then calling the height of the atmosphere $H$ and
the earth's radius $R$, and neglecting refraction (\textit{i.e.}, supposing the lines
$1\,m$ and $5\,m$ to be straight), we have from the right-angled triangle
$1\,C\,m$, $C\,m = 1\,C × \sec 9°$, or $R+H = R × \sec 9°$; whence $H = R\,
(\sec 9° - 1) = 0.0125\, R$, or almost exactly fifty miles. This must
be diminished about one-fifth part on account of the curvature of
the lines $1\,m$ and $5\,m$ by refraction, making the height of the atmosphere
about forty miles.

The result must not, however, be accepted too confidently. It only
proves that we get no sensible \textit{twilight illumination} from air at a
greater height: above that elevation the air is either too rare, or too
\textit{pure} from foreign particles, to send us any perceptible reflection.
There is abundant evidence from the phenomena of meteors that the
atmosphere extends to a height of 100~miles at least, and it cannot
be asserted positively that it has \textit{any} definite upper limit.

\begin{fineprint}
\nbarticle{99.} \nbparatext{Aberration.}---There is yet one more correction which has to be
applied in order to get the true direction of the line which at the instant of
observation joins the eye of the observer to the star he is pointing at. The
\textit{aberration of light} is an apparent displacement of the object observed, due
to the combination of the earth's orbital motion with the progressive motion
of light. It can be better discussed, however, in a different connection (see
\chapref{CHAPTERVI}{Chap.~VI.}), and we content ourselves with merely mentioning it here.
\end{fineprint}
\chelabel{CHAPTERIII}
%% -----File: 081.png---Folio 70-------

\Chapter{IV}{Problems of Practical Astronomy}
\nbchapterhang{PROBLEMS OF PRACTICAL ASTRONOMY, LATITUDE, TIME,
LONGITUDE, AZIMUTH, AND THE RIGHT ASCENSION AND
DECLINATION OF A HEAVENLY BODY.}

\sloppy
\nbarticle{100.} \textsc{There} are certain problems of Practical Astronomy which
have to be solved in obtaining the fundamental facts from which we
deduce our knowledge of the earth's form and dimensions, and other
astronomical data.

\fussy
The first of these problems is that of the

\section*{LATITUDE.}

The latitude (\textit{astronomical}) of a place (\artref{Art.}{30}) is simply \textit{the altitude
of the celestial pole $($Polhöhe$)$}, or, what comes to the same thing, as is
evident from \figref{illo007}{Fig.~7} (\artref{Art.}{33}), it is the \textit{declination of the zenith}. It may
also be defined, from the mechanical point of view, as \textit{the angle between
the plane of the earth's equator and the observer's plumb-line or vertical}.

\begin{fineprint}
Neither of these definitions assumes anything as to the form of the earth,
and we shall find farther on that this \textit{astronomical} latitude is seldom identical
with the \textit{geocentric}, nor even with the \textit{geodetic} latitude of a place. It is,
however, the only kind of latitude which can be \textit{directly} determined from
astronomical observations, and its determination is one of the most important
operations of what may be called Economic Astronomy.
\end{fineprint}

\nbarticle{101.} \nbparatext{Determination of Latitude.}---\nbparasubtext{First:}{By Circumpolars.} The
most obvious method of determining the latitude is to observe, with the
meridian circle or some analogous instrument, the altitude of a circumpolar
star at its upper culmination, and again, twelve hours later, at
its lower. Each of the observations must be corrected for \textit{refraction},
and then the \textit{mean of the two corrected altitudes will be the latitude}.

\begin{fineprint}
This method has the advantage of being an \textit{independent} one; \textit{i.e.}, it does
not require any data (such as the declination of the stars used) to be accepted
on the authority of previous observers. But to obtain much accuracy
it requires considerable time and a large fixed instrument. In low latitudes
the refraction is also very troublesome.
\end{fineprint}
%% -----File: 082.png---Folio 71-------

\nbarticle{102.} \nbparasubtext{Second:}{By the meridian altitude or zenith distance of a
body of known declination.}

In \figref{illo037}{Fig.~37} the \DPtypo{semicircle}{semi-circle} $AQPB$ is the meridian, $Q$ and $P$ being
respectively the equator and the pole, and $Z$ the zenith. $QZ$ is
the \textit{declination of the zenith}, or the observer's latitude ($=PB=\phi$).
Suppose now that we observe $Zs$ ($=\zeta_s$), the zenith distance of a
star $s$ (south of the zenith), as it crosses the meridian, and that its
declination $Qs$ ($=\delta_s$) is known; then evidently $\phi = \delta_s + \zeta_s$.

In the same way, if the star were at $n$, between zenith and pole,
$\phi = \delta_n - \zeta_n$.

\begin{fineprint}
If we use the meridian circle, we can always select stars that pass near
the zenith where the refraction will be small; moreover, we can select them
in such a way that some will be as much north of the zenith as others are
south, and thus \textit{eliminate} the refraction errors. But we have to get our star
declinations out of catalogues made by previous observers, and so the method
is not an \textit{independent} one.
\end{fineprint}

\includegraphicsouter{illo037}{\textsc{Fig.~37.}---Determination of Latitude.}

\nbarticle{103.} \nbparatext{At Sea} the latitude is usually obtained by \textit{observing with the sextant
the sun's maximum altitude}, which
of course occurs at noon. Since at sea
it is seldom that one knows beforehand
precisely the moment of local noon,
the observer takes care to begin to observe
the sun's altitude some ten or
fifteen minutes earlier, repeating his
observations every minute or two. At
first the altitude will keep increasing, but immediately after noon it
will begin to decrease. The observer uses therefore the \textit{maximum}\footnote
  {On account of the sun's motion in declination, and the northward or southward
  motion of the ship itself, the sun's maximum altitude is usually attained
  not \textit{precisely} on the meridian, but a few seconds earlier or later. This requires a
  slight correction to the deduced latitude, explained in books on Navigation or
  Practical Astronomy.}
altitude obtained, which, corrected for refraction, parallax, semi-diameter,
and dip of the horizon, will give him the true latitude of
his ship, by the formula $\phi = \delta \pm \zeta$.

\begin{fineprint}
\sloppy
\nbarticle{104.} \nbparasubtext{Third:}{By Circum-meridian Altitudes.}---If the observer knows
his time with reasonable accuracy, he can obtain his latitude from observations
made when the body is \textit{near} the meridian, with practically the same
precision as at the moment of meridian passage. It would take us a little
%% -----File: 083.png---Folio 72-------
too far to explain the method of reduction, which is given with the necessary
tables in all works on Practical Astronomy. The great advantage of this
method is that the observer is not restricted to a single observation at each
meridian-passage of the sun or of the selected star, but can utilize the half-hours
preceding and following that moment. The meridian-circle cannot
be used, as the instrument must be such as to make extra-meridian
observations possible. Usually the sextant or universal instrument is
employed. This method is much used in the French and German geodetic
surveys.
\end{fineprint}

\nbarticle{105.} \nbparasubtext{Fourth:}{The Zenith Telescope Method.}---{\footnotesize(Sometimes known as
the American method, because first practically introduced by Captain Talcott of the
United States Engineers, in a boundary survey in 1845.)}

\includegraphicsouter{illo038}{\textsc{Fig.~38.}---Principle of the Zenith Telescope.}

The essential characteristic of the method is the \textit{micrometric} measurement
of the \textit{difference} between the
nearly equal zenith distances of two stars
which culminate within a few minutes
of each other, one north and the other
south of the zenith, and not very far
from it: such pairs of stars can always
be found. When the method was first
introduced, a special instrument, known
as the zenith telescope, was generally
employed, but at present a simple transit
instrument, with declination micrometer,
and a delicate level attached to the telescope
tube, is ordinarily used.

The telescope is set at the proper altitude
for the star which first comes to the
meridian, and the ``latitude level,'' as it is
called, is set horizontal; as the star passes
through the field of view its distance
north or south of the central wire is measured by the micrometer.
The instrument is then reversed, and so set by turning the telescope
up or down (\textit{without, however, disturbing the angle} $\theta$ (\figref{illo038}{Fig.~38}) \textit{between
the level and telescope}), that the level is again horizontal. After this
reversal and adjustment, the telescope tube is then evidently elevated
at exactly the same angle, $\zeta$, as before, but on the opposite side of the
zenith. As the second star passes through the field, we measure
with the micrometer \textit{its} distance north or south of the centre of the
field; the comparison of the two micrometer measures gives the
\textit{difference} of the two zenith distances.
%% -----File: 084.png---Folio 73-------

From \figref{illo037}{Fig.~37} we have
\begin{align*}
\text{for star \textit{south} of zenith, } &\phi = \delta_s + \zeta_s; \\
\text{for star \textit{north} of zenith, } &\phi = \delta_n - \zeta_n.
\end{align*}

Adding the two equations and dividing by $2$, we have
\[
  \phi = \left( \frac{\delta_s + \delta_n}{2} \right)
       + \left( \frac{\zeta_s  -  \zeta_n}{2} \right).
\]

The star catalogue gives us the declinations of the two stars
$( \delta_s + \delta_n )$; and the difference of the zenith distances $( \zeta_s - \zeta_n
)$ is determined
with great accuracy by the micrometer measures.

\includegraphicsouter{illo039}{\textsc{Fig.~39.}---Latitude by Prime Vertical Transits.}

\begin{fineprint}
The great advantage of the method consists in its dispensing with a
\textit{graduated circle}, and in avoiding almost wholly the errors due to \textit{refraction};
it virtually utilizes the circles of the fixed observatories by which
the star declinations have been measured, without requiring them to be
brought into the field. Forty years ago it was not always easy to find
accurate determinations of the declinations
of the stars employed, but at present the star
catalogues have been so extended and improved
that this difficulty has practically disappeared,
so that this method of determining the
latitude is now not only the most convenient
and rapid, but is quite as precise as any, if the
level is sufficiently sensitive. Evidently the
limit of accuracy depends upon the exactness
with which the level measures the slight, but
inevitable, difference between the inclinations
of the instrument when pointed on the two stars.

\nbarticle{106.} \nbparasubtext{Fifth:}{By the Prime Vertical Instrument} {\normalsize(p.~\pageref{fig:illo020}).}---We observe
simply the moment when a known star passes the prime vertical on the
eastern side, and again upon the western side. Half the interval will give
the \textit{hour-angle} of the star when on the prime vertical; \textit{i.e.,} the angle $ZPS$
in \figref{illo039}{Fig.~39}, where $Z$ is the zenith, $P$ the pole, and $SZS'$ the prime vertical.
The distance $PS$ of the star from the pole is the complement of the star's
declination; and $PZ$ is the complement of the observer's latitude. Since
the prime vertical is perpendicular to the meridian at the zenith, the triangle
$PZS$ will be right-angled at $Z$, and from Napier's rule of circular
parts (taking $ZPS$ as the middle part) we shall have
\begin{flalign*}
&&\cos ZPS  &= \tan PZ \cot PS,&&\\
&\text{or}  &\cos t  &= \cot \phi \tan \delta;&&\\
&\text{whence}  &\tan \phi  &= \tan \delta \sec t.&&\phantom{whence}
\end{flalign*}
%% -----File: 085.png---Folio 74-------

If $\delta$ nearly equals $\phi$, $t$  will be small, and a considerable error in the observation
of $t$ will then produce very little change in its secant or in the computed
latitude.

The observations are not so convenient and easy as in the case of the
zenith telescope, and the number of stars available is less; but the method
presents the great advantage of requiring nothing but an ordinary transit
instrument, without any special outfit of micrometer and latitude level.
It also entirely evades the difficulties caused by refraction.
\end{fineprint}

\includegraphicsouter{illo040}{\textsc{Fig.~40.}---Latitude by the Gnomon.}

\sloppy
\nbarticle{107.} \nbparasubtext{Sixth:}{By the Gnomon.}---The ancients had no instruments
such as we have hitherto described, and of course could not use any
of the preceding methods of finding the latitude. They were, however,
able to make a very respectable approximation by means of the
simplest of all astronomical instruments, \textit{the gnomon}. This is merely
a vertical shaft or column of known height erected on a perfectly
horizontal plane; and the
observation consists in noting
the length of the shadow
cast at noon at certain times
of the year.

\fussy
Suppose, for instance, that
on the \textit{day of the summer
solstice}, at noon, the length
of the shadow is $AC$, \figref{illo040}{Fig.~40}.
The height $AB$ being given,
we can easily compute in
the right-angled triangle the
angle $ABC$, which equals
$SBZ$, the sun's zenith distance
when farthest north. Again observe the length $AD$ of the
shadow at \textit{noon of the shortest day in winter}, and compute the angle
$ABD$, which is the sun's corresponding zenith distance when farthest
south. Now, since the sun travels equal distances north and south
of the celestial equator, the mean of the two results will give the
angular distance between the equator and the zenith; \textit{i.e.}, the \textit{declination
of the zenith}, which (\artref{Art.}{100}) is the latitude of the place.

\begin{fineprint}
The method is an independent one, like that by the observation of circumpolar
stars, requiring no data except those which the observer determines
for himself. Evidently, however, it does not admit of much accuracy, since
the penumbra at the end of the shadow makes it impossible to measure its
length very precisely.

It should be noted that the ancients, instead of designating the position
%% -----File: 086.png---Folio 75-------
of a place by means of its latitude, used its \textit{climate} instead; the climate
(from $ \kappa \lambda \acute \iota \mu \alpha $) being the \textit{slope} of the plane of
the celestial equator, the angle
$AEB$, which is the complement of the latitude.

It is supposed, indeed known, that many of the Egyptian obelisks were
erected primarily to serve as gnomons, and were used for that purpose.

\nbarticle{108.}  \nbparatext{Possible Variations of the Latitude.}---It is an interesting question
whether the position of the earth's axis is \textit{fixed} with reference to its
mass and surface. Theoretically it is hardly possible that it should be,
because any change in the arrangement of the matter of the earth, by
denudation, subsidence, or elevation, would almost necessarily disturb it.
If so disturbed, the latitudes of places toward which the pole approached
would be increased, and those on the opposite side would be decreased. At
present we can only say that if such disturbance has occurred, it must have
been extremely slight for the last 200 years, not exceeding 40 or 50 feet at
most; but there are suspicions of a minute and progressive change of the
latitude of some of the observatories (notably Pulkowa), which have drawn
attention to the matter, and the subject is under investigation.
\end{fineprint}

\section*{TIME AND ITS DETERMINATION.}

\nbarticle{109.}  One of the most important problems presented to the astronomer
is the determination of \textit{Time}. By universal consent the apparent
rotation of the heavens is made to furnish the standard, and the
determination of time is effected by ascertaining by observation the
\textit{hour-angle of the object selected to mark the beginning of the day by
its transit across the meridian}. In practice, three kinds of time are
now recognized, viz., \textit{sidereal time, apparent solar time}, and \textit{mean
solar time}.

\nbarticle{110.} \textit{Sidereal Time}.---As has already been explained (\artref{Art.}{26}), the
sidereal time at any moment is \textit{the hour-angle of the vernal equinox at
that moment;} or, what comes to the same thing, though it sounds differently,
it is \textit{the time marked by a clock which is so set and adjusted
as to show noon, or $0^\mathrm{h}\: 00^\mathrm{m} \:00^\mathrm{s}$, at each transit of the vernal equinox.}
The sidereal \textit{day}, thus defined, is the time intervening between two
successive transits of the same \textit{star;} at least, it is so within the
hundredth part of a second, though on account of the precession of
the equinoxes (and the proper motions of the stars) the agreement is
not absolute, the difference amounting to about one day in twenty-six
thousand years.

\nbarticle{111.} \textit{Apparent Solar Time}.---Just as sidereal time is the hour-angle
of the vernal equinox, so at any moment \textit{the apparent solar time is
the hour-angle of the sun}. It is the time \textit{shown by the sun-dial,} and
%% -----File: 087.png---Folio 76-------
its noon is when the sun crosses the meridian. On account of the
annual eastward motion of the sun among the stars (due to the
earth's orbital motion), this day is about four minutes longer than
the sidereal; and moreover, because the sun's motion in right ascension
is not uniform, the apparent solar days are not all of the same
length, nor, consequently, its hours, minutes, or seconds. December
23d is fifty-one seconds longer from (apparent) noon to noon than
Sept.\ 16th. For this reason, apparent solar time is not satisfactory
for scientific use, and has long been discarded in favor of mean
solar time. Until within about a hundred years, however, it was the
only kind of time commonly employed, and its use in the city of Paris
was not discontinued until the year 1816.

{\nbarticle{112.} \textit{Mean Solar Time}.---A ``\textit{fictitious sun}'' is therefore imagined,
which moves \textit{uniformly and in the celestial equator}, completing its
annual course in exactly the same time as that in which the actual sun
makes the circuit of the ecliptic. It is mean noon when this ``fictitious
sun'' crosses the meridian, and at any moment \textit{the hour-angle
of this ``fictitious sun'' is the mean time} for that moment.}

\begin{fineprint}
Sidereal time will not answer for business purposes, because its noon (the
transit of the vernal equinox) occurs at all hours of night and daylight in
different seasons of the year. Apparent solar time is scientifically unsatisfactory,
because of the variation in the length of its days and hours. And
yet we have to live by the sun; its rising and setting, daylight and night,
control our actions. In mean solar time we find a satisfactory compromise,
an invariable time unit, and still an agreement with sun-dial time close enough
for convenience. It is the time now used for all purposes except in certain
astronomical work. The difference between apparent time and mean time,
never amounting to more than about a quarter of an hour, is called the equation
of time, and will be discussed hereafter in connection with the earth's
orbital motion, \chapref{CHAPTERVI}{Chap.~VI.}
\end{fineprint}

The nautical almanac furnishes data by means of which the sidereal
time may be deduced from the corresponding solar, or \textit{vice versa}, by
a very brief and simple calculation.

\nbarticle{113.} In practice the problem of determining the time always takes
the form of \textit{ascertaining the error of a time-piece;} that is, the amount
by which a clock or watch is fast or slow of the time it ought to show.
The methods most in use by astronomers are the following:---

First. \textit{By means of the transit instrument}. Since the right ascension
of a star is the sidereal time of its passage across the meridian
%% -----File: 088.png---Folio 77-------
(\artref{Art.}{26}), it is obvious that the difference between the right ascension
of a known star and the time shown by a sidereal clock at the instant
when the star crosses the middle wire of an accurately adjusted
transit instrument, is the error of the clock at that moment. Practically,
it is usual to observe a number of stars (from eight to ten),
reversing the instrument once at least, so as to eliminate the collimation
error (\artref{Art.}{60}). With a good instrument a skilled observer can
determine this clock error or ``correction'' within about one-thirtieth
of a second of time, provided proper means are taken to ascertain
and allow for his ``personal equation.''

\nbarticle{114.} \textit{Personal Equation}.---It is found that every observer has his
own peculiarities of time observation with a transit, and his \textit{``personal
equation''} is the amount to be added (algebraically) to the time
observed by him, in order to get the actual moment of transit as it
would be recorded by some supposable arrangement, which should
automatically register the moment when the star's image was bisected
by the wire.

\begin{fineprint}
This personal equation differs for different observers, but is reasonably
(though never strictly) constant for one who has had much practice. In the
case of observations with the chronograph, it is usually less than $ \pm 0^{\text{s}}.2$. It
can be determined by an apparatus in which an artificial star, resembling
the real stars as much as possible in appearance, is made to traverse the field
of view and to telegraph its arrival at certain wires, while the observer notes
the moments for himself.

One of the most important problems of practical astronomy now awaiting
solution is the contrivance of some practical method of time observation
free from this annoying human element, the personal equation, which is
always more or less uncertain and variable.
\end{fineprint}

If mean time is wanted, it can be deduced from the sidereal time
by the data of the almanac.

The sun can also be observed instead of the stars, the moment of
the sun's transit being that of apparent noon; but this observation,
for many reasons, is far less accurate and satisfactory than observations
of the stars.

\nbarticle{115.}  Second. \textit{The method of equal altitudes}.---If we observe with
a sextant in the forenoon the time shown by the chronometer when the
sun attains the height indicated by a certain reading of the sextant, and
then in the afternoon, the time when the sun again reaches the same
%% -----File: 089.png---Folio 78-------
altitude, the moment of apparent noon will be half-way between the
two observed times; provided, of course, that the chronometer runs
uniformly during the interval, and also provided that proper correction
is made for the sun's slight motion in declination---a correction
easily computed.

The advantage of this method is that the errors of graduation of
the sextant have no effect, nor is it necessary for the observer to
know his latitude except approximately.

\includegraphicsouter{illo041}{\textsc{Fig.~41.}---Determination of Time by a Single Altitude.}

\textit{Per contra}, there is, of course, danger that the afternoon observations
may be interfered with by clouds; and, moreover, both observations
must be made at the same place.

A modification of this method is now coming into extensive use,
in which two different stars of known right ascension and of nearly
the same declination are used, at equal altitudes east and west of the
meridian.

\nbarticle{116.} \nblabel{pg:78}Third. \textit{By a single altitude of the sun, the observer's latitude
being known}.---This is the method usual at sea. The altitude of the sun
having been measured with the sextant, and the corresponding time
shown by the chronometer having been accurately noted, we compute
the hour-angle of the sun, $P$,
from the triangle $ZPS$ (Fig.\
41), and this hour-angle corrected
for the equation of
time, gives the true mean
time at the observed moment.
The difference between this
and that shown by the chronometer
is \textit{the error of the
chronometer.} In the triangle
$ZPS$ all three of the sides are given, viz.: $PZ$ is the complement
of the latitude $\phi$, which is supposed to be known; $PS$ is the complement
of the declination $\delta$, which is found in the almanac, as is
also the equation of time; while $ZS$ or $\zeta$, is the complement of the
sun's altitude, as measured by the sextant, and corrected for semi-diameter,
refraction, and parallax. The formula is
\[
\sin \tfrac{1}{2}P = \left( \cfrac{\sin \tfrac{1}{2} \left[ \zeta + \left( \phi - \delta
\right) \right] \sin \tfrac {1}{2} \left[ \zeta - \left( \phi - \delta \right) \right] }{
\cos \phi \cos \delta} \right) ^{\frac {1}{2}}.
\]

In order to accuracy, it is desirable that the sun should be on the
prime vertical, or as near it as practicable. It should \textit{not} be near the
%% -----File: 090.png---Folio 79-------
meridian. Any slight error in the assumed latitude produces no
sensible effect upon the result, if the sun is exactly east or west at
the time the observation is taken. The disadvantage of the method
is that any error of graduation of the sextant enters into the result
with its full effect.

\begin{fineprint}
In some cases a person is so situated that it is necessary to determine his
time roughly, without instruments; and this can be done within about a
half a minute by establishing a noon-mark, which is nothing but a line
drawn exactly north and south, with a plumb-line, or some vertical edge, like
the edge of a door-frame or window-sash, at its southern extremity. The
shadow will then always fall upon the meridian line at \textit{apparent} noon.
\end{fineprint}

\nbarticle{117.} \nbparatext{The Civil and the Astronomical Day.}---The \textit{astronomical} day
begins at mean noon. The \textit{civil} day begins at midnight, twelve hours
earlier. Astronomical mean time is reckoned round through the whole
twenty-four hours, instead of being counted in two series of twelve
hours each. Thus, 10~\textsc{a.m.} of Wednesday, May~2, \textit{civil reckoning}, is
Tuesday, May~1, $22^\text{h}$ by \textit{astronomical reckoning}. Beginners need to
bear this in mind in using the almanac.

\section*{LONGITUDE.}

\nbarticle{118.}  Having now methods of obtaining the true local time, we can
attack the problem of longitude, which is perhaps the most important
of all the economic problems of astronomy. The great observatories
at Greenwich and at Paris were established simply for the purpose
of furnishing the observations which could be made the basis of the
accurate determination of longitude at sea.

The longitude of a place on the earth is \textit{the angle at the pole between
the meridian of Greenwich and the meridian passing through the observer's
place;} or it is the arc of the equator intercepted between
these meridians; or, what comes to the same thing, since this arc is
measured by the time required for the earth to turn sufficiently to
bring the second meridian into the same position held by the first, it is
simply \textit{the difference of their local times},---the amount by which the noon
at Greenwich is earlier or later than at the observer's place. It is now
usually reckoned in hours, minutes, and seconds, instead of degrees.

Since it is easy for the observer to find his own local time by the
methods which have been given, the knot of the problem is really
this: \textit{being at any place, to find the corresponding local time at Greenwich
without going there}.
%% -----File: 091.png---Folio 80-------

The methods of finding the longitude may be classed under three
different heads:

\textbf{First}, By means of signals simultaneously observable at the places
between which the difference of longitude is to be found.

\textbf{Second}, By making use of the moon as a clock-hand in the sky.

\textbf{Third}, By purely mechanical means, such as chronometers and the
telegraph.

\nbarticle{119.} Under the first head we may make use of

[A] \textit{A Lunar Eclipse}.---When the moon enters the shadow of the
earth, the phenomenon is seen at the same moment, no matter where
the observer may be. By noting, therefore, his own local time at the
moment, and afterwards comparing it with the time at which the phenomenon
was observed at Greenwich, he will obtain his longitude
from Greenwich. Unfortunately, the edge of the earth's shadow is
so indistinct that the progress of events is very gradual, so that
sharp observations are impossible.

[B] \textit{Eclipses of the satellites of Jupiter} may be used in the same
way, with the advantage that they occur very frequently,---almost every
night, in fact; but the objection to them is the same as to the lunar
eclipses,---they are not sudden.

[C] \textit{The appearance and disappearance of meteors} may be and has
been used to determine the difference of longitude between places
not more than two or three hundred miles apart, and gives very accurate
results. (Now superseded by the telegraph.)

[D] \textit{Artificial signals}, such as flashes of powder and rockets, can
be used between two stations not too far distant. Early in the century
the difference of longitude between the Black Sea and the Atlantic
was determined by means of a chain of signal stations on the
mountain tops; so also, later, the difference of longitude between the
eastern and western extremities of the northern boundary of Mexico.
This method is now superseded by the telegraph.

\nbarticle{120.} \textsc{Second}, \textit{the moon regarded as a clock}.

Since the moon revolves around the earth once a month, it is, of
course, continually changing its place among the stars; and as the
laws of its motion are now well known, and as the place which
it will occupy is predicted for every hour of every Greenwich day
three years in advance in the nautical almanac, it is possible to
deduce the corresponding Greenwich time by any observation which
will determine the place of the moon among the stars. The almanac
%% -----File: 092.png---Folio 81-------
place, however, is the place at which the moon would be seen by an
observer \textit{at the centre of the earth}, and consequently the actual observations
are in most cases complicated with very disagreeable
reductions for parallax before they can be made available.

The simplest lunar method is,

[A]  \textit{That of Moon Culminations}.---We merely observe with a
transit instrument the time when the moon's bright limb crosses the
meridian of the place; and immediately after the moon we observe
one or more stars with the same instrument, to give us the error
of our clock. As the moon is observed on the meridian, its parallax
does not affect its right ascension, and accordingly, by a simple
reference to the almanac, we can ascertain the Greenwich time at
which the moon had the particular right ascension determined by
the observation. The method has been very extensively used, and
would be an admirable one were it not for the effects of personal
equation.

\begin{fineprint}
It seldom happens that the personal equation of an observer is the same
for such an object as the limb of the moon as it is for a star; and since the
moon's motion among the stars is very slow, the effect of such a difference
is multiplied by about 30 (roughly the number of days in a month) in its
effect upon the longitude deduced.
\end{fineprint}

[B] \textit{Lunar-Distances.}---At sea it is, of course, impossible to
observe the moon with a transit instrument, but we can observe its
distance from the stars near its path by means of a sextant. The
distance observed will not be the same that it would be if the
observer were at the centre of the earth, but by a mathematical
process called ``clearing a lunar'' the distance as seen from the
centre of the earth can be easily deduced, and compared with the
distance given in the almanac. From this the longitude can be
determined. Any error, however, in measuring a lunar-distance
entails an error about thirty times as great in the resulting longitude,
and the method is at present very little used, the moon having been
superseded by the chronometer for such purposes.

[C] \textit{Occultations}.---Occasionally, in its passage through the sky,
the moon over-runs a star, or ``\textit{occults}'' it. The star vanishes instantaneously,
and, of course, at the moment of its disappearance the
distance from the centre of the moon to the star is precisely equal
to the apparent semi-diameter of the moon; we thus have a ``lunar-distance''
self-measured.

Observations of this kind furnish one of the most accurate methods
%% -----File: 093.png---Folio 82-------
of determining the difference of longitude between widely separated
places, the only difficulty arising from the fact that the edge of the
moon is not smooth, but more or less mountainous, so that the distance
of a star from the moon's centre is not always the same at
the moment of its disappearance.

[D] \textit{In the same way a solar eclipse may be employed by observing
the moment when the moon's limb touches that of the sun.}

\begin{fineprint}
It will be noticed that these two last methods (the methods of occultation
and solar eclipse) do not belong in the same class with the method of lunar
eclipse, because the phenomena are not seen at the same instant at different
places, but the calculation of longitude depends upon the determination of
the moon's place in the sky at the given time, as seen from the earth's
centre.
\end{fineprint}

There are still other methods, depending upon measurements of
the moon's position by observations of its altitude or azimuth. In
all such cases, however, every error of observation entails a vastly
greater error in the final results. Lunar methods (excepting occultations)
are only used when better ones are unavailable.

\nbarticle{121.} Finally we have what may be called the \textit{mechanical methods}
of determining the longitude.

[A] \textit{By the chronometer}; which is simply an accurate watch that
has been set to indicate Greenwich time before the ship leaves port.
In order to find the longitude by the chronometer, the sailor has to
determine its ``error'' upon local time by an observation of the altitude
of the sun when near the prime vertical, as indicated on page~\pageref{pg:78}.
If the chronometer indicates true Greenwich time, \textit{the error deduced
from the observation will be the longitude}. Usually, however, the indication
of the chronometer face requires correction for the rate and
run of the chronometer since leaving port.

\begin{fineprint}
Chronometers are only imperfect instruments, and it is important, therefore,
that several of them should be used to check each other. It requires
three at least, because if only \textit{two} chronometers are carried and they disagree,
there is nothing to indicate which one is the delinquent.

On very long voyages the errors of chronometers are cumulative, and the
error is found to accumulate, not merely in proportion to the time, but \textit{more
nearly in proportion to the square of the time;} \textit{i.e.,} if the error to be feared in
the use of a chronometer in longitude determinations at the end of a
week is about two seconds of time, at the end of the month it would be, not
eight seconds, but about thirty-two seconds.

If, therefore, a ship is to be at sea, without making port, more than three
%% -----File: 094.png---Folio 83-------
or four months at a time, the method becomes untrustworthy, and it may be
necessary to recur to lunar distances; for voyages of less than a month the
method is now, practically, all that could be desired.
\end{fineprint}

[B] But the method which, wherever it is applicable, has superseded
all others, is that of \textit{The Telegraph}. When we wish to find the
longitude between two stations connected by telegraph, the process
is usually as follows: The observers at both stations, after ascertaining
that they both have clear weather, proceed to determine their
own local time by extensive series of star observations with the
transit instrument. Then, at a agreed-upon time, the observer at
Station A ``switches his clock'' into the telegraphic circuit, so that
its beats are communicated along the line and received upon the chronograph
of the other, say the western station. After the eastern clock
has thus sent its signals, say for two minutes, it is switched out of
the circuit, and the western observer now switches his clock into the
circuit, and its beats are received upon the eastern chronograph. The
operation is closed by another series of star observations.

We have now upon each chronograph sheet an accurate comparison
of the two clocks, showing the amount by which the western clock is
slow of the eastern. If the transmission of electric signals were
instantaneous, the difference shown upon the two chronograph sheets
would agree precisely. Practically, however, there will always be a
small discrepancy amounting to twice the time occupied in the transmission
of the signals; but the mean of the two differences will be
the true difference of longitude of the places after the proper corrections
have been applied. \textit{Especial care must be taken to determine
with accuracy, or to eliminate, the personal equations of the observers}.

\begin{fineprint}
It is customary to make observations of this kind on not less than five
or six evenings in cases where it is necessary to determine the difference of
longitude with the highest accuracy. The \textit{astronomical} difference of longitude
between two places can thus be telegraphically determined within about
the one-hundredth part of a second of time; \textit{i.e.}, within about ten feet or so,
in the latitude of the United States.

It may be noted here that the time occupied by the transmission of electric
signals in longitude operations is not to be taken as the real measure of
``the velocity of the electric fluid'' upon the wires, as was once supposed.
The time apparently consumed in the transmission is simply the time required
for the current at the receiving station (which current probably
\textit{begins} at the very instant the key is touched at the other end of the line) to
become \textit{strong enough} to do its work in making the signal; and this time
depends upon a multitude of circumstances.
%% -----File: 095.png---Folio 84-------

\nbarticle{122.}  \nbparatext{Local and Standard Time}.---In connection with time and
longitude determinations, a few words on this subject will be in place. Until
recently it has always been customary to use only \textit{local} time, each observer
determining his own time by his own observations. Before the days of the
telegraph, and while travel was comparatively slow and infrequent, this was
best; but the telegraph and railway have made such changes that, for many
reasons, it is better to give up the old system of local times in favor of a
system of standard time. It facilitates all railway and telegraphic business
in a remarkable degree, and makes it practically easy for every one to
keep accurate time, since it can be daily wired from some observatory to
every telegraph office.

According to the system that is now established in this country, there are
five such standard times in use,---the colonial, the eastern, the central, the
mountain, and the Pacific,---which differ from Greenwich time by exactly
four, five, six, seven, and eight hours respectively, \textit{the minutes and seconds being
identical everywhere}. At most places only one of these times is employed;
but in cities where different systems join each other, there are two standard
times in use, differing from each other by exactly one hour, and from the
local time by about half an hour. In some such places the local time also
maintains its place.

In order to determine the standard time by observation, it is only necessary
to determine the local time by one of the methods given, and correct
it according to the observer's longitude from Greenwich.

\nbarticle{123.} \nbparatext{Where the Day Begins.}---If we imagine a traveller starting
from Greenwich on Monday noon, and journeying westward as swiftly as the
earth turns to the east under his feet, he would, of course, keep the sun exactly
on the meridian all day long, and have continual noon. But what noon?
It was Monday when he started, and when he gets back to London, twenty-four
hours later, it is Tuesday noon there, and there has been no intervening
sunset. When does Monday noon become Tuesday noon? The convention
is that \textit{the change of date occurs at the $180$th meridian from Greenwich.}
Ships crossing this line \textit{from the east} skip one day in so doing. If it is
Monday forenoon when the ship reaches the line, it becomes Tuesday forenoon
the moment it passes it, the intervening twenty-four hours being
dropped from the reckoning on the log-book. \textit{Vice versa}, when a vessel
crosses the line \textit{from the western side}, it counts the same day twice, passing
from Tuesday forenoon back to Monday, and having to do its Tuesday over
again.

This $180$th meridian passes mainly over the ocean, hardly touching land
anywhere. There is a little irregularity in the date upon the different
islands near this line. Those which received their earliest European inhabitants
\textit{via} the Cape of Good Hope have, for the most part, the Asiatic date,
belonging to the west side of the $180$th meridian; while these that were approached
\textit{via} Cape Horn have the American date.
%% -----File: 096.png---Folio 85-------

When Alaska was transferred from Russia to the United States, it was
necessary to drop one day of the week from the official dates.
\end{fineprint}

\section*{THE PLACE OF A SHIP AT SEA.}

\nbarticle{124.} The determination of the place of a ship at sea is commercially
of such importance that, at the risk of a little repetition, we
collect together here the different methods available for its determination.
The methods employed are necessarily such that observations
can be made with the sextant and chronometer, the only
instruments available under the circumstances.

\medskip
The \textbf{Latitude} is usually obtained by observations of the sun's
altitude at noon, according to the method explained in \artref{Art.}{103}.

\medskip
The \textbf{Longitude} is usually found by determining the error upon local
time of the chronometer, which carries Greenwich time. The necessary
observations of the sun's altitude should be made when the
sun is near the prime vertical, as explained in \artref{Art.}{116}.

In the case of long voyages, or when the chronometer has for any
reason failed, the longitude may also be obtained by measuring a
lunar-distance and comparing it with the data of the nautical almanac.

\medskip
By these methods separate observations are necessary for the latitude
and for the longitude.

\nbarticle{125.}  \nbparatext{Sumner's Method.}---Recently a new method, first proposed
by Captain Sumner, of Boston, in 1843, has been coming largely into
use. In this method, each observation of the sun's altitude, with the
corresponding chronometer time, is made to define the position of the
ship upon a certain line, called \textit{the circle of position}. Two such observations
will, of course, determine the exact place of the vessel at
one of the intersections of the two circles.

At any moment the sun is vertically over some point upon the
earth's surface, which may be called the \textit{sub-solar point}. An observer
there would have the sun directly overhead. Moreover, if at any
point on the earth an observer measures the altitude of the sun with
his sextant, \textit{the zenith distance of the sun} (which is the complement
of this altitude) \textit{will be his distance from the sub-solar point at the
moment of observation, reckoned in degrees of a great circle}.

If, then, I take a terrestrial globe, and, opening the dividers so as
to cover an arc equal to this observed zenith distance of the sun,
put one foot of the dividers upon the sub-solar point, and sweep a
%% -----File: 097.png---Folio 86-------
circle on the surface of the globe around that point, the observer
must be \textit{somewhere on the circumference of that circle}; and moreover,
if to the observer the sun is in the \textit{southwest}, he himself must be in
the opposite direction from this sub-solar point; \textit{i.e., northeast} of it.
In other words, the \textit{azimuth of the sun} at the time of observation
informs him upon \textit{what part of the circle} he is situated.

Suppose a similar observation made at the same place a few hours
later. The sub-solar point, and the zenith distance of the sun, will
have changed; and we shall obtain a new circle of position, with its
centre at the new sub-solar point. The observer must be at one of
its two intersections with the first circle---which of the two intersections
is easily determined from the roughly observed azimuth of
the sun.

If the ship moves between the two observations, the proper allowance
must be made for the motion. This is easily done by shifting
upon the chart that part of the first circle of position where the
ship was situated, carrying the line forward parallel to itself, by an
amount just equal to the ship's run between the two observations,
as shown by the log. The intersection with the second circle then
gives the ship's place \textit{at the time of the second observation}.

The only problem remaining is to find the position of the ``sub-solar
point'' at any given moment. Now, the \textit{latitude} of this point is obviously
the \textit{declination of the sun} (which is found in the almanac).
If the sun's declination is zero, the sun is vertically over some point
upon the equator. If its declination is  $+20°$, it is vertically over
some point on the twentieth parallel of north latitude, etc.

In the next place, its \textit{longitude} is equal to the \textit{Greenwich apparent
solar time} at the moment of observation; and this is given by the
chronometer (which keeps Greenwich mean solar time), by simply adding
or subtracting the equation of time; so that, by looking in his
almanac and at his chronometer, the observer has the position of the
sub-solar point immediately given him. (See note, page~\pageref{pg:90}.)

\includegraphicsmid{illo042}{\textsc{Fig.~42.}---Sumner's Method.}

\begin{fineprint}
Suppose, for example, that on May 20 (the sun's declination being $+20°$),
at 11~\textsc{a.m.}, Greenwich \textit{apparent} time (\textit{i.e.}, May 19, $23^\text{h}$ by astronomical reckoning),
according to the chronometer, the sun is observed to have an altitude
of $40°$ by a ship in the North Atlantic. The sub-solar point will then be
(\figref{illo042}{Fig.~42}) at a point in Africa having a latitude of $+20°$, and an east longitude
of $15°$---at $A$ in the \figref{illo042}{figure}. And the radius of the ``circle of position,''
\textit{i.e.,} the distance from $A$ to $C$---will be $50°$.

Again, a second observation is made three hours later, when the sun's
altitude is found to be $65°$. The sub-solar point will then be at $B$, latitude
%% -----File: 098.png---Folio 87-------
$20°$, longitude $30°$~W., and the radius of the circle of position $BC$ be
$25°$, $C$ being the ship's place.

Of course it would be impracticable to carry on a vessel a terrestrial
globe large enough for the accurate working out of the graphical operation
indicated, but tables are provided, by which the necessary portions of the
position circles can be easily drawn upon the ordinary charts.
\end{fineprint}

\nbarticle{126.} The peculiar advantage of the method is, that a single observation
is used for all it is worth, giving \textit{accurately} the position of a line
upon which the ship is somewhere situated, and \textit{approximately} (by the
rough observation of the sun's azimuth) the part of that line upon
which its place will be found. In approaching the American coast,
for instance, if an observation be taken in the forenoon when the
sub-solar point is over the continent of Africa, the ship's position
circle will lie nearly parallel to the coast, and then a single observation
will give approximately the distance of the ship from land, which
may be all the sailor wishes to know. The observations need not be
taken at any particular time. We are not limited to observations at
noon, or to the time when the sun is on the prime vertical. It is to
be noted, however, \textit{that everything depends upon the chronometer}, as
much as in the ordinary chronometric determination of longitude.

\nbarticle{127.} \nbparatext{Determination of Azimuth.}---A problem, important, though
not so often encountered as that of latitude and longitude determinations,
%% -----File: 099.png---Folio 88-------
is that of determining \textit{the azimuth, or true bearing, of a line upon
the earth's surface}. The process is this: With a theodolite having
an accurately graduated horizontal circle the observer points alternately
upon the pole star and upon a distant
signal erected for the purpose; the
signal being an artificial star consisting of
a small hole in a plate of metal, with a
bull's-eye lantern or other light behind it.
It is desirable that it should be at least
a mile away from the observer, so that
any small displacement of the instrument
will be harmless. The theodolite must
be carefully adjusted for collimation, and
especial pains must be taken to have the
telescope perfectly level.

\includegraphicsouter{illo043}{\textsc{Fig.~43.}---Determination of Azimuth.}

The next morning by daylight the observer
measures the angle or angles between the night-signal and the
objects whose azimuth is required.

If the pole star were exactly at the pole, the mere difference
between the two readings of the circle, obtained when the telescope
is pointed on the star and on the signal, would directly give the
azimuth of the signal, As this is not the case, however, the \textit{time}
at which each observation of the pole star is made must be noted,
and the azimuth of the star must be computed for that moment.
This can easily be done, as the right ascension and declination of
this star are given in the almanac for every day of the year.

\begin{fineprint}
Recurring to the Z.P.S. [zenith-pole-star] triangle, $N$ (\figref{illo043}{Fig.~43}) being the
north point of the horizon, $P$ the pole, and $NZ$ the meridian, we at once
see that the side $PS$ is the complement of the star's declination; the side
$PZ$ is the complement of the observer's latitude (which must be known);
and the angle at $P$ is the difference between the right ascension of the pole
star and the sidereal time of the observation; [$(t-a)$ if the star is west of
the meridian at the time, and $(a-t)$ if it is east.] This will come out in
hours, of course, and must be reduced to degrees before making the computation.
We thus have two sides of the triangle, viz., $PS$ and $PZ$, with
the included angle at $P$, from which to compute the angle $Z$ at the zenith.
This is the star's azimuth.

The pole star is used because, being so near the pole, any slight error in
the assumed latitude of the place or in the sidereal time of the observation
will hardly produce any effect upon the result, especially if the star be
caught between five and six hours before or after its upper culmination, at
%% -----File: 100.png---Folio 89-------
a time when it changes its azimuth very slowly (near $S'$ or $S''$ in the \figref{illo043}{figure}).
The sun, or any other heavenly body whose position is given in the almanac,
can also be used as a reference point in the same way, provided sufficient
pains \DPtypo{is}{are} taken to secure an accurate observation of the time at the instant
when the pointing is made. The altitude should not exceed thirty degrees
or so. But the results are usually rough compared with these obtained by
means of the pole star.
\end{fineprint}

\section*{DETERMINATION OF THE POSITION OF A HEAVENLY BODY.}

\nbarticle{128.} The position of a heavenly body is defined by its right
ascension and declination. These quantities may be determined---

\medskip
\sloppy
(1) \textbf{By the meridian circle}, provided the body is bright enough to
be seen by the instrument and comes to the meridian in the night-time.
If the instrument is in exact adjustment, the \textit{sidereal time
when the object crosses the middle wire of the reticle of the instrument is
directly} (according to \artref{Art.}{27}) \textit{the right ascension of the object}.

\fussy
The reading of the circle of the instrument, corrected for refraction
and parallax if necessary, gives the \textit{polar distance} of the object, if
the polar point of the circle has been determined (\artref{Art.}{66}); or it gives
the \textit{zenith distance} of the object if the nadir point has been determined
(\artref{Art.}{67}). In either case the \textit{declination} can be immediately
deduced, being the complement of the polar distance, and equal to
the latitude of the observer, minus the distance of the star south of
the zenith. One complete observation, then, with the meridian circle,
determines both the right ascension and declination of the object.

\medskip
If a body (a comet, for instance) is too faint to be observed by
the telescope of the meridian circle, which is seldom very powerful,
or if it does not come to the meridian during the night, we usually
accomplish our object---

\nbarticle{129.} (2) \nbparatext{By the Equatorial}, determining the position of the body
by measuring the \textit{difference of right ascension and declination} between
it and some neighboring star, whose place is given in a star
catalogue, and of course has been determined by the meridian circle
of some observatory.

\begin{fineprint}
In measuring this difference of right ascension and declination, we usually
employ a filar micrometer fitted like the reticle of a meridian circle. It carries
a number of wires which lie north and south in the field of view, and
these are crossed at right angles by one or more wires which can be moved
%% -----File: 101.png---Folio 90-------
by the micrometer screw. \textit{The difference of right ascension} between the star
and the object to be determined is measured by simply observing with the
chronograph the transits of the two objects across the north and south
wires; \textit{the difference of declination}, by bisecting each object with one of the
micrometer wires as it crosses the middle of the field of view. The observed
difference must be corrected for refraction and for the motion of
the body, if it is appreciable.

Other less complicated micrometers are also in use. One of them, called
the \textit{ring micrometer}, consists merely of an opaque ring supported in the field
of view either by being cemented to a glass plate or by slender arms of
metal. The observations are made by noting the transits of the comparison
star and of the object to be determined across the outer and inner edges of
the ring. If the radius of the ring is known in seconds of arc, we can
from these observations deduce the differences both of right ascension and
declination. The results are less accurate than those given by the wire
micrometer, but the ring micrometer has the advantage that it can be used
with any telescope, whether equatorially mounted or not, and requires no
adjustment.

There are also many other methods of effecting the same object.

\nbarticle{130.} \nbparatext{To Compute the Time of Sunrise or Sunset.}---To solve this problem,
it is only necessary to work out the Z.P.S. triangle and find the hour-angle
$P$, having given precisely the same data as in finding the time by a single
altitude of the sun (\artref{Art.}{116}). $PZ$ is the observer's co-latitude, $PS$ is the
complement of the sun's declination (given by the almanac); and the true
distance from the zenith to the centre of the sun at the moment when its
upper edge is at the horizon is $90°\, 50'$, which is made up of $90°$, $+16'$ (the
mean semi-diameter of the sun), plus $34'$ (the mean refraction at the horizon).
The resulting hour-angle $P$, corrected for the equation of time, gives the mean
time (\textit{local}) at which the sun's upper limb touches the horizon, under the
average circumstances of temperature and barometric pressure. If it is very
cold, with the barometer standing high, sunrise will be accelerated, or sunset
retarded, by a considerable fraction of a minute. If the sun rises or sets
over the sea-horizon, and the observer's eye is at any considerable elevation
above the sea-level, the dip of the horizon must also be added to the $90°\, 50'$
before making the computation.

The beginning and end of twilight may be computed in the same way
by merely substituting $108°$ for $90°\, 50'$.

\nbarticle{131.} \nblabel{pg:90}\nbparatext{Note to \artref{Art.}{125}.}---In the explanation of Sumner's method it is
assumed that the earth is a \textit{perfect sphere}. In the actual application of the
method certain corrections are therefore necessary to take into account the
earth's ellipticity.
\end{fineprint}
\chelabel{CHAPTERIV}
%% -----File: 102.png---Folio 91-------

\Chapter{V}{The Earth}
\nbchaptercenter{THE EARTH AS AN ASTRONOMICAL BODY.}
\nbchaptersubhang{\stretchyspace Approximate Dimensions---Proofs of its Rotation---Accurate
Determination of its Form and Size by Geodetic Operations
and Pendulum Observations---Astronomical, Geodetic and Geocentric
Latitude---Determination of the Earth's Mass and Density.}

\nbarticle{132.} \textsc{Having} discussed the methods of making astronomical observations,
we are now prepared to consider the earth in its astronomical
relations; \textit{i.e.}, those facts relating to the earth which are
ascertained by astronomical methods, and are similar to the facts
which we shall have to consider in the case of the other planets.
The facts are broadly these:---

\begin{asparaenum}[1.]
\item \textit{The earth is a great ball, about $7918$ miles in diameter.}

\item \textit{It rotates on its axis once in twenty-four sidereal hours.}

\item \textit{It is flattened at the poles}, the polar diameter being nearly
\textit{twenty-seven miles}, or one \textit{two hundred and ninety-fifth part} less than
the equatorial.

\item \textit{It has a mean density of about five and six-tenths times that of
water, and a mass represented in tons by six with twenty-one ciphers
after it $($or six sextillions of tons, according to the French numeration$)$.}

\item \textit{It is flying through space in its orbital motion around the sun,
with a velocity of about nineteen miles a second}; i.e., \textit{about seventy-five
times as swiftly as any cannon-ball.}
\end{asparaenum}

\section*{I.}

\nbarticle{133.} \nbparatext{The Earth's Approximate Form and Size.}---It is not necessary
to dwell upon the ordinary proofs of its globularity. We merely mention
them. \begin{inparaenum}[1.] \item It can be circumnavigated. \item The appearance of
vessels coming in from sea indicates that the surface is everywhere
convex. \item The fact that the sea-horizon, as seen from an eminence,
is everywhere depressed to the same extent below the level
line, shows that the surface is approximately spherical. \item The fact
that as one goes from the equator toward the north, the elevation of
%% -----File: 103.png---Folio 92-------
the pole increases proportionally to the distance from the equator,
proves the same thing. \item \textit{The shadow of the earth, an seen upon
the moon at the time of a lunar eclipse, is that which only a sphere
could cast}.
\end{inparaenum}

We may add as to the smoothness and globularity of the earth,
that if the earth be represented by an 18-inch globe, the difference
between the polar and equatorial diameter would only be about one-sixteenth
of an inch, the highest mountains upon the earth's surface
would be represented by about one-eightieth of an inch, and the average
elevation of the continents would be hardly greater than that of
a film of varnish. The earth is really relatively smoother and
rounder than most of the balls in a bowling-alley.

\includegraphicsouter{illo044}{\textsc{Fig.~44.}---Curvature of the Earth's Surface.}

\begin{fineprint}
\sloppy
\nbarticle{134.} An approximate measure of the diameter is easily obtained. Erect
upon a level plain three
rods in line, a mile apart,
and cut off their tops at
the same level, carefully
determined with a surveyor's
levelling instrument.
It will then be
found that the line $AC$,
Fig.~44, joining the extremities
of the two terminal rods, passes about eight inches below $B$, the
top of the middle rod.

\fussy
Suppose the circle $ABC$ completed, and that $E$ is the point on the circumference
opposite $B$, so that $BE$ equals the diameter of the earth $(=2\,R)$.
\begin{flalign*}
&\text{\indent By geometry,}& &BD:BA = BA:BE, &&\phantom{\text{\indent By geometry,}}\\
&\text{whence }& &BE = \frac{BA^{2}}{BD}, \text{ or }
  R = \frac{BA^{2}}{2BD}.  &&
\end{flalign*}

Now $BA$ is one mile, and $BD = \frac{2}{3}$ of a foot, or $\frac{1}{7920}$ of a mile.

Hence $2R = \dfrac{1^2}{\frac{1}{7920}}$, or 7920 miles: a very fair approximation.

On account of refraction, however, the result cannot be made \textit{exact} by
any care in observation. The line of sight, $AC$, is not \textit{strictly} straight,  but
curves slightly towards the earth, and differently as the weather changes.
\end{fineprint}

\nbarticle{135.} The best method of ascertaining the size of the earth---in
fact the only one of real value---is by measuring the arcs of the meridian
in order to ascertain \textit{the number of miles or kilometers in one degree},
from which we immediately get the circumference of the earth.
%% -----File: 104.png---Folio 93-------
This measure involves two distinct operations. One---the measure
of the number of miles---is purely \textit{geodetic}; the other---the determination
of the number of degrees, minutes, and seconds between
the two stations---is purely \textit{astronomical}.

We have to find by \textit{astronomical} observation the angle between two
radii drawn from the centre of the earth to the two stations (regarding
the earth as spherical); or, what is the same thing, the \textit{angular distance
in the sky between their respective zeniths}. The two stations being
on the same meridian, all that is necessary is to measure their \textit{latitudes}
by any of the methods which have been given in \chapref{CHAPTERIV}{Chapter~IV.} and take
the difference. This will be the angle wanted. If, for instance, the
distance between the two stations was found by measurement to be
120 miles, and the difference of latitude was found by astronomical
observations to be $1°\,44'.2$, we should get 69.27 miles for one degree.
Three hundred and sixty times this would be the circumference of
the earth, a little less than 25,000 miles, and the diameter would be
found by dividing this by $\pi$, which would give 7920 miles.

\begin{fineprint}
\nbarticle{136.}  Eratosthenes of Alexandria seems to have understood the matter
as early as 250~\textsc{b.c.} His two stations were Alexandria and Syene in Upper
Egypt. At Syene he observed that at noon of the longest day in summer
there was no shadow at the bottom of a well, the sun being then vertically
overhead. On the other hand, the gnomon at Alexandria, on the same day,
by the length of the shadow, gave him $\frac{1}{50}$ of a circumference, or $7°\,12'$ as the
distance of the sun from the zenith at that place, which, therefore, is the
difference of latitude between Alexandria and Syene.

The weak place in his work was in the measurement of the distance between
the two places. He states it as 5000 stadia, thus making the circumference
of the earth 250,000 stadia; but we do not know the length of his
stadium, nor does he give any account of the means by which he measured
the distance, if he measured it at all. There seem to have been as many
different stadia among the ancient nations as there were kinds of ``feet'' in
Europe at the beginning of this century.

The first really valuable measure of the arc of a meridian was that made
by Picard in Northern France in 1671---the measure which served Newton so
well in his verification of the idea of gravitation.
\end{fineprint}

\section*{II.}

\nbarticle{137.}  \nbparatext{The Rotation of the Earth.}---At the time of Copernicus the
only argument in favor of the earth's rotation\footnote
  {The word \textit{rotate} denotes a spinning motion like that of a wheel on its axis.
  The word \textit{revolve} is more general in its application, and may be applied either to
  describe such a spinning motion, or (and this is the more usual use in astronomy)
  to describe the motion of one body around another, as that of the earth around
  the sun.}
was that the hypothesis
%% -----File: 105.png---Folio 94-------
was \textit{more probable} than that the heavens themselves revolved.
All phenomena \textit{then known} would be sensibly the same on either
supposition. A little later, analogy could be adduced, for when the
telescope was invented, we could \textit{see} that the sun, moon, and several
of the planets are rotating globes.

At present we are able to adduce experimental proofs which absolutely
demonstrate the earth's rotation, and some of them even make
it visible.

\includegraphicsouter{illo045}{\textsc{Fig.~45.}\\
Eastward Deviation
of a Falling
Body.}

\nbarticle{138.} 1.~\textit{The Eastward Deviation of Bodies falling from a Great
Height}.---The idea that such a deviation ought to occur was first
suggested by Newton. Evidently, since the top of a tower, situated
anywhere but at the pole of the earth, describes every
day a larger circle than its base, it must move faster.
A body which is dropped from the top, retaining its excess
of eastward motion as it descends, must therefore
strike \textit{to the east} of the point which is vertically under
its starting-point, provided it is not deflected in its fall
by the resistance of the air or by air-currents. \figref{illo045}{Fig.~45}
illustrates the principle. A body starting from $A$, the
top of the tower, reaches the earth at $D$ ($BD$ being
equal very approximately to $AA'$), while during
its fall the \textit{bottom} of the tower has only moved from
$B$ to $B'$. The experiments are delicate, since the deviation
is very small, and it is not easy to avoid the
effect of air-currents. It is also extremely difficult to
get balls so perfectly spherical that they will not sheer
off to one side or the other in falling.

\begin{fineprint}
The best experiments of this kind so far have been these of Benzenberg,
performed at Hamburg in 1802, and those of Reich, performed in 1831, in
an abandoned mine shaft near Freiberg, in Saxony. The latter obtained a
free fall of 520 feet, and from the mean of 160 trials, the eastern deviation
observed was 1.12 inches, while theory would make it 1.08. The experiment
also gave a southern deviation of 0.17 of an inch, unexplained by theory.
It seems to indicate the probable error of observation. The balls in falling
sometimes deviated two or three inches one side or the other from the
average.
%% -----File: 106.png---Folio 95-------

\includegraphicsouter{illo046}{\textsc{Fig.~46.}---Foucault's Pendulum Experiment.}

The formula given by Worms in his treatise on ``The Earth and its
Mechanism,'' is
\[
  x = \frac{4\pi t(H-\frac{1}{2}\Delta)\cos{\phi}}{3T} ;
\]
where $x$ is the deviation, $t$ is the number of seconds occupied in falling, $T$
the number of seconds in a sidereal day, $H$ the height fallen through, and
$\Delta$ the difference between $H$ and the height through which a body would fall
in $t$ seconds if there were no resistance (so that $\Delta=\frac{1}{2}gt^2-H$). Finally, $\phi$ is
the latitude of the place of observation. In latitude $45°$ a fall of 576 feet
should give, neglecting the resistance of the air, a deviation of 1.47 inches.
The resistance would increase it a little.

It will be noted that \textit{at the pole}, where the cosine of the latitude equals
zero, \textit{the experiment fails}. The largest deviation is obtained at the equator.
\end{fineprint}

\nbarticle{139.} 2.~\textit{Foucault's Pendulum Experiment}.---In 1851 Foucault,
that most ingenious of French
physicists, devised and first executed
an experiment which actually
shows the earth's rotation to the
eye. From the dome of the Pantheon
in Paris he suspended a heavy
iron ball about a foot in diameter
by a wire more than 200 feet long
(\figref{illo046}{Fig.~46}). A circular rail some
twelve feet across, with a little
ridge of sand built upon it, was
placed under the pendulum in such
a way that a pin attached to the
swinging ball would just scrape
the sand and leave a mark at each
vibration. The ball was drawn
aside by a cotton cord and allowed
to come absolutely to rest; then
the cord was burned, and the pendulum
set to swinging in a true
plane; but this plane seemed to
\textit{deviate slowly towards the right}, cutting the sand in a new place at
each swing and shifting at a rate which would carry it completely
around in about thirty-two hours if the pendulum did not first come
to rest. In fact, the floor of the Pantheon was soon turning under
the plane of the pendulum's vibration. The experiment created
%% -----File: 107.png---Folio 96-------
great enthusiasm at the time, and has since been very frequently
performed and always with substantially the same results.

\nbarticle{140.} The approximate theory of the experiment is very simple.
Such a pendulum, consisting of a round ball hung by a \textit{round} wire or
else suspended on \textit{point, so as to be equally free to swing in any plane}
(unlike the common clock pendulum in this freedom), being set up
at the pole of the earth, would appear to shift around in twenty-four
hours. Really, the plane of vibration remains invariable and the
earth turns under it, the plane of vibration in this case being unaffected
by the motion of the earth. This can be easily shown
by setting up a similar apparatus, consisting of a ball hung by a
thread, upon a table, and then turning the table around with as little
jar as possible. The plane of the swing will remain unchanged by
the motion of the table.

\includegraphicsouter{illo047}{\textsc{Fig.~47.}\\
Explanation of the Foucault
Pendulum Experiment.}

It is easy to see, further, that at the equator there would be no such
tendency to shift. In any other latitude the effect will be intermediate,
and the time required for the pendulum to complete the revolution
of its plane will be \textit{twenty-four hours
divided by the sine of the latitude}. The northern
edge of the floor of a room (in the northern
hemisphere) is nearer the axis of the earth
than its southern edge, and therefore is carried
more slowly eastward by the earth's rotation.
Hence it must \textit{skew around} continually,
like a postage stamp gummed upon a whirling
globe anywhere except at the globe's equator.
The southern extremity of every north and
south line on the floor continually works toward
the east faster than the northern extremity,
causing the line itself to shift its direction
accordingly, compared with the direction
it had a few minutes before. A free pendulum,
set at first to swing along such a line, must therefore apparently
deviate continually at the same rate in the opposite direction. In
the \textit{northern} hemisphere its plane moves \textit{dextrorsum}; \textit{i.e.}, with the
hands of a watch: in the \textit{southern}, its motion is \textit{sinistrorsum}.

\includegraphicsouter{illo048}{\textsc{Fig.~48.}---Developed Cone.}

\begin{fineprint}
\nbarticle{141.} Suppose a parallel of latitude drawn through the place in question,
and a series of tangent lines drawn toward the north at points an inch or so
apart on this parallel. All these tangents would meet at some point, $V$, \figref{illo047}{Fig.~47},
which is on the earth's axis produced; and taken together these tangents
%% -----File: 108.png---Folio 97-------
would form a cone with its point at $V$. Now if we suppose this cone cut
down upon one side and opened up (technically, ``\textit{developed}''), it would give
us a sector of a circle, as in \figref{illo048}{Fig.~48}, and the angle of the sector would be the
sum total of the angles between all the adjacent
meridians tangent to the earth on that parallel.
Now it is easy to prove that the angle of this sector
equals $360°× \sin{\phi}$ ($\phi$ being the latitude).
(1)~The circumference of the parallel $AB$ (\figref{illo047}{Fig.~47})
$= 2\pi× R\cos{\phi}$, since $R×\cos{\phi}=AD$, which is
the radius of the parallel, $R$ being $AC$, the radius
of the globe, and the angle $ACD$ equal to $90°-\phi$.
(2)~The side $AV$ of the cone (which will be the
radius of the sector when the cone is developed)
$= R\cot{\phi}$; so that the circumference of the circle
which has $VA$ for its radius would be $2\pi× R\cot{\phi}$.

Hence ($ABA'$ being the circumference of the
parallel forming the edge of the developed sector $ABA'V$ in \figref{illo048}{Fig.~48}), the
angle of the sector $AVA'$ (greater than $180°$ in the \figref{illo048}{figure}): $360°=arc\
ABA'$: whole circumference $ABA'm$; or angle $V : 360°=R\cos{\phi} : R\cot{\phi}$;
\begin{flalign*}
&\text{whence }&&
  V = 360°\frac{\cos{\phi}}{\cot{\phi}}
    = 360°\sin{\phi}.  &&\phantom{whence }
\end{flalign*}

$V$ is the total angle described by the plane of the pendulum in a day.

At the pole the cone produced by the tangent lines becomes a little
``button,'' a complete circle. At the equator it becomes a \textit{cylinder},
and the angle is zero.

In order to make the experiment successfully, many precautions must be
taken. It is specially important that the pendulum should vibrate in a true
plane, without any lateral motion. To secure this end, it must be carefully
guarded against all jarring motion and air-currents. To diminish the effect
of all such disturbances, which will always occur to a certain extent, the
pendulum should be very heavy and very long, and of course the suspended
ball must be truly round and smooth. Ordinary clock-work cannot be used
to keep the pendulum in vibration, since it must be free to swing in every
plane. Usually, the apparatus once started is left to itself until the vibrations
cease of their own accord; but Foucault contrived a most ingenious
electrical apparatus, which we have not space to describe, by means of which
the vibration could be kept up for days at a time without producing any
hurtful disturbance whatever.

It will be noticed that this experiment is most effective precisely where
the experiment of the falling bodies fails. This is best near the pole, the
other at the equator.
\end{fineprint}

\includegraphicsouter{illo049}{\textsc{Fig.~49.}---Foucault's Gyroscope.}

\nbarticle{142.} 3.~\textit{By the Gyroscope}, an experiment also due to Foucault
and proposed and executed soon after the pendulum experiment.
%% -----File: 109.png---Folio 98-------

The instrument shown in \figref{illo049}{Fig.~49} consists of a wheel so mounted in
gimbals that it is free to turn in every direction, and so delicately
balanced that it will stay in any position if undisturbed. If the
wheel be set to rotating rapidly, \textit{it will maintain the direction of its
axis invariable, unless acted upon by extraneous force}. If, then,
we set the axis horizontal and arrange a microscope to watch a
mark upon one of the gimbals,
it will appear slowly to
shift its position as the earth
revolves, in the same way as
the plane of the pendulum
behaves.

\begin{fineprint}
\nbarticle{143.} 4.~There are many other
phenomena which depend upon
and really demonstrate the earth's
rotation. We merely mention
them:---

\begin{asparaenum}[\itshape a.\normalfont]
\item \textit{The Deviation of Projectiles.}
In the northern hemisphere a
projectile always deviates towards
the right; in the southern hemisphere
toward the left.

\item \textit{The Trade Winds.}

\item \textit{The Vorticose Revolution of
the Wind in Cyclones.} In the
northern hemisphere the wind in
a cyclone moves spirally towards
the centre of the storm, whirling
\textit{counter clock-wise}, while in the
southern, the spiral motion is \textit{with the hands of a watch}. The motion is
explained in either case by the fact that currents of air, setting out for the
centre of disturbance where the cyclone is formed, deviate like projectiles,
to the right in the northern hemisphere, and towards the left in the southern
hemisphere, so that they do not meet squarely in the centre of disturbance.

\item \textit{The Ordinary Law of Wind-change;} that is, in the northern hemisphere
the north wind, under ordinary circumstances, changes to a northeast, a
northeast wind to an east, east to southeast, etc. When the wind changes
in the opposite direction, it is said to ``\textit{back}'' around. In the southern
hemisphere it of course \textit{usually} backs around, much to the disconcertment
of the early Australian settlers.
\end{asparaenum}
\end{fineprint}

It might seem at first time the rotation of the earth, which occupies
twenty-four hours, is not a very rapid motion. A point on the equator,
%% -----File: 110.png---Folio 99-------
however, has to move nearly one thousand miles an hour, which
is about fifteen hundred feet per second, and very nearly the speed
of a cannon-ball.

\nbarticle{144.} \nbparatext{Invariability of the Earth's Rotation.}---It is a question of
great importance whether the day changes its length. Theoretically it
must almost necessarily do so. The friction of the tides, and the
deposits of meteoric matter upon the earth both tend to lengthen it;
while on the other hand, the earth's loss of heat by radiation and
consequent shrinkage must tend to shorten it. Then geological
changes, the elevation and subsidence of continents, and the transportation
of matter by rivers, act, some one way, some the other. At
present it can only be said that the change, if any has occurred since
astronomy became accurate, has been too small to be detected. The
day is certainly not longer or shorter by $\frac{1}{100}$ of a second than in the
days of Ptolemy, and \textit{probably} has not changed by $\frac{1}{1000}$ of a second.
The criterion is found in comparing the \textit{times} at which celestial
phenomena, such as eclipses, transits of Mercury, etc., occur.

\section*{III.}

\nbarticle{145.} \nbparatext{The Earth's Form}, more accurately stated, \textit{is that of a
spheroid of revolution, having an equatorial radius of} 6,377,377 \textit{metres,
and a polar radius of} 6,355,270 \textit{metres, according to Listing $(1873);$
or of} 6,378,206.4 \textit{and} 6,356,583.8 respectively, \textit{according to Clarke}.\footnote
  {This is Clarke's spheroid of 1866, and is adopted by the United States Coast
  and Geodetic Survey. See \hyperref[app:spheroid]{Appendix} for his spheroid of 1878.}
It must be understood, also, that this statement is only a second
\textit{approximation} (the first being that the earth is a globe). Owing
to mountains and valleys, etc., the earth's surface does not strictly
correspond to that of any geometrical solid whatever.

The flattening at the poles is the necessary consequence of the
earth's rotation, and might have been cited in the preceding section
as proving it.

\includegraphicsouter{illo050}{\textsc{Fig.~50.}---A Triangulation.}

\nbarticle{146.} There are two ways of determining the form of the earth:
one, by \textit{measurement of distances upon its surface in connection with
the latitudes and longitudes of the points of observation}. This gives
not only the \textit{form}, but the \textit{dimensions}. The other method is by the
observation of the \textit{varying force of gravity at various points},---observations
which are made by means of a pendulum apparatus of some
kind, and determine \textit{only the form}, but not the size of the earth.
%% -----File: 111.png---Folio 100-------

\nbarticle{147.} 1.~\textit{Measurements of Arcs of Meridian in Different Latitudes}.---To
determine the size of the earth regarded as a sphere, a \textit{single}
arc of meridian in any latitude is sufficient. Assuming, however,
that the earth is not a sphere, but a spheroid with elliptical meridians,
we must measure at least \textit{two} such arcs, one of which should be near
the equator, the other near the pole.

The \textit{astronomical work} consists simply in finding with the greatest
possible accuracy the \textit{difference of latitude} between the terminal stations
of the meridian arc. The \textit{geodetic work} consists in measuring
their \textit{distance} from each other in miles, feet, or metres, and it is this
part of the work which consumes the most time and labor. The
process is generally that known as triangulation.

\begin{fineprint}

Two stations are selected for the extremities of a
\textit{base line} six or seven miles long, and the ground
between them is levelled as if for a railroad. The
distance between these stations ($A$ and $B$ in \figref{illo050}{Fig.~50})
is then carefully measured by an apparatus especially
designed for the purpose and with an error not to
exceed half an inch or so in the whole distance. A
third station, $1$, is then chosen, so situated that it will
be visible from both $A$ and $B$, and all the angles of
the triangle $AB\,1$ are measured with great care by a
theodolite. A fourth station, $2$, is then selected, such
that it will be visible from $A$ and $1$ (and if possible
from $B$ also), and the angles of the triangle $A\,1\,2$ are
measured in the same way. In this manner the whole
ground between the two terminal stations is covered
with a network of triangulation, the two terminal stations themselves being
made two of the triangulation points. Knowing \textit{one distance and all the
angles} in this system, it is possible to compute with great accuracy the exact
length of the line $1\,5$ and its direction.
\end{fineprint}

The sides of the triangles are usually from twenty-five to thirty
miles in length, though in a mountainous country not infrequently
much longer ones are available. Generally speaking, the fewer the
stations necessary to connect the extremities of the arc, and the
longer the lines, the greater will be the ultimate accuracy. In this
way it is possible to measure distances of 200 or 300 miles with a
probable error not exceeding two or three feet.

\begin{fineprint}
Many arcs of meridians have been measured in this way,---not less than
twenty or thirty in different parts of the earth, the most extensive being the
%% -----File: 112.png---Folio 101-------
so-called Anglo-French arc, extending more than twelve degrees in length;
the Indian arc, nearly eighteen degrees long; and the great Russo-Scandinavian
arc, more than twenty-five degrees in length, and reaching from
Hammerfest to the mouth of the Danube. One short arc has been measured
in South America and one in South Africa.
\end{fineprint}

In a general way, it appears that the higher the latitude the longer
the arc. Thus, near the equator the length of a degree has been
found to be 362,800 feet in round numbers, while in northern Sweden,
in latitude $66°$, it is 365,800 feet; in other words, the earth's surface
is \textit{flatter near the poles}. It is necessary to travel 3000 feet further in
Sweden than in India to increase the latitude one degree, as measured
by the elevation of the celestial pole.

\nbarticle{148.}
The deduction of the exact form of the earth from such
measurements is an abstruse problem. Owing to errors of observation
and local deviations in the direction of gravity, the different arcs do
not give strictly accordant results, and the best that can be done is to
find the result \textit{which most nearly satisfies all the observations}.

If we assume that the form is that of an \textit{exact spheroid of revolution},
with all the meridians true ellipses and all exactly alike, the problem
is simplified somewhat, though still too complicated for discussion
here. Theory indicates that the form of a revolving mass, fluid
enough to yield to the forces acting in such a case, \textit{might, and probably
would, be such a spheroid}; but other forms are also theoretically
possible, and some of the measurements rather indicate that
the equator of the earth is not a true circle, but an oval flattened by
nearly half a mile. On the whole, however, astronomers are disposed
to take the ground that since no regular geometrical solid
whatsoever can \textit{absolutely} represent the form of the earth, we may as
well assume a regular spheroid for the standard surface, and consider
all variations from it as local phenomena, like hills and valleys.

\begin{fineprint}
\includegraphicsouter{illo051}{\textsc{Fig.~51.}\\
Radii of Curvature of the Meridian.}

\nbarticle{149.}  Each measurement of a degree of latitude gives the ``\textit{radius of curvature},''
as it is called, of the meridian at the degree measured. The length
of a degree from $44°\,30'$ to $45°\,30'$, multiplied by 57.29 (the number of degrees
in a radian), gives the radius of the ``\textit{osculatory circle},'' which would
just fit the curve of the meridian at that point, Having a table giving
the actual length of each degree of latitude, we could construct the earth's
meridian graphically as follows:---

Draw the line $AX$, \figref{illo051}{Fig.~51}. On it lay off $Aa$, equal to the radius of curvature
of the first measured degree (that is, 57.3 times the length of the degree),
%% -----File: 113.png---Folio 102-------
and with $a$ as centre, describe an arc $AB$, making the angle $AaB$ just one
degree. Next produce the line $Ba$ to $b$,
making $Bb$ the radius of curvature of
the second degree, and draw this second
degree-arc; and so proceed until the
whole ninety have been drawn. This
will give one quarter of the meridian, and
of course the three other quarters are all
just like it. $a$, $b$, $c$, etc., are called the
``centres of curvature'' of the different
degrees.

If we assume the curve to be an
ellipse, then the equatorial \DPtypo{semidiameter}{semi-diameter}
$AO$, and the polar, $PO$, are given respectively
by the two formulas, $AO = \sqrt[3]{qp^{2}}$
and $PO = \sqrt[3]{q^{2}p}$, $q$ and $p$ being the radii
of curvature ($Aa$ and $Pe$ in the \figref{illo051}{figure}) at the equator and pole.
\end{fineprint}

\nbarticle{150.} The ``\textit{ellipticity}'' or ``\textit{oblateness}'' of an ellipse is the fraction
found by dividing the difference of the polar and equatorial diameters
by the equatorial, and is expressed by the equation
\[
d = \frac{A - B}{A}.
\]
In the case of the earth this is $\frac{1}{295}$, according to Clarke's spheroid,
of 1866. Until within the last few years Bessel's smaller value,
viz., $\frac{1}{299}$, was generally adopted. Listing's larger value, $\frac{1}{288}$, is now
preferred by some.

The \textit{ellipticity} of an ellipse must not be confounded with its
\textit{eccentricity}. The latter is
\[
e = \frac{\sqrt{A^{2} - B^{2}}}{A},
\]
and is always a much larger numerical quantity than the ellipticity.
In the case of the earth's meridian, it is $\frac{1}{12.1}$ as against $\frac{1}{295}$. Its
symbol is usually $e$.

\begin{fineprint}
\nbarticle{151.} \textit{Arcs of longitude} are also available for determining the earth's form
and size. On a spherical earth a degree of longitude measured along any
parallel of latitude would be equal to one degree of the equator multiplied by
the cosine of the latitude. On an oblate or orange-shaped spheroid (the surface
of which lies wholly within the sphere having the same equator) the degrees
%% -----File: 114.png---Folio 103-------
of longitude are evidently everywhere shorter than on the sphere,
the difference being greatest at a latitude of $45°$.

In fact, \textit{arcs in any direction between stations of which both the latitude and
longitude are known} can be utilized for the purpose; and thus the extensive
surveys that have been made in different countries have given us a pretty
accurate knowledge of the earth's dimensions. It is very desirable, that in
some way the chain of actual measurements should be extended from the
eastern continent to the western, but the immense difficulties of so doing
are obvious.

At present the distance from a point on the earth's surface (say the observatory
at Washington) to any other point in the opposite hemisphere (say
the observatory at the Cape of Good Hope) is uncertain to perhaps the
extent of a quarter of a mile.
\end{fineprint}

\nbarticle{152.} 2.~\textit{Pendulum Experiments}.---Since
\[
  t = \pi\sqrt{\frac{l}{g}}\ \text{(Physics, p.~72)},\
  g = \frac{\pi^{2}l}{t^{2}};
\]
we can therefore measure the variations of the force of gravity, $g$,
at different parts of the earth, either by taking a pendulum of invariable
length and determining $t$, the time of its vibration; or by
measuring the length, $l$, of a pendulum which will vibrate seconds.
Extensive surveys of this sort have been made, and are still in progress,
and it is found that the \textit{force of gravity at the pole exceeds that at
the equator by about $\frac{1}{190}$ part}. In other words, a person who weighs
190 pounds at the equator (\textit{by a spring balance}) would, if carried to
the pole, show 191 pounds by the same balance.

\begin{fineprint}
The apparatus most used at present for the purpose of measuring the
force of gravity is a modification of the so-called Kater's pendulum. The
pendulum itself usually consists of a brass tube about an inch in diameter
and about four feet long, carrying a ball three or four inches in diameter at
each end, \textit{both balls being exactly of the same size, but one solid while the other
is hollow}. Two knife edges are inserted through the rod at right angles,
one near the heavy ball and the other at just the same distance from the
lighter one, and the weights and dimensions of the apparatus are so adjusted
that the \textit{time of vibration will be very approximately the same whether the pendulum
is swung heavy end up or light end up, and will be not far from one second}.
The distance between the knife edges will then, according to the theory of
the pendulum, be very nearly equal to the length of a simple pendulum
vibrating in the same time; and the small difference can be accurately calculated
when we know the exact time of vibration, each end up. The
knife edges swing on agate planes which are fastened upon a firm support;
%% -----File: 115.png---Folio 104-------
and great pains must be taken to have the support really firm. Professor
Peirce of our Coast Survey a few years ago detected important errors in a
majority of the earlier pendulum observations, due to insufficient care in
this respect.

\nbarticle{153.} The observations consist in comparing the pendulum with a clock,
either by noting the ``\textit{coincidences},'' or by an electrical record automatically
made on a chronograph. A pin attached to the end of the pendulum
touches a globule of mercury (which is momentarily raised for the purpose
once in eight or ten minutes), and so records the swing upon the chronograph
sheet. The observations need to be carefully corrected for \textit{temperature}
(which, of course, affects the distance between the knife edges), for the
\textit{length of arc} through which the pendulum is swinging, and for the \textit{resistance
of the air}. The observations determine the ``\textit{force of gravity}'' (French
``\textit{pesanteur}'') at the station. This ``force of gravity,'' however, thus determined,
is not simply the earth's \textit{attraction}, but includes also the effects of the
centrifugal force, due to the earth's rotation, which we must consider and
allow for.
\end{fineprint}

\nbarticle{154.}
At the equator the centrifugal force acts vertically in direct
opposition to gravity, and is given by the well-known formula
\[
C=\frac{V^2}{R}
\]
(see Physics, p.~62), in which $V$ is the velocity of the earth's surface
at the equator, and $R$ the earth's radius. Since $V$ is equal to
the earth's circumference divided by the number of seconds in a
sidereal day, we have
\[
V=\frac{2\pi R}{t}, \text{ and } C = \frac{4\pi^2 R}{t^2}.
\]
\includegraphicsouter{illo052}{\textsc{Fig.~52.}\\
The Earth's Centrifugal Force.}
Now $R$, the radius of the earth, equals 20,926,000 feet; and $t$ equals
86,164 mean-time seconds. $C$, therefore, comes out to 0.111 feet, which
is $\frac{1}{289}$ of $g$, $g$ being $32\frac{1}{6}$ feet.

\begin{fineprint}
We may remark in passing that if the rate of rotation were seventeen
times as great, $C$ would be $17^2$, or 289 times greater than now, and would
equal gravity; so that on that supposition bodies at the equator would weigh
absolutely nothing, and any greater velocity of rotation would send them
flying.
\end{fineprint}

At any other latitude, since $MN = OQ\cos{MOQ}$,\footnote
  {This is not \textit{exact}, since $MN$ in an oblate spheroid is less than $OQ×\cos{MOQ}$;
  but the difference is unimportant in the case of the earth.}
the centrifugal
%% -----File: 116.png---Folio 105-------
force, $c$, equals $C \cos {\phi}$, acting at right angles to the axis of the earth
and parallel to the plane of the equator. Now, this centrifugal
force $c$ is not \textit{wholly} effective in diminishing
the weight of a body, but only
that portion of it ($MR$ in \figref{illo052}{Fig.~52}) which
is directed vertically. $c$ is $MT$ in the
\figref{illo052}{figure}, and $MR$ is equal to $c$ multiplied
by the cosine of $\phi$, which finally gives
us $C×\cos^2\phi$ for the amount by which
the centrifugal force diminishes gravity
at a station whose latitude is $\phi$.

Every observation, therefore, of the
``force of gravity,'' obtained by the pendulum, needs to be increased
by the quantity
\[
\frac{g}{289}×\cos^2{\phi},
\]
in order to get the real value of the earth's \textit{gravitational attraction}
at the point of observation.

\begin{fineprint}
The other component of $c$ (viz.~$MS$) acts at right angles to gravity and
parallel to the earth's surface, and is given by the formula
\[
C\cos{\phi} \sin{\phi} = \tfrac{1}{2}C\sin{2\phi}.
\]
The direction of still water is determined by the resultant of the earth's
attraction combined with this deflecting force acting towards the equator;
so that this surface is not perpendicular to a line drawn towards the centre
of the earth anywhere excepting at the equator and the poles.
\end{fineprint}

\nbarticle{155.} Having a series of pendulum observations, we can then form
a table showing the force of gravity at each station; and correcting
this by adding the amount of the centrifugal force at each place, we
shall have the force of the earth's attraction. This is greater the
nearer each station is to the centre of the earth; but unfortunately
there is no simple relation connecting the force with the distance. The
attraction depends not only on the distance from the centre of the earth,
but also upon the form of the earth and the constitution of its interior,
and the arrangement of its strata of different density. We may safely
assume, however, that the earth is made up \textit{concentrically}, so to speak;
the strata of equal density being arranged like the coats of an onion.
On this hypothesis Clairaut, in 1742, demonstrated the relation given
below, which is always referred to as Clairaut's equation.
%% -----File: 117.png---Folio 106-------

Let $w$ be the loss of weight between the equator and the pole, and
$C$ the centrifugal force at the planet's equator, both being expressed
as fractions of the equatorial force of gravity, and let $d$ be the
ellipticity of the planet.

Then, as Clairaut proved,
\begin{flalign*}
&&&                d + w = 2\tfrac{1}{2}× C;  &&\\
&\text{whence }&&  d = 2\tfrac{1}{2}C-w.  &&\phantom{whence }
\end{flalign*}

In the case of the earth we have
\begin{flalign*}
&&&   d = 2\tfrac{1}{2}×\frac{1}{289}-\frac{1}{190},  &&\\
&\text{which gives }&&  d = \frac{1}{292.8}.  &&\phantom{which gives }
\end{flalign*}

Considering all the data, the most that can safely be said as to $d$ is
that it lies between the fractions $\frac{1}{290}$ and $\frac{1}{295}$. (Clarke's later values
for $d$ are larger than that adopted by the Coast Survey.)

\nbarticle{156.}
\nbparatext{Astronomical, Geographical, and Geocentric Lati\-tudes.}---The
astronomical latitude of a place has been defined as \textit{the elevation of
the pole}, or, what comes to the same thing, it is \textit{the angle between the
plane of the equator and the direction of gravity} at that place, however
that direction may be affected by local causes.

The \textit{geocentric} latitude, on the other hand, is the angle made at
the centre of the earth (as the word implies)
between the plane of the equator and a line
drawn from the observer to the centre of
the earth, which line of course does not
coincide with the direction of gravity, since
the earth is not spherical.

The \textit{geographical} or \textit{geodetic} latitude of
a station is the angle formed with the plane
of the equator by a line drawn from the
station \textit{perpendicular to the surface of the
standard spheroid}.

\begin{fineprint}

\includegraphicsouter{illo053}{\textsc{Fig.~53.}\\
Astronomical and Geocentric
Latitude.}

If the earth's surface were \textit{strictly spher\-oidal, and there were no local variations
of gravity}, the astronomical latitude and the geographical latitude
would coincide---and they never differ greatly; but the geocentric latitude
differs from them by a very considerable quantity---as much as $11'$ in latitude
$45°$. The geocentric latitude is but little used except in certain astronomical
calculations where parallax is involved.
%% -----File: 118.png---Folio 107-------

In \figref{illo053}{Fig.~53}, the angle $MOQ$ is the geocentric latitude of $M$, while $MNQ$ is
the geographical latitude. $MNQ$ is also the astronomical latitude, unless
there is some local disturbance of the direction of gravity. The angle $OMN$,
which is the difference between the geocentric and astronomical latitudes, is
called ``\textit{the angle of the vertical}.''
\end{fineprint}

\nbarticle{157.} It will be noticed that the astronomical latitude of a place is
the only one of these three latitudes which is \textit{determined directly by
observation}. In order to know the \textit{geocentric} and \textit{geographical} latitudes
of a place, we must know the form and dimensions of the earth,
which are ascertained only by the help of observations made elsewhere.

The geocentric degrees are longer near the equator than near the
poles, and it is worth noticing that if we form a table giving the length
of each degree of \textit{geographical} latitude from the equator to the pole,
the same table, read \textit{backwards}, gives the length of \textit{geocentric} degrees.

Since the earth is ellipsoidal instead of spherical, it is evident that
lines of ``level'' on the earth's surface are affected by the earth's rotation.
If this rotation were to cease, the direction of gravity would
be so much changed that the Gulf of Mexico would run up the Mississippi
River, because the distance from the centre of the earth to
the head of the river is less by some thousands of feet than the
distance from the mouth of the river to the centre of the earth.

\begin{fineprint}
\nbarticle{158.} \nbparatext{Station Errors.}---The irregularities in the direction of gravity
are by no means insensible as compared with the accuracy of modern astronomical
observation, and the difference between the astronomical latitude
and longitude of a place and the geographical latitude and longitude of the
same place constitute what is called the ``\textit{station error}.'' In the eastern part
of the United States these station errors, according to the Coast Survey
observations, average about $1\frac{1}{2}''$. Errors of from $4''$ to $6''$ are not uncommon,
and in mountainous countries, as for instance in the Caucasus and in
Northern India, these errors occasionally amount to $30''$ or $40''$. They are
not ``errors'' in the sense that the astronomical latitude of the place has not
been determined correctly, but are merely the effects of the irregular distribution
of matter in the crust of the earth in altering the direction of gravity.
Pendulum observations show local variations in the \textit{force} of gravity quite
proportional to the deviations which the station-errors show in its \textit{direction}.
\end{fineprint}

\section*{IV.}

\nbarticle{159.} \nbparatext{The Earth's Mass and Density.}---The `\textit{mass}' of a body is the
\textit{quantity of matter} that it contains, the unit of mass being the quantity
of matter contained in a certain arbitrary body which is taken as a
standard. For instance, a ``kilogram'' is the quantity of matter
%% -----File: 119.png---Folio 108-------
contained in the block of platinum preserved at Paris as the standard
of mass.\footnote
  {This was meant to be just equal to the mass contained in a cubic decimeter
  of water at its maximum density, and is so very nearly indeed.}
A pound is similarly defined by reference to the prototypes
at Washington and London.

Two masses of matter are defined as equal which \textit{require the same
expenditure of energy to give them the same velocity}; or \textit{vice versa,
those are equal which, when they have the same velocity, possess the
same energy, and, in giving up their motion and coming to rest, do
the same amount of work}.

\begin{fineprint}
Masses can therefore be compared \textit{by placing them in the same field of
force and comparing the energies developed in them when they have moved equal
distances under the action of the force}. This method, however, is seldom
convenient.
\end{fineprint}

\nbarticle{160.} \nbparatext{Proportionality of Mass to Weight.}---Newton showed by his
experiments with pendulums of different substances, that at any
given point the attraction of the earth for a body of any kind of
matter is proportional to the mass of that body; the attraction being
measured as a pull or ``stress'' in this case, and called ``\textit{the weight}''
of the body. In other and more common language, \textit{the mass of a
body is proportional to its weight} (we must not say it \textit{is} its weight),
provided the weighing of the bodies thus compared is done, in cases
where scientific accuracy is essential, at the same place on the earth's
surface. Practically, therefore, we \textit{usually measure the masses of
bodies by simply weighing them}. It is to be carefully observed, however,
that the words ``kilogram,'' ``pound,'' ``ton,'' etc., have also
a secondary meaning, as denoting units of pull and push,---of
``\textit{stress}'' speaking strictly and technically,---or of ``force,'' as that
much abused word is very generally used.

\begin{fineprint}
It is, from a literary point of view, just as proper to speak of a \textit{stress or a
pull} of a hundred pounds\footnote
  {Of course the student will remember that we have a unit of stress,---the
  \textit{dyne},---which is wholly free from this objection of ambiguity.}
as of a \textit{mass} of a hundred pounds, but the word
``pound'' means an entirely different thing in the two cases. At the surface
of the earth the relation between the ideas, however, is so close that
the way in which the ambiguity came about is perfectly obvious, and it is
hardly probable that language will ever change so as to remove it. To a
certain extent it is admittedly unfortunate, and the student must always be
on his guard against it. At the earth's surface a \textit{mass} of 100 pounds always
%% -----File: 120.png---Folio 109-------
``\textit{weighs}'' very nearly 100 pounds; but, to anticipate slightly, at an elevation
of 4000 miles above the surface, the same mass would ``weigh'' only 25
pounds; at the distance of the moon about half an ounce; while on the
surface of the sun it would ``weigh'' nearly 2800 pounds.
\end{fineprint}

\nbarticle{161.} \nbparatext{Gravity.}---The law of gravitation discovered by Newton declares
that \textit{any particle of matter attracts any other particle with a force}
(``\textit{stress},'' if the bodies are prevented from moving) \textit{proportional
inversely to the square of the distance between them, and directly to the
product of their masses;} or, as a formula, we may write,
\[
F = k\frac{M_{1}× M_{2}}{d^{2}},
\]
in which $M_{1}$ and $M_{2}$ are the two masses, and $d$ the distance between
them, while $k$ is a constant numerical factor depending upon the units
employed.\footnote
  {It will not do to write the formula
\[
F = \frac{M_{1}× M_{2}}{d^{2}}
\]
(omitting the $k$), unless the units are so chosen that the unit of force shall be
equal to the attraction between two masses each of one unit, at a distance of one
\textit{unit}. It is not true that the attraction between two particles, each having a
mass of one \textit{pound}, at a distance of \textit{one foot}, is equal to a stress of \textit{one pound} (of
force), as would rather naturally be inferred if we should write the equation
without the constant factor.}

\begin{fineprint}
We must not imagine the word ``\textit{attract}'' to mean too much. It merely
states the fact that there is a \textit{tendency} for the bodies to move toward each
other, without including or implying any explanation of the fact. So far,
no explanation has appeared which is less difficult to comprehend than
the fact itself. Whether bodies are \textit{drawn} together by some outside action,
or \textit{pushed} together; or whether they themselves can act across space with
mathematical intelligence,---in what way it is that ``attraction'' comes
about, is still unknown,---apparently as inscrutable as the very nature and
constitution of an atom of matter itself; \textit{it is simply a fundamental fact}.
\end{fineprint}

\nbarticle{162.} When the distance between attracting bodies is large as compared
with their own magnitude, then reckoning the distance between
their centres of mass as their true distance, the formula is sensibly
true for them as it would be for mere particles. When, however, the
distance is not thus great, the calculation of the attraction becomes a
very serious problem, involving what is known as a ``double integration.''
%% -----File: 121.png---Folio 110-------
We must find the attraction of each particle of the first body
upon each particle of the other body, and take the sum of all these
infinitesimal stresses. Newton, however, showed that if the \textit{bodies are
spheres, either homogeneous or of concentric structure, then they attract
and are attracted precisely as if the matter in them were wholly collected
at their centres}. The earth, for instance, attracts a body at its surface
very nearly as if it were all collected at its own centre, 4000
miles distant; not exactly so, because the earth is not strictly
spherical; but in what follows we shall neglect this slight inaccuracy.

\nbarticle{163.}
In order, then, to find the mass of the earth in kilograms,
pounds, or tons, we must find some means of accurately comparing its
attraction for some object on its own surface with the attraction of the
same object by some body of known mass, at a measured distance.
The difficulty lies in the fact that the attraction produced by any body,
not too large to be handled conveniently, is so excessively small that
only the most delicate operations serve to detect and measure it.

The first successful attack upon the problem was made in 1774 by
Maskelyne, the Astronomer Royal, by means of what is now usually
referred to as,---

\nbarticle{164.} 1.~``\textsc{The Mountain Method},'' because, in fact, the earth
in this operation is weighed against a mountain.

\includegraphicsouter{illo054}{\textsc{Fig.~54.}\\
The Mountain Method of Determining the
Earth's Density.}

Two stations were chosen on the same meridian, one north and one
south of the mountain Schehallien, in Scotland. In the first place, a
careful topographical survey was
made of the whole region, giving the
precise distance between the stations,
as well as the exact dimensions
of the mountain, which is a
``hog-back'' of very regular contour.
From the known dimensions of the
earth and the measured distance,
the difference of the \textit{geographical} latitudes of the two places $M$ and $N$
(\figref{illo054}{Fig.~54})
can be accurately computed; \textit{i.e.}, the angle which the plumb
lines at $M$ and $N$ would have made if there were no mountain there.

\includegraphicsouter[8]{illo055}{\textsc{Fig.~55.}}

In this case it was $41''$. The next operation was to observe the
\textit{astronomical} latitude at each station. This astronomical difference of
latitude, \textit{i.e.}, the angle which the plumb lines actually do make, was
found to be $53''$, the plumb lines at $M$ and $N$ being drawn inward
out of their normal position by the attraction of the mountain to the
%% -----File: 122.png---Folio 111-------
extent of $6''$ on each side; so that the astronomical difference of
latitude was increased by $12''$ over the geographical.

\begin{fineprint}
Now, in such a case the ratio of gravity to the deflecting force,
according to the laws of the composition of forces, is that of
$aM$ to $aA'$ in the figure (\figref{illo055}{Fig.~55}), or the ratio of 1 to the tangent
of the deflection, $\delta$; that is, calling the deflecting force $f$,
we have $\dfrac{g}{f}=\cot{\delta},=\cot{6}''$ in this case.

By the law of gravitation, the earth's attracting force at its
surface is given by the formula
\begin{align*}
g&=k\frac{E}{R^{2}},\\
\intertext{where $E$ is the mass of the earth (the unknown quantity of our problem),
and $R$ its radius, 4000 miles. Similarly, if $C$ in the \figref{illo055}{figure} is the centre
of attraction of the mountain, we have}
f&=k\frac{m}{d^{2}},\\
\intertext{$m$ being the mass of the mountain, and $d$ the distance from $C$ to the station.
Combining this with the preceding, we get}
\frac{E}{m} &= \left(\frac{g}{f}\right) \left(\frac{R}{d}\right)^{2},\\
\intertext{or, in this case,}
\qquad\qquad %compensate for illustration on right hand side
\frac{E}{m} &= \cot{6''} \left(\frac{R}{d}\right)^{2}.
\end{align*}
\end{fineprint}

We thus get the \textit{ratio of the earth's mass to that of the mountain};
and provided we can find the mass of the mountain in tons
or any other known unit of mass, the problem will be completely
solved. By a careful geological survey of the mountain, with deep
borings into its strata, the mass of the mountain was determined as
accurately as it could be (though here is the weakest point of the
method), and thus the \textit{mass} of the earth was finally computed.

Now, knowing the diameter of the earth, its volume in cubic feet is
easily found, and from the volume and the known number of mass-pounds,
($62\frac{1}{2}$ nearly) in a cubic foot of water, the weight the earth
would have, if composed of water, follows. Comparing this with the
mass actually found, we get the density, which in this experiment came
out 4.71.

A repetition of the work in 1832 at Arthur's Seat, near Edinburgh,
gave 5.32.
%% -----File: 123.png---Folio 112-------

\includegraphicsouter{illo056}{\textsc{Fig.~56.}---Plan of the Torsion Balance.}

\nbarticle{165.} 2.~Much more trustworthy results, however, are obtained by
the method of the \textsc{torsion balance}, first devised by Michell, but first
employed by Cavendish in 1798. A light rod, carrying two small
balls at its extremities, is suspended horizontally at its centre by a
long fine metallic wire. If it be allowed
to come to rest, and then a very
slight deflecting force be applied, the
rod will be pulled out of position by
an amount depending on the stiffness
and length of the wire, as well as
the force itself. When the deflecting
force is removed, the rod will vibrate
back and forth until brought to rest
by the resistance of the air. The
``\textit{torsional coefficient},'' as it is called
(\textit{i.e.}, the stress corresponding to a torsion
of one revolution), can be accurately
determined by observing the time
of vibration when the dimensions and
weight of the rod and balls are known.
If, now, two large balls $A$ and $B$ are
brought near the smaller ones, as in
Fig.~56, a deflection will be produced by their attraction, and the
small balls will move from $a$ and $b$ to $a'$ and $b'$. By shifting the
large balls to the other side at $A'$ and $B'$, we get an equal deflection
in the opposite direction, \textit{i.e.}, to $a''$ and $b''$, and the difference between
the two positions assumed by the small balls, \textit{i.e.}, $a'a''$ and
$b'b''$, will be twice the deflection.

\begin{fineprint}
It is not necessary, nor even best, to wait for the balls to come to rest.
We note the extremities of their swing. The middle point of the swing
gives the point of rest, and the time occupied by the swing is the time of
vibration, which we need in determining the coefficient of torsion. We
must also measure accurately the distance, $Aa'$ and $Bb'$ between the centre
of each of the large balls and the point of rest of the small ball when
deflected.
\end{fineprint}

The \textit{earth's attraction} on each of the small balls of course equals
\textit{the ball's weight}. The \textit{attractive force of the large ball} on the small one
near it is found directly from the experiment. If the deflection, for
instance, is $1°$ and the coefficient of torsion is such that it takes
\textit{one grain} to twist the wire around one whole revolution, then the
%% -----File: 124.png---Folio 113-------
deflecting force, which we will call $f$ as before, will be $\frac{1}{360}$ of a grain.
Call the mass of the large ball $B$, and let $d$ be the measured distance
from its centre to that of the deflected ball. We shall then have
\begin{align*}
f &= k\frac{B}{d^{2}},\\
\intertext{also, $w$ being the weight of the small ball,}
w &= k\frac{E}{R^{2}},\\
\intertext{whence we get, very much as in the preceding case,}
\frac{E}{B} &= \frac{w}{f}\left(\frac{R}{d}\right)^{2} \!\! ,
\end{align*}
which gives the mass of the earth in terms of $B$.

The method differs from the preceding in that we use a large ball
of metal instead of a mountain, and measure its deflecting force by a
laboratory experiment instead of comparing astronomical observations
with geodetic measurements.

\begin{fineprint}
\nbarticle{166.} In the earlier experiments by this method the small balls were
of lead, about two inches in diameter, at the extremities of a light wooden
rod, five or six feet long, enclosed in a case with glass ends, and their position
and vibration was observed by a telescope looking directly at them from
a distance of several feet. The attracting masses, $B$, were balls also of lead,
about one foot in diameter, mounted on a frame pivoted in such a way
that they could be easily brought to the required positions.

Great difficulty was caused by air currents in the case, and it was necessary
to enclose the whole apparatus in a small room of its own which was
covered with tin-foil on the outside, and to avoid going near the room or
allowing any radiant heat to strike it for hours before the observations.
Baily, in England, and Reich, in Germany, between 1838 and 1842, made
very extensive series of observations of this kind. Baily obtained 5.66
for the earth's density, and Reich 5.48.

The experiment was repeated in 1872 by Cornu, in Paris, with a modified
apparatus.

The horizontal bar was in this case only half a metre long, of aluminium,
with small platinum balls at the end. For the large balls, glass globes were
used, which could be pumped full of mercury or emptied at pleasure. The
whole was enclosed in an air-tight case, and the air exhausted by an air-pump.
The deflections and vibrations were observed by means of a telescope
watching the image of a scale reflected in a small mirror attached to
the aluminium beam near its centre, according to the method first devised
%% -----File: 125.png---Folio 114-------
by Gauss and now so generally used in galvanometers and similar apparatus.
Cornu obtained 5.56 as the result, and showed that Baily's figure required a
correction which, when applied, would reduce it to 5.55.
\end{fineprint}

\nbarticle{167.} 3.~\textsc{Potsdam Observations}.---During 1886 and 1887 another
series of observations was made by Wilsing, at Potsdam, with apparatus
similar in principle to the torsion balance, except that the bar carrying
the balls to be attracted was \textit{vertical}, and turned on knife edges
very near its centre of gravity. The knife edges, like those of an
ordinary balance, rested upon agate planes, and the centre of gravity
of the apparatus was so adjusted that one vibration of the pendulum,
under the influence of gravity alone, would occupy from two to four
minutes. The deflecting weights in this case were large cylinders of
cast iron, suspended in such a way that they could be brought opposite
the small balls, first on one side and then on the other. The
whole was set up in a basement, and carefully and very effectually
guarded against all changes of temperature, the arrangements being
such that all manipulations and observations could be effected from
the outside without entering the room. The deflections and vibrations
were observed by a reflected scale, as in Cornu's observations.
The result obtained was 5.59.

Several other methods have been used; of less scientific value,
however.

\begin{fineprint}
\nbarticle{168.}  \textit{a}. The mass of the earth can he deduced by ascertaining \textit{the force
of gravity at the top of a mountain} and \textit{at its base, by means of pendulum experiments}.
The mass of the mountain must be determined by a survey, just as
in the Schehallien method, which makes the method unsatisfactory. At the
top of a mountain the height of which is $h$, and the distance of its centre of
attraction from the top is $d$, gravity will be made up of two parts, one the
attraction of the earth at a distance from its centre equal to $R + h$, and the
other the attraction of the mountain alone considered. Calling the mass of
the mountain $m$, and gravity at its summit $g'$ ($g$ being the force of gravity at
the earth's surface), we shall have the proportion
\[
g:g' = \frac{E}{R^{2}}:\left[\frac{E}{(R+h)^{2}}+\frac{m}{d^{2}}\right],
\]
the second fraction in the last term of the proportion being the attraction
of the mountain. When $g$ and $g'$ are ascertained by the pendulum experiments,
$E$ remains as the only unknown quantity, and can be readily found. Observations
of this kind were made by Carlini, in 1821, at Mt.~Cenis, and
the result was 4.95.
%% -----File: 126.png---Folio 115-------

\nbarticle{169.} \textit{b. By means of pendulum observations at the earth's surface compared
with those at the bottom of a mine of known depth}. This method was employed
by Airy in 1843, at Harton Colliery, 1200 feet deep; result, 6.56. In this case
the principle involved is somewhat different. \textit{At any point within a hollow,
homogeneous, spherical shell, gravity is zero}, as Newton has shown. The
attraction balances in all directions. If, then, we go down into a mine,
the effect on gravity is the same as if a shell composed of all that part of the
earth above our level had been removed. At the same time our distance
from the earth's centre has been decreased by $d$, the depth of the mine.
\begin{flalign*}
&\text{At the surface }&
  g  &= k\frac{E}{R^{2}},\text{ as before.}  &&
\\[1ex]
&\text{At the bottom of the mine }&
  g' &= k\frac{E - \text{``shell''}}{(R - d)^{2}}.
&&\phantom{\text{At the bottom of t}}
\end{flalign*}

Comparing the two equations, we find $E$ in the terms of the shell, since
the ratio of $g$ to $g'$ is given by pendulum observations. Obviously, however,
the mass of the ``shell'' is difficult to determine with accuracy. And it is by
no means homogeneous, so that there is no great reason for surprise at the
discordant result.  $g'$ was found to be actually greater than $g$, showing that
although at the centre of the earth the attraction necessarily becomes zero,
yet as \textit{we descend below the surface, gravity increases for a time} down to some
unknown but probably not very great depth, where it becomes a maximum.

\nbarticle{170.}  \textit{c}. By \textit{experiments with a common balance}. If a body be hung from
one of the scale-pans of a balance, its apparent weight will obviously be
increased when a large body is brought very near it underneath; and this
increase can be measured. Poynting in England and Jolly in Germany
have recently used this method, and have obtained results agreeing very
fairly with these got from the torsion balance. The experiment, with some
modifications, is soon to be tried again on a very large scale in Germany.
\end{fineprint}

\nbarticle{171.} \nbparatext{Constitution of the Earth's Interior.}---Since the average density
of the earth's crust does not exceed three times that of water,
while the mean density of the whole earth is about 5.58 (taking the
average of all the most trustworthy results), it is obvious that at the
centre the density must be very much greater than at the surface,---very
likely as high as eight or ten times that of water, and equal to
the density of the heavier metals. There is nothing in this that might
not have been expected. If the earth were ever fluid, it is natural to
suppose that in the solidification the densest materials would settle
towards the interior.

\begin{fineprint}
Whether the interior of the earth is solid or fluid it is difficult to say with
certainty. Certain tidal phenomena, to be discussed hereafter, have led Sir
%% -----File: 127.png---Folio 116-------
William Thomson and the younger Darwin to conclude that the earth as a
whole is solid throughout, and ``more rigid than glass,'' volcanic centres
being mere pustules in the general mass. To this many geologists demur.

As regards the temperature at the earth's centre, it is hardly an astronomical
question, though it has very important astronomical relations. We can only
take space to say that the temperature appears to increase from the surface
downward at the rate of about one degree Fahrenheit for every fifty or sixty
feet, so that at the depth of a few miles the temperature must be very
high.
\end{fineprint}
\chelabel{CHAPTERV}
%% -----File: 128.png---Folio 117-------

\Chapter{VI}{The Earth's Orbital Motion}
\nbchapterhang{THE APPARENT MOTION OF THE SUN AMONG THE STARS,
AND THE EARTH'S ORBITAL MOTION.---THE EQUATION
OF TIME, PRECESSION, NUTATION, AND ABERRATION.---VARIOUS
KINDS OF ``YEAR.''---THE CALENDAR.}

\nbarticle{172.} \nbparatext{The Annual Motion of the Sun.}---The apparent \textit{annual motion
of the sun} must have been one of the earliest noticed of all astronomical
phenomena. Its discovery antedates history.

As seen by the people in Europe and Asia, the sun, starting in
the spring, mounts higher in the sky each day at noon for three
months, appears to stand still for a few days at the summer solstice,
and then descends towards the south, reaching in the autumn
the same noonday elevation it had in the spring. It keeps on its
southward course to a winter solstice in December, and then returns
to its original height at the end of the year, marking and causing the
seasons by its course. A year, the interval between the successive
returns of the sun to the same position, was very early found to
consist of a little more than three hundred and sixty days.

Nor is this all. The sun's motion is \textit{not merely a north-and-south
motion}, but it also moves \textit{eastward among the stars}; for in the
spring the stars which are rising in the eastern horizon at sunset are
different from these which are found there in the summer or winter.
In the spring, the most conspicuous of the eastern constellations at
sunset are Leo and Boötes; a little later, Virgo appears; in the summer,
Ophiuchus and Libra; still later, Scorpio; and in mid-winter,
Orion and Taurus are in the eastern sky.

\nbarticle{173.} So far as mere appearances go, everything would be explained
by assuming that the earth is at rest and the sun moving
around it; but equally by the converse supposition,---for if the earth
as seen from the sun appears at any point in the heavens, the sun as
seen from the earth must appear in exactly the opposite point, and
must keep opposite, moving through the same path in the sky (but
%% -----File: 129.png---Folio 118-------
six months behind), and always in the same ``angular direction,'' if
we may use the expression. (Just as two opposite teeth on a gear-wheel
move in the same \textit{angular direction}, though at any moment
they are moving in opposite \textit{linear} directions.)

\nbarticle{174.} That it is really the earth which moves, and not the sun, is
absolutely demonstrated by two phenomena, too minute and delicate
for pre-telescopic observations, but accessible enough to modern
methods. We can only mention them here, leaving their fuller discussion
for the present. One of them is \textit{the aberration of light}, the
other \textit{the annual parallax of the fixed stars}. These can be explained
only by the actual motion of the earth.

\nbarticle{175.} \nbparatext{The Ecliptic.}---By observing with a meridian circle daily
the declination of the sun, and the difference between its right
ascension and that of some star (Flamsteed used $\alpha$~Aquilę for the
purpose), we shall obtain a series of positions of the sun's centre
which can be plotted on a celestial globe; and we can thus make out
the path of the sun among the stars, and find the place where it
cuts the celestial equator, and the angle it makes. This path turns
out to be a \textit{great circle}, as is shown by its cutting the equator at two
points just $180°$ apart (the so-called equinoctial points or equinoxes),
and makes an angle with it of approximately $23\frac{1}{2}°$. This great circle
is called the \textsc{Ecliptic}, because, as was early discovered, eclipses
happen only when the moon is crossing it. It may be defined as \textit{the
trace of the plane of the earth's orbit upon the celestial sphere}, just as
the celestial equator is the trace of the plane of the terrestrial equator
on the same sphere.

\nbarticle{176.} \nbparatext{Definitions.}---The angle which the ecliptic makes with the
equator is called the \textit{Obliquity of the ecliptic}, and the points midway
between the equinoxes are called the \textit{Solstices} (\textit{sol-stitium}), because at
these points the sun ``\textit{stands}'' or stops moving in declination for a
short time.

Two circles parallel to the equator, drawn through the solstices, are
called the \textit{Tropics} (Greek $\tau\rho\acute{\epsilon}\pi\omega$), or ``\textit{turning-lines},'' because there
the sun turns from its northward motion to a southward, or \textit{vice versa}.
The obliquity is, of course, simply equal to the \textit{sun's maximum declination},
or greatest distance from the equator, which is reached in
June and December.
%% -----File: 130.png---Folio 119-------

\begin{fineprint}
The ancients were accustomed to determine it by means of the gno\-mon\footnote
  {The Chinese claim to have made an observation of this kind about 4000~\textsc{b.c.},
and the result given is very nearly what it should have been at that time. (The
obliquity changes slightly in centuries.) If their observation is genuine, it is
  by far the oldest of all astronomical records.}
(\artref{Art.}{107}). The length of the shadow at noon on the solstitial days determines
the zenith distance of the sun on these days, and the difference of the
zenith distances at the two solstices is twice the angle desired. The gnomon
also determined for the ancients the length of the year, it being only necessary
to observe the interval between days in the spring or autumn, when the
shadow had the same length at noon.
\end{fineprint}

\nbarticle{177.} \nbparatext{The Zodiac and its Signs.}---A belt $16°$ wide, $8°$ on each side
of the ecliptic, is called the \textit{Zodiac}. The name is said to be derived
from $\zeta\acute{\omega}o\nu$, a living creature, because the constellations in it (except
Libra) are all figures of animals. It was taken of that particular
width by the ancients simply because the moon and the then known
planets never go further than $8°$ from the ecliptic.

\begin{fineprint}
This belt is divided into the so-called \textsc{signs}, each $30°$ in length, having
the following names and symbols:---
\[
\begin{array}{l@{\;}ll@{\;}l}
\text{Spring} &\left\{
 \begin{array}{ll}
    \text{Aries,}  & \aries  \\
    \text{Taurus,} & \taurus \\
    \text{Gemini,} & \gemini \\
 \end{array}
\right.
%
&\text{Autumn} &\left\{
 \begin{array}{ll}
    \text{Libra,}                                    & \libra \\
    \text{Scorpio,}                                  & \scorpio  \\
    \rlap{Sagittarius,}\phantom{\text{Capricornus,}} & \sagittarius \\ %phantom text to ensure width matches lower half
 \end{array}
\right.
\\
\text{Summer} &\left\{
 \begin{array}{ll}
    \rlap{Cancer,}\phantom{\text{Gemini,}} & \cancer \\ %phantom text to ensure width matches upper half
    \text{Leo,}                            & \leo    \\
    \text{Virgo,}                          & \virgo  \\
 \end{array}
\right.
%
&\text{Winter} &\left\{
 \begin{array}{ll}
    \text{Capricornus,} & \capricornus \\
    \text{Aquarius},    & \aquarius    \\
    \text{Pisces},      & \pisces      \\
 \end{array}
\right.
\end{array}
\]

The symbols are for the most part conventionalized pictures of the objects.
The symbol for Aquarius is the Egyptian character for water. The
origin of the signs for Leo, Virgo, and Capricornus is not quite clear. It
has been suggested that $\leo$ is simply a ``\textit{cursive}'' form for $\Lambda$, the initial of
$\Lambda\acute{\epsilon}\omega\nu$; $\virgo$ for $\Pi\alpha\rho$ ($\Pi\alpha\rho\Theta\acute{\epsilon}\nu o\varsigma$), and $\capricornus$ for T$\rho$ (T$\rho\acute{\alpha}\gamma o\varsigma$).
\end{fineprint}

\section*{CELESTIAL LATITUDE AND LONGITUDE.}

\nbarticle{178.} Since the moon and all the principal planets always keep
within the zodiac, the ecliptic is a very convenient circle of reference,
and was used us such by the ancients. Indeed, until the invention
of pendulum clocks, it was on the whole more convenient than the
equator, and more used.

The two points in the heavens $90°$ distant from the ecliptic are called
the \textit{Poles of the ecliptic}. The northern one is in the constellation
%% -----File: 131.png---Folio 120-------
of Draco, about half-way between the stars $\delta$ and $\zeta$~Draconis. Now,
suppose a set of great circles drawn, like meridians, through these
poles of the ecliptic, and hence perpendicular to that circle; these
are \textit{Circles of latitude} or \textit{secondaries to the ecliptic}. The \textsc{Longitude}
of a star or any other heavenly body is, then, \textit{the angle made at the
pole of the ecliptic, between the circle of latitude, which passes through
the vernal equinox, and the circle of latitude passing through the body};
or, what comes to the same thing, it is \textit{the arc of the ecliptic included
between the vernal equinox and the foot of the circle of latitude passing
through the body}. Celestial longitude is always reckoned \textit{eastward}
from the vernal equinox, completely around the ecliptic, so that
the longitude of the sun when $10°$ \textit{west} of the vernal equinox would be
written as $350°$, and not as $-10°$.

The \textsc{Latitude} of a star is simply \textit{its distance north or south of
the ecliptic measured on the star's circle of latitude}.

\nbarticle{179.} It will be seen that \textit{longitude differs from right ascension in
being reckoned on the ecliptic instead of
on the equator}, nor can it be reckoned
in \textit{time}, but only in degrees, minutes,
and seconds. \textit{Latitude differs from declination
in that it is reckoned from the
ecliptic instead of from the equator}.

\includegraphicsouter{illo057}{\textsc{Fig.~57.}\\
Relation between Celestial Latitude and
Longitude, and Right Ascension and
Declination.}

The relation between right ascension
and declination on the one hand, and
longitude and latitude on the other,
may be made clearer by the accompanying
diagram (\figref{illo057}{Fig.~57}), in which
$EC$ is the ecliptic and $EQ$ the equator,
$E$ being the vernal equinox. $S$
being a star, its right ascension ($\alpha$) is
$ER$ and its declination ($\delta$) is $SR$; its longitude ($\lambda$) is $EL$, and its latitude
($\beta$) is $SL$. $P$ and $K$ are the poles of the equator and ecliptic
respectively, and the circle $KPCQ$ is the \textit{Solstitial Colure}, so called.

\begin{fineprint}
The student can hardly take too great care to avoid confusion of celestial
latitude and longitude with right ascension and declination or with \textit{terrestrial}
latitude and longitude. It is, of course, unfortunate that latitude in
the sky should not be analogous to latitude upon the earth, or celestial longitude
to terrestrial. The terms right ascension and declination are, however,
of comparatively recent introduction, and found the ground preoccupied,
celestial latitude and longitude being much older.
%% -----File: 132.png---Folio 121-------

\nbarticle{180.} \nbparatext{Conversion of $\lambda$ and $\beta$ into $\alpha$ and $\delta$, or Vice Versa.}---Right
ascension and declination can, of course, always be converted into longitude
and latitude by a trigonometrical calculation. We proceed as follows: In
the triangle $ERS$, right-angled at $R$, we have given $ER$ and $RS$  ($\alpha$ and $\delta$),
from which we find the hypothenuse $ES$ and the angle $RES$. Next in the
triangle $ELS$, right-angled at $L$, we have the hypothenuse $ES$ and the angle
$LES$, which is equal to $RES-LEQ$ ($LEQ$ being {$\omega$}, the obliquity of the
ecliptic). Hence we easily find $EL$ and $LS$.
\end{fineprint}

\nbarticle{181.} \nbparatext{The Earth's Orbit in Space.}---The \textit{ecliptic is not the earth's
orbit}, and must not be confounded with it. It is a \textit{great circle} of the
infinite celestial sphere, the \textit{trace} made upon the sphere by the plane
of the earth's orbit, as was stated in its definition. The fact that
it is a great circle gives us no information about the earth's orbit,
except that \textit{the orbit all lies in one plane passing through the sun}. It
tells us nothing as to its real form and size.

By reducing the observations of the sun's right ascension and
declination through the year to longitude and latitude (the latitude
will always be zero, of course, except for some slight perturbations)
and combining them with observations of the sun's apparent diameter,
we can, however, ascertain the real form of the earth's orbit and the
law of its motion in this orbit. But the \textit{size} of the orbit---the scale
of miles---cannot be fixed until we can find the sun's distance.

\includegraphicsouter{illo058}{\textsc{Fig.~58.}\\
Determination of the Form of the Earth's Orbit.}

\nbarticle{182.} \nbparatext{To find the Form of the Orbit}, we may proceed thus: Take
a point $S$ for the sun and draw
from it a line $SO$, \figref{illo058}{Fig.~58},
directed towards the vernal
equinox as the origin of longitudes.
Lay off from $S$ indefinite
lines, making angles with
$SO$ equal to the earth's longitude
on each of the days observed
through the year; \textit{i.e.},
the angle $OS$\:10, is the longitude
at the time of the 10th
observation; and so on. We
shall thus get a sort of ``spider,'' showing the \textit{directions}
as seen from the earth on these days.

Next, as to \textit{relative distances}. While the apparent diameter of the
sun does not tell us its real distance from the earth, unless we first
%% -----File: 133.png---Folio 122-------
know the sun's real diameter in miles, the changes in the apparent
diameter do inform us as to the \textit{relative} distance of the earth at
different times, since the nearer we are, the larger the sun appears,---the
distance being inversely proportional to the apparent diameter
(\artref{Art.}{6}). If, then, we lay off on the arms of our ``spider'' distances
inversely proportional to the number of seconds of arc in the
sun's measured diameter at each date, these distances will be \textit{proportional}
to the true distance of the earth from the sun, and the curve
joining the points thus obtained will be a true map of the earth's
orbit, though without any scale of
miles upon it.

\includegraphicsouter{illo059}{\textsc{Fig.~59.}---The Ellipse.}

When the operation is performed,
we find that the orbit is an ellipse
of small eccentricity (about one-sixtieth),
with the sun, not in the
centre, but at one focus.

\begin{fineprint}
\nbarticle{183.} For the benefit of any who
may not have studied conic sections
we define the ellipse. It is a curve such
that \textit{the sum of the two distances from any point on its circumference to
two points within, called the foci, is always constant}, and equal to what is
called the major-axis of the ellipse, $SP + PF = AA'$, in \figref{illo059}{Fig.~59}. $AC$ is
called the semi-major-axis, and is usually denoted by $A$ or $a$. $BC$ is the
semi-minor-axis, denoted by $B$ or $b$. The eccentricity, denoted by $e$, is the
fraction $\dfrac{SC}{AC}$.
\begin{flalign*}
&\text{Since $BS$ is equal to $A$,}&&
  SC = \sqrt{A^{2} - B^{2}};  &&\phantom{\text{Since $BS$ is equal to $A$,}}
\\[1ex]
&\text{and }&&
  e = \frac{\sqrt{A^{2} - B^{2}}}{A}.  &&
\end{flalign*}
\end{fineprint}

The points where the earth is nearest to and most remote from the
sun are called respectively \textit{perihelion} and \textit{aphelion} and the line
that joins them is, of course, the major axis of the orbit. This line, considered
as indefinitely produced in both directions, is called the \textit{line
of apsides},---the major-axis being a limited piece or ``sect'' of the
line of apsides.

\begin{fineprint}
\nbarticle{184.} The variations of the sun's diameter are too small to be detected
without a telescope (amounting to only about three per cent), so that
the ancients were unable to perceive them. Hipparchus, however, about
%% -----File: 134.png---Folio 123-------
150~\textsc{b.c.}, discovered that the earth is not in the centre of the circular orbit
which he supposed the sun to describe around it. Everybody assumed, on
\textit{ą priori} grounds, never disputed until the time of Kepler, that the sun's orbit
must be a circle and described with a uniform motion, because a circle is
the only ``perfect'' curve, and uniform motion the only perfect motion.
Obviously, however, the sun's \textit{apparent} motion is not uniform, because it
takes 186 days for the sun to pass from the vernal equinox to the autumnal
through the summer months, and only 179 days to return during the winter.
Hipparchus explained this difference by the hypothesis that the earth
is out of the centre of the sun's path.
\end{fineprint}

\nbarticle{185.} \nbparatext{To find the Eccentricity of the Orbit.}---Having the greatest
and least apparent diameters of the sun, the eccentricity, $e$, is easily
found. In \figref{illo059}{Fig.~59}, since, by definition, $e = CS÷ CA$, we have $CS =
CA× e$, or $Ae$. The perihelion distance $AS$ is therefore equal to
$A× (1-e)$, and the aphelion distance $SA'$ to $A (1+e)$. Suppose
now that the greatest and least measured diameters of the sun are $p$
and $q$. This gives us the proportion $p:q = A(1+e):A(1-e)$,
since the diameters are \textit{inversely} proportional to the distances. From
this we get
\[
e = \frac{p-q}{p+q}.
\]
The actual values of $p$ and $q$ are $32'\,36''.4$ and $31'\,31''.8$, which give
$e = 0.01678:$ this is about $\frac{1}{60}$, as has been stated.

\includegraphicsouter{illo060}{\textsc{Fig.~60.}\\
Equable Description of Areas.}

\nbarticle{186.} \nbparatext{To find the Law of the Earth's Motion.}---By comparing the
measured apparent diameter with the differences of longitude from day
to day, we can also deduce the \textit{law} of the earth's motion. On making
a table of daily motions and apparent diameters, we find that these
\textit{daily motions} vary \textit{directly as the squares of the diameters}; from which
it directly follows that the earth moves
in such a way that its \textit{radius-vector
describes areas proportional to the times}
(a law which Kepler first brought to
light in 1609). The radius-vector is
the line which joins the earth to the
sun at any moment.

\begin{fineprint}
\nbarticle{187.}  Consider a small elliptical sector,
$dSc$ (\figref{illo060}{Fig.~60}), described by the earth
in a unit of time. Regarding it as a triangle,
its area is given by the formula $\frac{1}{2}SC× SD\sin{cSd}$; and calling this
angle $\theta$ (which will be very small), and considering that in so short a time
%% -----File: 135.png---Folio 124-------
$Sd$ and $Sc$ would remain sensibly equal, each being equal to $R$ (the radius-vector
at the middle point of the arc), this formula becomes,
\begin{flalign*}
&&  &\text{Area of sector } = \tfrac{1}{2}R^{2}\theta.  &&\phantom{whence}\\
\rlap{\text{\indent Now, calling the sun's apparent diameter $D$, we have:}}\\
&&  &R = \frac{k}{D},  &&\\
\rlap{\text{($k$ being a constant, and depending on the sun's diameter in miles);}}\\
&\text{whence }&  &R^{2} = \frac{k^{2}}{D^{2}}.  &&\\
\rlap{\text{But our measurements show that $\theta = k_{1}D^{2}$, $k_{1}$ being another constant.}}\\
\rlap{\text{Substitute these values of $R^{2}$ and $\theta$ in the formula for the area, and we have}}\\
&&  &\text{Area of sector } = \tfrac{1}{2}\frac{k^{2}}{D^{2}}× k_{1}D^{2} = \tfrac{1}{2}k^{2}k_{1},
\end{flalign*}
a constant; that is, the area described by the radius-vector in a unit of time
is always the same. The planet near perihelion moves so much faster, that
the areas $aSb, cSd$, and $eSf$ are all equal to each other, if the arcs are described
in the same time.
\end{fineprint}

\includegraphicsouter{illo061}{\textsc{Fig.~61.}---Kepler's Problem}

\nbarticle{188.} \nbparatext{\stretchyspace Kepler's Problem.}---{\stretchyspace As the case} stands so far, this is a mere
fact of observation; but as we shall see hereafter, and as was demonstrated
by Newton, the fact shows
that the earth moves under the action
of a force \textit{always directed in line
with the sun}. In such a case the
``equable description of areas'' is a
necessary mechanical consequence.
It is true in every case of elliptical
motion, and enables us to find the
position of the earth or any planet
in its orbit at any time, when we
once know the time of its orbital revolution (technically the period),
and the time when it was at perihelion. Thus, the angle $ASP$ (\figref{illo061}{Fig.~61}),
which is called the \textit{Anomaly} of the planet, must be such that the area of
the elliptical sector $ASP$ will be that portion of the whole ellipse which
is represented by the fraction $\dfrac{t}{T}$, $t$ being the number of days since the
planet last passed the perihelion, and $T$ the number of days in the
whole period. For instance, if the earth last passed perihelion on
Dec.~31 (which it did), its place on May~1 must be such that the
%% -----File: 136.png---Folio 125-------
sector $ASP$ will be $\frac{121}{365\frac{1}{4}}$ of the whole of the earth's orbit; since from
Dec.~31 to May~1 is 121 days. The solution of this problem, known
as ``\textit{Kepler's problem},'' leads to transcendental equations, and lies
beyond our scope.

\begin{fineprint}
See Watson's ``Theoretical Astronomy,'' pp.~53 and 54, or any other similar
work.
\end{fineprint}

\nbarticle{189.} \nbparatext{Anomaly and Equation of the Centre.}---The angle $ASP$,
which has been termed simply the ``\textit{Anomaly},'' is strictly the \textit{true}
Anomaly, as distinguished from the \textit{mean} Anomaly. The \textit{former}
may be defined as \textit{the angle actually made at any time by the radius-vector
of a planet with the line of apsides}, the angle being reckoned
from the perihelion point completely around in the direction of the
planet's motion. The \textit{mean} Anomaly is what the Anomaly \textit{would be
at the given moment if the planet had moved with uniform} angular
\textit{velocity, completing the orbit in the same period, and passing perihelion
at the same time}, as it actually does. The difference between the
two anomalies is called the \textit{Equation of the Centre}. This is zero at
perihelion and aphelion, and a maximum midway between them. In
the case of the sun, its greatest value is nearly $2°$, the sun getting
alternately that amount ahead of, and behind, the position it would
occupy if its apparent daily motion were uniform.

\nbarticle{190.} \nbparatext{The Seasons.}---The earth in its motion around the sun
always keeps its axis parallel to itself, for the mechanical reason that
a revolving body necessarily maintains the direction of its axis invariable,
unless disturbed by extraneous force, as is very prettily illustrated
by the gyroscope. About March 20 the earth is so situated
that the plane of its equator passes through the sun, the sun's declination
being zero on that day.

At that time, the line which separates the illuminated portions of
the earth passes through the two poles, and day and night are everywhere
equal. The same is again true of the 22d of September, when
the sun is at the autumnal equinox on the opposite side of the orbit.

About the 21st of June the earth is so situated that its \textit{north} pole
is inclined towards the sun by about $23\frac{1}{2}°$, which is the sun's northern
declination on that date. The south pole is then in the obscure half
of the earth's globe, while the north pole receives sunlight all day
long; and in all portions of the northern hemisphere the day is longer
than the night, the difference between the day and night depending
upon the latitude of the place, while in the southern hemisphere the
%% -----File: 137.png---Folio 126-------
days are shorter than the nights. At the time of the winter solstice
these conditions are reversed. At the equator (of the earth) the day
and night are equal at all times of the year. The sun when in northern
declination of course always rises at a point on the horizon \textit{north
of east}, and sets at a point north of west, so that for a portion of
the time each day it shines on the north side of a house.

\begin{fineprint}
\nbarticle{191.} \nbparatext{Diurnal Phenomena near the Pole.}---At the north pole, where
the celestial pole is in the zenith, and the diurnal circles are parallel with
the horizon, the sun will maintain the same elevation all day long, except
for the slight change caused by the variation of its declination in twenty-four
hours. The sun will appear on the horizon at the date of the vernal
equinox (in fact, about three days before, on account of refraction), and
slowly wind upward in the sky until it reaches its maximum elevation of
$23\frac{1}{2}°$ on June 21. Then it will retrace its course until a day or two after
the autumnal equinox, when it sinks out of sight.

At points between the north pole and the polar circle the sun will appear
above the horizon earlier in the year than March 20, and will rise and set
daily until its declination becomes equal to the observer's distance from the
pole, when it will make a complete circuit of the heavens, touching the horizon
at midnight at the northern point; and after that never setting again
until it reaches the same declination in its southward course after passing
the solstice. From that time it will again rise and set daily until it reaches
a southern declination just equal to the observer's polar distance, when the
long night begins; to continue until the sun, having passed the southern
solstice, returns again to the same declination at which it made its appearance
in the spring. At the polar circle itself (or, more strictly speaking,
owing to refraction, about one-half a degree south of it) the ``\textit{midnight sun}''
will be seen on just one day in the year, the day of the summer solstice;
and there will also be one absolutely sunless day, viz., the day of the winter
solstice. The same remarks apply in the southern hemisphere, by making
the obvious changes.
\end{fineprint}

\nbarticle{192.} \nbparatext{Effects on Temperature.}---The changes in the duration of
``\textit{insolation}'' (exposure to sunshine) at any place involve changes
of temperature and other climatic conditions, thus producing the seasons.
Taking as a standard the amount of heat received in twenty-four
hours on the day of the equinox, it is clear that the surface of the
soil at any place in the northern hemisphere will receive more than
this average amount of heat whenever the sun is north of the celestial
equator, for two reasons.

\includegraphicsouter{illo062}{\textsc{Fig.~62.}\\
Effect of Sun's Elevation on Amount
of Heat Imparted to the Soil.}

1. Sunshine lasts more than half the day.

2. The \textit{mean elevation} of the sun during the day is greater than
%% -----File: 138.png---Folio 127-------
when it is at the equinoxes, since it is higher at noon, and in any
case reaches the horizon at rising and setting. Now, the more obliquely
the rays strike, the less heat they bring to each square inch of surface,
as is obvious from \figref{illo062}{Fig.~62}. A beam of sunshine having the cross-section
$ABCD$, when it strikes the
surface at an angle $h$ (equal to the
sun's altitude) is spread over a much
larger surface, $Ac$, and of course
the amount of heat per square inch
is proportionately reduced. If $Q$
is the amount of heat per square
inch brought by the ray when falling
perpendicularly, as on the surface
$AC$, then on $Ac$ the amount
per square inch will be $Q×\sin{h}$,
since $AB = Ab×\sin{h}$. This difference in favor of the more nearly
vertical rays is exaggerated by the absorption of heat in the atmosphere,
because rays that are nearly horizontal have to traverse a
much greater thickness of air before reaching the ground.

For these two reasons, at a place in the northern hemisphere, the
temperature rises rapidly as the sun comes north of the equator, thus
giving us our summer.

\nbarticle{193.} \nbparatext{Time of Highest Temperature.}---We, of course, receive the
most heat \textit{per diem} at the time of the summer solstice; but this is
not the hottest time of the summer, for the obvious reason that the
weather is then all the time \textit{getting hotter}, and the maximum will not
be reached \textit{until the increase ceases}; that is, not until the \textit{amount of
heat lost in the night} equals that \textit{stored up by day}.

\begin{fineprint}
If the earth's surface threw off the same amount of heat hourly whether
it were hot or cold, then this maximum would not come until \textit{the autumnal
equinox}. This, however, is not the case. The soil loses heat faster when
warm than it does when cold, the loss being nearly proportional to the difference
between the temperature of the soil and that of surrounding space;
(Newton's law of cooling); and so the time of the maximum is made to
come not far from the end of July, or the first of August, in our latitude.
For similar reasons the minimum temperature of winter occurs about Feb.~1,
about half-way between the solstice and the vernal equinox. Since, however,
our weather is not entirely ``made on the spot where it is used,'' but
is affected by winds and currents that come from great distances, the actual
time of the maximum temperature cannot be determined by any mere astronomical
considerations, but varies considerably from year to year.
\end{fineprint}
%% -----File: 139.png---Folio 128-------

\nbarticle{194.} \nbparatext{\stretchyspace Difference between Seasons in Northern and Southern Hemispheres.}---Since
in December the distance of the earth from the sun
is about three per cent less than it is in June, the earth (as a whole)
receives hourly about six per cent more heat in December than in
June, the heat varying inversely as the \textit{square} of the distance. For
this reason the southern summer, which occurs in December and
January, is \textit{hotter} than the northern. It is, however, seven days
\textit{shorter}, because the earth moves more rapidly in that part of its
orbit. The total amount of heat per acre, therefore, received during
the summer is sensibly the same in each hemisphere, the shortness
of the southern summer making up for its increased warmth.

\begin{fineprint}
\nbarticle{195.} The southern \textit{winter}, however, is both longer and colder than the
northern; and it is maintained by certain geologists, Mr.~Croll especially,
that, on the whole, the mean annual temperature of the hemisphere which
has its winter at the time when the earth is in aphelion is lower than that
of the opposite one. It has been attempted to account for the glacial epochs
in this way. It is certain that at present, at any place in the southern hemisphere,
the difference between the maximum temperature of summer and
the minimum of winter must be greater than in the case of a station in the
northern hemisphere, similarly situated as to elevation, etc. We say ``at
present'' because, on account of certain slow changes in the earth's orbit, to
be spoken of immediately, the state of things will be reversed in about ten
thousand years, the northern summer being then the hotter and shorter one.
\end{fineprint}

\nbarticle{196.} \nbparatext{Secular Changes in the Orbit of the Earth.}---The orbit of
the earth is not absolutely unchangeable in \textit{form} or \textit{position}, though
it is so in the long run as regards the \textit{length of its major axis} and the
\textit{duration of the year}.

\sloppy
\nbarticle{197.} 1.~\textit{Change in Obliquity of the Ecliptic}.---The ecliptic slightly
and very slowly shifts its position among the stars, thus altering
the latitudes of the stars and the angle between the ecliptic and
equator, \textit{i.e.}, the obliquity of the ecliptic.  This obliquity is at
present about $24'$ less than it was 2000 years ago, and is still
\textit{decreasing} about half a second a year. It is computed that this
diminution will continue for about 15,000 years, reducing the obliquity
to $22\frac{1}{4}°$, when it will begin to increase. The whole change,
according to J.~Herschel, can never exceed about $1°\, 20'$ on each side
of the mean.

\fussy
\nbarticle{198.} 2.~\textit{Change of Eccentricity.}---At present the eccentricity of
the earth's orbit (which is now 0.0168) is also slowly diminishing.
%% -----File: 140.png---Folio 129-------
According to Leverrier, it will continue to decrease for about 24,000
years, until it becomes 0.003, and the orbit will be almost circular.
Then it will increase again for 40,000 years, until it becomes
0.02.

\begin{fineprint}
In this way the eccentricity will oscillate backwards and forwards, always,
however, remaining between zero and 0.07; but the oscillations are not
equal either in amount or time, and so cannot properly be compared to the
``vibrations of a mighty pendulum,'' which is rather a favorite figure of
speech.
\end{fineprint}

\nbarticle{199.} 3.~\textit{Revolution of the Apsides of the Earth's Orbit.}---The line
of apsides of the orbit, which now stretches in both directions towards
the constellations of Sagittarius and Gemini, is also slowly and
steadily moving eastward, and at a rate which will carry it around
the circle in about 108,000 years.

\nbarticle{200.} These so-called ``\textit{secular}'' changes are due to the action of
the other planets upon the earth. Were it not for their attraction, the
earth would keep her orbit with reference to the sun strictly unaltered
from age to age, except that possibly in the course of millions of
years the effects of falling meteoric matter and the attraction of the
nearer fixed stars might make themselves felt.

\begin{fineprint}
Besides these secular perturbations of the earth's orbit, the earth itself is
continually being slightly disturbed in its orbit. On account of its connection
with the moon, it oscillates each month a few hundred miles above and
below the true plane of the ecliptic, and by the action of the other planets it is
sometimes set forwards or backwards to the extent of a few thousand miles.
Of course every such change produces a corresponding slight change in the
apparent position of the sun.
\end{fineprint}

\nbarticle{201.} \nbparatext{Equation of Time.}---We have stated a few pages back
(\artref{Art.}{111}), that the interval between the successive passages of the
sun across the meridian is somewhat variable, and that for this
reason apparent solar, or sun-dial, days are unequal. On this account
mean time has been adopted, which is kept by a ``\textit{fictitious}''
or ``\textit{mean}'' sun moving uniformly in the equator at the same
average rate as that of the real sun in the ecliptic. The hour-angle
of this mean sun is, as has been already explained, \textit{the
local mean time} (or clock time); while the hour-angle of the
real sun is the \textit{apparent} or \textit{sun-dial time}. The \textit{Equation of Time}
is the difference between these two times, reckoned as \textit{plus} when
%% -----File: 141.png---Folio 130-------
the sun-dial is \textit{slower} than the clock, and \textit{minus} when it is faster. It
is the \textit{correction which must be added} (algebraically) \textit{to apparent time
in order to get mean time}. As it is the difference between the two
hour-angles, it may also be defined as \textit{the difference between the
right ascensions of the mean sun and the true sun}; or as a formula
we may write: $E = \alpha_{t} - \alpha_{m}$, in which $\alpha_{m}$ is the right ascension of the
mean sun, and $\alpha_{t}$, of the true sun.

The principal causes of this difference are two:---

\includegraphicsouter{illo063}{\textsc{Fig.~63.}
Effect of Obliquity of Ecliptic in producing
Equation of Time.}

\nbarticle{202.} 1.~\textit{The Variable Motion of the Sun in the Ecliptic, due to the
Eccentricity of the Earth's Orbit.}---Near perihelion, which occurs
about Dec.~31, the sun's motion in longitude is most rapid. Accordingly,
at this time the apparent solar days exceed the sidereal
by more than the average amount, making the sun-dial days longer
than the mean. (The average solar day, it will be remembered, is $3^{\text{m}}\, 56^{\text{s}}$
longer than the sidereal.) The sun-dial will therefore lose time at
this season, and will continue to do so for about three months, until
the angular motion of the sun falls to its mean value. Then it will
gain until aphelion, when, if the clock and the sun were started
together at perihelion, they will once more be together. During
the next half of the year the action will be reversed. Thus, twice
a year, so far as the eccentricity of the earth's orbit is concerned,
the clock and sun would be together at perihelion and aphelion,
while half-way between they would differ by about \textit{eight minutes};
the equation of time (so far as due to this cause only) being $+8$
minutes in the spring, and $-8$ minutes in the autumn.

\nbarticle{203.} 2.~\textit{The Inclination of the
Ecliptic to the Equator.}---Even if
the \textit{sun's} (\textit{apparent}) \textit{motion in longitude}
(\textit{i.e.}, along the ecliptic) were
uniform, its motion in \textit{right ascension}
would be variable. If the true
and fictitious suns started together
at the equinox, they would indeed
be together at the solstices and at
the other equinox, because it is just
$180°$ from equinox to equinox, and
the solstices are exactly half-way
between them. But at \textit{intermediate points}, between the equinoxes
and solstices, they would not be together on the same hour-circle.
%% -----File: 142.png---Folio 131-------
This is best seen by taking a celestial globe and marking \textit{on the ecliptic}
a point, $m$, half-way between the vernal equinox and the solstice,
and also marking a point $n$ \textit{on the equator}, $45°$ from the equinox. It
will at once be seen that the former point, $m$ in \figref{illo063}{Fig.~63},\footnote
  {Fig.~63 represents a celestial globe viewed from the \textit{west} side, the axis being
vertical, and $K$, the pole of the ecliptic, on the meridian, while $E$ is the vernal
  equinox.}
is \textit{west} of $n$,
so that $m$ in the daily westward motion of the sky will come to the
meridian first; in other words, when the sun is half-way between
the vernal equinox and the summer solstice, the \textit{sun-dial} is \textit{faster}
than the clock, and the equation of time is \textit{minus}. The difference,
measured by the arc $m'n$, amounts to nearly ten minutes; and of
course the same thing holds, \textit{mutatis mutandis}, for the other quadrants.

\includegraphicsmid{illo064}{\textsc{Fig.~64.}---The Equation of Time.}

\nbarticle{204.} \nbparatext{Combination of the Effects of the Two Causes.}---We can represent
graphically these two components of the equation of time and
the result of their combination as follows (\figref{illo064}{Fig.~64}):---

The central horizontal line is a scale of dates one year long, the
letters denoting the beginning of each month. The dotted curve
shows the equation of time due to the \textit{eccentricity} of the earth's orbit,
above considered. Starting at perihelion on Dec.~31, this component
is then zero, rising from there to a value of about $+8^{\text{m}}$ on
April~2, falling to zero on June~30, and reaching a second maximum
of $-8^{\text{m}}$ on Oct.~1. In the same way the broken-line curve denotes
the effect of the \textit{obliquity of the ecliptic}, which, by itself alone
considered, would produce an equation of time having \textit{four} maxima
of, approximately, $10^{\text{m}}$ each, about the 6th of February, May,
%% -----File: 143.png---Folio 132-------
August, and November (alternately $+$ and $-$), and reducing to zero
at the equinoxes and solstices.

The full-lined curve represents their combined effect, and is constructed
by making its ordinate at each point equal to the sum
(algebraic) of the ordinates of the two other curves. At the 1st of
February, for instance, the distance $F$,~3, in the \figref{illo064}{figure} $=F,\,1 + F,\,2$.
So, also, $M,\, 6 = M,\, 4 + M,\, 5$; the components, however, in this case
have opposite signs, so that the \textit{difference} is actually taken.

The equation of time is zero \textit{four} times a year, viz.: on April~15,
June~14, Sept.~1, and Dec.~24. The maxima are February 11, $+14^{\text{m}}\,32^{\text{s}}$;
May 14, $-3^{\text{m}}\,55^{\text{s}}$; July 26, $+6^{\text{m}}\,12^{\text{s}}$, and Nov.~2, $-16^{\text{m}}\,18^{\text{s}}$.
But the dates and amounts vary slightly from year to year.

\begin{fineprint}
The two causes above discussed are only the \textit{principal} ones effective in
producing the equation of time. Every perturbation suffered by the earth
comes in with its own effect; but all other causes combined never alter the
equation by more than a few seconds.
\end{fineprint}

\nbarticle{205.} \nbparatext{Precession of the Equinoxes.}---The length of year was
found in two ways by the ancients:---

1.~By the gnomon, which gives the time of the equinox and solstice;
and

2.~By observing the position of the sun with reference to the
stars,---their rising and setting at sunrise or sunset.

Comparing the results of observations made by these two methods
at long intervals, Hipparchus (120~\textsc{b.c.}) found that the two do not
agree; the former year (from equinox to equinox) being $20^{\text{m}}\,23^{\text{s}}$
shorter than the other (according to modern data). The equinox
is plainly moving \textit{westward} on the ecliptic, as if it \textit{advanced to
meet the sun} on each annual return. He therefore called the motion
the ``\textit{precession}'' of the equinoxes.

On comparing the \textit{latitudes} of the stars in the time of the ancient
astronomers with the present latitudes, we find that they have
changed very slightly indeed; and we know therefore that the ecliptic
and the plane of the earth's orbit maintains its position sensibly
unaltered. On the other hand, the \textit{longitudes} of the stars have been
found to increase regularly at the rate of about $50''.2$ annually,---fully
$30°$ in the last 2000 years. Since longitudes are reckoned
from the equinox (the intersection between the ecliptic and equator),
and since the ecliptic does not move, it is evident that the
motion must be in the \textit{celestial equator}; and accordingly we find that
both the \textit{right ascension and the declination} of the stars are constantly
changing.
%% -----File: 144.png---Folio 133-------

\nbarticle{206.} \nbparatext{Motion of the Pole of the Equator around the Pole of the
Ecliptic.}---The obliquity of the ecliptic, which equals the distance in
the sky between the pole of the equator and the pole of the ecliptic
(\artref{Art.}{178}), has remained nearly constant. Hence the pole of the
equator must be describing a circle around the pole of the ecliptic in a
period of about 25,800 years ($360°$ divided by $50''.2$). The pole of
the ecliptic has remained practically fixed among the stars, but the
other pole has changed its position materially. At present the pole
star is about $1\frac{1}{4}°$ from the pole. At the time of the star catalogue of
Hipparchus it was $12°$ distant from it, and during the next century
it will approach to within about $30'$, after which it will recede.

\begin{fineprint}

\nbarticle{207.} If upon a celestial globe we take the pole of the ecliptic as a centre,
and describe about it a circle with a radius of $23\frac{1}{2}°$, we shall get the
track of the celestial pole among the stars, and shall find that the circle
passes very near the star $\alpha$~Lyrę, on the opposite side of the pole of the
ecliptic from the present pole star. About 12,000 years hence $\alpha$~Lyrę
will be the pole star. Reckoning backwards, we find that some 3000
years ago $\gamma$~Draconis was the pole star; and it is a curious circumstance
that certain of the tunnels in the pyramids of Egypt face exactly to the
north, and slope at such an inclination that this star at its lower culmination
would have been visible from their lower end at the date when the pyramids
are supposed to have been built. It is probable that these passages were
arranged to be used for the purpose of observing the transits of their then
pole star.

\nbarticle{208.} \nbparatext{Effect of Precession upon the Signs of the Zodiac.}---Another
effect of precession is that the \textit{signs} of the zodiac do not now agree with the
\textit{constellations} which bear the same name. The sign of Aries is now in the
constellation of Pisces; and so on, each sign having ``backed,'' so to speak,
into the constellation west of it.
\end{fineprint}

\includegraphicsouter{illo065}{\textsc{Fig.~65.}\\
Effect of Attraction
on a Spheroid.}

\nbarticle{209.} \nbparatext{Physical Cause of Precession.}---The \textit{physical cause} of this
slow conical rotation of the earth's axis around the pole of the ecliptic
lies in the two facts that \textit{the earth is not exactly spherical}, and that
\textit{the attractions of the sun and moon\footnote
  {The \textit{planets}, by their action upon the plane of the earth's orbit (\artref{Art.}{197}),
slightly disturb the equinox \textit{in the opposite direction}. This effect amounts to about
  $0''.16$ annually.}
act upon the equatorial ring of matter
which projects above the true sphere, tending to draw the plane of the
equator into coincidence with the plane of the ecliptic} by their greater
attraction on the nearer portions of the ring. The action is just what
it would be if a spheroidal ball of iron of the shape of the earth had
%% -----File: 145.png---Folio 134-------
a magnet brought near it. The magnet, as illustrated in \figref{illo065}{Fig.~65},
would tend to draw the plane of the equator into the line $CM$ joining its
pole with the centre of the globe, because it attracts the nearer portion
of the equatorial protuberance at $E$ more strongly than the remoter at
$Q$. If it were not for the earth's rotation, this attraction would bring the
two planes of the ecliptic and equator together; but since the earth is
spinning on its axis, we get the same result that we do with the whirling
wheel of a gyroscope by hanging a weight at one end of the axis.
We then have the result of the combination of two
rotations at right angles with each other, one the
whirl of the wheel, the other the ``tip'' which the
weight tends to give the axis. (See Brackett's Physics,
pp.~53--56.)

\nbarticle{210.} In this case, if the wheel of the gyroscope is turning swiftly
\textit{clock-wise}, as seen from above (\figref{illo066}{Fig.~66}), the weight at the (lower)
end of the axis will make the axis move slowly around, \textit{counter-clockwise},
without at all changing its inclination. If we regard the horizontal
plane passing through the gyroscope as representing the ecliptic,
and the point in the ceiling vertically above the gyroscope as the
pole of the ecliptic, the line of the axis of the wheel produced upward
would describe on the ceiling a circle around this imaginary
ecliptic pole, acting precisely as does the pole of the earth's axis in
the sky. The swifter the wheel's rotation, the slower would be this
``precessional'' motion of its axis; and of course, the rate of motion
also depends upon the magnitude of the suspended weight.

\includegraphicsmid{illo066}{\textsc{Fig.~66.}\\
Precession Illustrated by the Gyroscope.}

\begin{fineprint}
\nbarticle{211.} A full treatment of the subject would be too complicated for our
pages. An elementary notion of the way the action takes place, correct as
%% -----File: 146.png---Folio 135-------
far as it goes, is easily obtained by reference to \figref{illo067}{Fig.~67}. Let $XY$ be the axis of
the gyroscope, the wheel being seen in section edge-wise, and the eye being
on the same level as the centre of the wheel; the wheel turning so that the
point $B$ is coming towards the observer. Now, suppose a weight hung on
the lower end of the axis. If the wheel
were not turning, the point $B$ would
come to some point $F$ in the same time
it now takes to reach $C$ (that is, after a
quarter of a revolution). By combination
of the two motions it will come to a
point $K$ at the end of the same time,
having crossed the horizontal plane $AD$
at $L$; and this can be effected only by
a backward ``skewing around'' of the
whole wheel, axis and all. This does not,
of course, explain why the inclination of
the axis does not change under the action
of the weight, but is only a very partial illustration, showing merely why
the plane of the wheel regresses. A complete discussion would require the
consideration of the motion of every point on the wheel by a thorough and
difficult analytical treatment, in order to give the complete explanation of
the reason why the depressing weight, however heavy, does not cause the end
of the axis to fall perceptibly. (See article, ``Gyroscope,'' in Johnson's
Cyclopędia.)
\end{fineprint}

\nbarticle{212.} \nbparatext{Why Precession is so Slow.}---The slowness of the precession
depends on three things: (\textit{a})~the enormous ``moment of rotation''
of the earth---a point on the equator moves with the speed of a
cannon ball; (\textit{b})~the smallness of the mass (compared with that of
the whole earth) of the protuberant ring to which precession is due;
and (\textit{c})~the minuteness of the force which tends to bring this ring into
coincidence with the ecliptic, a force which is not constant and persistent,
like the weight hung on the gyroscope axis, but very variable.

\begin{fineprint}
\nbarticle{213.} \nbparatext{The Equation of the Equinox.}---Whenever the sun is in the
plane of the equator (which is twice a year, at the time of the equinoxes), the
\textit{sun's} precessional force disappears entirely, its attraction then having no
tendency to draw the equator out of its position. The moon's action, on account
of her proximity, is still more powerful than that of the sun; on the
average two and a half times as great. Now, the moon crosses the celestial
equator twice every month, and at these times \textit{her} action ceases.

\includegraphicsouter{illo067}{\textsc{Fig.~67.}\\
Regression of the Gyroscope Wheel.}

There is still another cause for variation in the effectiveness of the \textit{moon's}
attraction. As we shall see hereafter, she does not move in the ecliptic, but
in a path which cuts the ecliptic at an angle of about $5°$, at two points called
the \textit{Nodes}; the \textit{ascending} node being the point where she crosses the ecliptic
%% -----File: 147.png---Folio 136-------
from south to north. These nodes move westward on the ecliptic (\artref{Art.}{455}),
making the circuit once in about nineteen years. Now, when the ascending
node of the moon's orbit is at $B$ (\figref{illo068}{Fig.~68}), near the autumnal equinox $F$, its
inclination to the equator will
be, as the \figref{illo068}{figure} shows, \textit{less}
than the obliquity of the ecliptic
by about $5°$; \textit{i.e.}, it will
be only about $18°$. On the
other hand, nine and a half
years later, when the node has
backed around to a point $A$, near the vernal equinox, the inclination of the
moon's orbit to the equator will be nearly $28°$. When the node is in this
position, the moon will produce nearly twice as much precessional movement
each month as when the node was at $B$.

\includegraphicsouter{illo068}{\textsc{Fig.~68.}\\
Variation in the Inclination of Moon's Orbit to Equator.}

The precession, therefore, is not uniform, but variable, almost ceasing at
some times and at others becoming rapid. The average amount, as has
been stated, is $50''.2$ a year; and the variation is taken account of in what
is called the \textit{equation of the equinox}, which is the difference between the
actual position of the equinox at any time and the position it would have at
that moment if the precession had been all the time going on \textit{uniformly}.
\end{fineprint}

\nbarticle{214.} \nbparatext{Nutation.}---Not only does the precessional force vary in
amount at different times, but in most positions of the disturbing
body with respect to the earth's equator there is a \textit{slight thwartwise
component of the force, tending directly to accelerate or retard} the precessional
movement of the pole---just as if one should gently draw
the weight $W$ (\figref{illo063}{Fig.~63}) horizontally.  The consequence is what
is called \textit{Nutation} or ``nodding.'' The axis of the earth, instead of
moving smoothly in a circle, nods in and out a little with respect to
the pole of the ecliptic, describing a wavy curve resembling that
shown in \figref{illo069}{Fig.~69}, but with nearly 1400 indentations in the entire circumference
traversed in 26,000 years.

\begin{fineprint}

\includegraphicsouter{illo069}{\textsc{Fig.~69.}---Nutation.}

\nbarticle{215.}  We distinguish three of these nutations, (\textit{a})~The \textit{Lunar Nutation},
depending upon the motion of the moon's nodes.
This has a period of a little less than nineteen
years, and amounts to $9''.2$, (\textit{b})~The \textit{Solar Nutation},
due to the changing declination of the sun.
Its period is a year, and its amount $1''.2$. (\textit{c})~The
\textit{Monthly Nutation}, precisely like the solar nutation,
except that it is due to the \textit{moon's} changes of declination
during the month. It is, however, too
small to be certainly measured, not exceeding one-tenth
of a second.
%% -----File: 148.png---Folio 137-------

Nutation was detected by Bradley in 1728, but not fully explained until
1748.

\textit{Neither} precession \textit{nor} nutation \textit{affects the} latitudes \textit{of the stars, since they
are not due to any change in the position of the ecliptic, but only to displacements
of the earth's axis}. \textit{The} longitudes \textit{alone are changed by them}.

The \textit{right ascension} and \textit{declination} of a star are both affected.
\end{fineprint}

\nbarticle{216.} \nbparatext{The Three Kinds of Year.}---In consequence of the motion
of the equinoxes caused by precession, the \textit{sidereal} year and the
equinoctial or ``\textit{tropical}'' year do not agree in length. Although the
sidereal year is the one which represents the earth's \textit{true} orbital revolution
around the sun, it is not used as the year of chronology and
the calendar, because the \textit{seasons} depend on the sun's place in relation
to the equinoxes. The tropical year is the year usually employed,
unless it is expressly stated to the contrary. The length of the
Sidereal year is $365^{\text{d}}\ 6^{\text{h}}\ 9^{\text{m}}\ 9^{\text{s}}$; that of the Tropical year is about
$20^{\text{m}}$ less, $365^{\text{d}}\ 5^{\text{h}}\ 48^{\text{m}}\ 46^{\text{s}}$.

The third kind of year is the \textit{anomalistic} year, which is the time
from perihelion to perihelion again. As the line of apsides of the
earth's orbit moves always slowly towards the east, this year is a little
longer than the sidereal. Its length is $365^{\text{d}}\ 6^{\text{h}}\ 13^{\text{m}}\ 48^{\text{s}}$.

\nbarticle{217.} \nbparatext{The Calendar.}---The natural units of time are \textit{the day, the
month}, and \textit{the year}. The day, however, is too short for convenient
use in designating extended periods of time, as for instance in
expressing the age of a man. The month meets with the same
objection, and for all chronological purposes, therefore, the year is
the unit practically employed. In ancient times, however, so much
regard was paid to the month, and so many of the religious beliefs
and observances connected themselves with the times of the new and
full moon, that the early history of the calendar is largely made up
of attempts to fit the month to the year in some convenient way.
Since the two are incommensurable, the problem is a very difficult,
and indeed strictly speaking, an impossible, one.

In the earliest times matters seem to have been wholly in the hands
of the priesthood, and the calendar then was predominantly \textit{lunar},
with months and days intercalated from time to time to keep the
seasons in place. The Mohammedans still use a purely lunar calendar,
having a year of twelve lunar months, and containing alternately
354 and 355 days. In their reckoning the seasons fall, of course,
continually in different months, and their calendar gains about one
year in thirty-three upon the reckoning of Christian nations.
%% -----File: 149.png---Folio 138-------

\nbarticle{218.} \nbparatext{The Metonic Cycle.}---Among the Greeks the discovery of
the so-called lunar or Metonic cycle by Meton, about 433~\textsc{b.c.}, considerably
simplified matters. This cycle consists of 235 \textit{synodic
months} (from new moon to new again), which is very approximately
equal to 19 common years of $365\frac{1}{4}$ days.

\begin{fineprint}
\sloppy
235 months equal $6939^{\text{d}}\ 16^{\text{h}}\ 31^{\text{m}}$; 19 tropical years equal $6939^{\text{d}}\ 14^{\text{h}}\ 27^{\text{m}}$;
so that at the end of the 19 years, the new and full moon recur again on the
same days of the year, and at the same time of day within about two
hours. The calendar of the phases of the moon, for instance, for 1889 is the
same as for 1870 and 1908 (except that intervening leap-years may change
the dates by one day).

\fussy
The ``\textit{Golden number}'' of a year is its number in this Metonic cycle, and
is found by adding 1 to the ``date-number'' of the year and dividing by 19.
The remainder, unless zero, is the ``golden number'' (if it comes out zero, 19
is taken instead). Thus the golden number for 1888 is found by dividing
1889 by 19, and the remainder 8, is the golden number of the year.

This cycle is still employed in the ecclesiastical calendar in finding the
time of Easter.

{\footnotesize
For further information on the subject, consult Johnson's Encyclopędia, or Sir
Edmund Becket's ``Astronomy without Mathematics.''
}
\end{fineprint}

\nbarticle{219.} \nbparatext{Julian Calendar.}---Until the time of Julius Cęsar the Roman
calendar seems to have been based upon the lunar year of twelve
months, or 355~days, and was substantially like the modern Mohammedan
calendar, with arbitrary intercalations of months and days made by
the priesthood and magistrates from time to time in order to bring it
into accordance with the seasons. In the later days of the Republic,
the confusion had become intolerable. Cęsar, with the help of the astronomer
Sosigenes, whom he called from Alexandria for the purpose,
reformed the system in the year 45~\textsc{b.c.}, introducing the so-called
``Julian calendar,'' which is still used either in its original shape or
with a very slight modification. He gave up entirely the attempt to coordinate
the month with the year, and adopting $365\frac{1}{4}$ days as the true
length of the tropical year, he ordained that every fourth year should
contain an extra day, the \textit{sixth day before the Kalends of March on that
year being counted twice}, whence the year was called ``\textit{bissextile}''
Before his time the year had begun in March (as indicated by the
Roman names of the months,---September, seventh month; October,
eighth month, etc.), but he ordered it to begin on the 1st of January,
which in that year (45~\textsc{b.c.}) was on the day of the new moon next following
the winter solstice. In introducing the change it was necessary
to make the preceding year 445~days long, and it is still known in
%% -----File: 150.png---Folio 139-------
the annals as ``the year of confusion.'' He also altered the name of
the month Quintilis, calling it ``July'' after himself.

\begin{fineprint}
There was some irregularity in the bissextile years for a few years after
Cęsar's death, from a misunderstanding of his rule for the intercalary day;
but his successor Augustus remedied that, and to put himself on the same
level with his predecessor, he took possession of the month Sextilis, calling
it ``August''; and to make its length as great as that of July, he robbed
February of a day.

From that time on, the Julian calendar continued unbrokenly in use until
1582; and it is still the calendar of Russia and of the Greek Church.
\end{fineprint}

\nbarticle{220.} \nbparatext{The Gregorian Calendar.}---The Julian calendar is not quite
correct. The true length of the tropical year is 365~days 5~hours
48~minutes and 46~seconds, and this leaves a difference of 11~minutes
and 14~seconds by which the Julian calendar year is the longer, being
exactly $365\frac{1}{4}$~days. As a consequence, the date of the equinox comes
gradually earlier and earlier by about three days in 400~years.
($400× 11\frac{1}{6}^{\text{m}} = 4467\ \text{minutes} = 3^{\text{d}}\ 2^{\text{h}}\ 27^{\text{m}}$.) In the year 1582, the
date of the vernal equinox had fallen back 10~days to the 11th of
March, instead of occurring on the 21st of March, as at the time of
the Council at Nice, 325~\textsc{a.d.}
Pope Gregory, therefore, acting under
the advice of the Jesuit astronomer, Clavius, ordered that the day
following Oct.~4 in the year 1582 should be called not the 5th, but
the 15th, and that the rule for leap-year should be slightly changed
so as to prevent any such future displacement of the equinox. The
rule now stands: \textit{All years whose date-number is divisible by four
without a remainder are leap-years, unless they are century years} (1700,
1800, etc.). \textit{The century years are not leap-years unless their date-number
is divisible by} 400, \textit{in which case they are}: that is, 1700, 1800,
and 1900 are not leap years; but 1600, 2000, and 2400 are.

\nbarticle{221.} \nbparatext{Adoption of the New Calendar.}---The change was immediately
adopted by all Catholic nations; but the Greek Church and
most of the Protestant nations, rejecting the Pope's authority, declined
to accept the correction. In England it was at last adopted
in the year 1752, at which time there was a difference of eleven days
between the two calendars. (The year 1600 was a leap-year according
to the Gregorian system as well as the Julian, but 1700 was not.)
Parliament in 1751 enacted that the day following the 2d of September,
in the year 1752, should be called the 14th instead of the 3d;
and also that this year (1752), and all subsequent years, should begin
on the first of January.
%% -----File: 151.png---Folio 140-------

\begin{fineprint}
The change was made under very great opposition, and there were violent
riots in consequence in different parts of the country, especially at Bristol,
where several persons were killed. The cry of the populace was, ``Give us
back our fortnight,'' for they supposed they had been robbed of eleven days,
although the act of Parliament was carefully framed to prevent any injustice
in the collection of interest, payment of rents, etc.

At present, since the year 1800 was not a leap-year according to the
Gregorian calendar, while it was so according to the Julian, the difference
between the two calendars amounts to twelve days; thus in Russia the 19th
of August would be reckoned as the 7th. In Russia, however, for scientific
and commercial purposes \textit{both} dates are very generally used, so that the date
mentioned would be written Aug.~$\frac{7}{19}$. When Alaska was annexed to the
United States, its calendar had to be altered by \textit{eleven} days. (See \artref{Art.}{123}.)

\nbarticle{222.} \nbparatext{The Beginning of the Year.}---The beginning of the year has
been at several different dates in the different countries of Europe. Some
have regarded it as beginning at Christmas, the 25th of December; others,
on the 1st of January; others still, on the 1st of March; others, on the
25th; and others still, at Easter, which may fall on any day between the
22d of March and the 25th of April.

In England previous to the year 1752 the legal year commenced on the
25th of March, so that when the change was made, the year 1751 necessarily
lost its months of January and February, and the first twenty-four days of
March. Many were slow to adopt this change, and it becomes necessary,
therefore, to use considerable care with respect to English dates which occur
in the months of January, February, or March about that period. The
month of February, 1755, for instance, would by some writers be reckoned
as occurring in 1754. Confusion is best avoided by writing, Feb.~$\frac{1754}{1755}$.

\nbarticle{223.} \nbparatext{First and Last Days of the Year.}---Since the ordinary civil
year consists of 365 days, which is 52 weeks and one day, the last day of
each common year falls on the same day as the first; so that any given date
will fall one day later in the week than it did on the preceding year, unless
a 29th of February has intervened, in which case it will be \textit{two} days later;
that is, if the 3d of January, 1889, falls on Thursday, the same date in 1890
will fall on Friday.
\end{fineprint}

\nbarticle{224.} \nbparatext{Aberration.}---{\footnotesize Although
in strictness the discussion of aberration does
not belong to a chapter describing the earth and its motions, yet since it is a phenomenon
due to the earth's motion, and affects the right ascension and declination of
the stars in much the same ways as do precession and nutation, it may properly enough
be considered here.}

\textit{Aberration is the apparent displacement of a star, due to the combination
of the motion of light with the motion of the observer.}

The direction in which we have to point our telescope in observing
a star is not the same that it would be if the earth were at rest. It
%% -----File: 152.png---Folio 141-------
lies beyond our scope to show that according to the wave theory of
light the apparent direction of a ray will be affected by the observer's
motion precisely in the same way (within very narrow limits) as it
would be if light consisted of corpuscles shot off from a luminous
body, as Newton supposed. This
is the case, however, as Doppler
and others have shown; and assuming
it, the explanation of
aberration is easy:---

\includegraphicsouter{illo070}{\textsc{Fig.~70.}---Aberration of a Raindrop.}

Suppose an observer standing
at rest with a tube in his hand in
a shower of rain where the drops
are falling vertically. If he wishes
to have the drops descend axially
through the tube without touching
the sides, he must of course keep
it vertical; but if he advances in
any direction, he must draw back the bottom of the tube by an
amount which equals the advance he makes in the time while the
drop is falling through the tube, so that when the drop falling from
$B$ reaches $A'$, the bottom of the tube will be there also; \textit{i.e., he must
incline the tube forward by an angle $\alpha$, such that} $\tan{\alpha} = u÷ V$,
where $V$ is the velocity of the raindrop and $u$ that of his own motion.
In \figref{illo070}{Fig.~70} $BA'= V$ and $AA'=u$.

\includegraphicsouter{illo071}{\textsc{Fig.~71.}---Aberration of Light.}

\nbarticle{225.} Now take the more general case.
Suppose a star sending us light with a
velocity $V$ in the direction $SP$, \figref{illo071}{Fig.~71},
which makes the angle $\theta$ with the line of
the observer's motion. He himself is
carried by the earth's orbital velocity in
the direction $QP$. In pointing the telescope
so that the light may pass exactly
along its optical axis, he will have to
draw back the eye-end by an amount
$QP$ which just equals the distance he is
carried, by the earth's motion during the
time that the light moves from $O$ to $P$. The star will thus apparently
be displaced towards the point towards which he is moving,
the angle of displacement $POQ$, or $\alpha$, being determined by the relative
length and direction of the two sides $OP$ and $QP$ of the triangle
%% -----File: 153.png---Folio 142-------
$OPQ$. These sides are respectively proportional to the velocity of
light, $V$, and the orbital velocity of the earth, $u$.

\begin{fineprint}
The angle at $P$ being $\theta$, the angle $OQP$ will be ($\theta - \alpha$), and we shall
have from trigonometry the proportion $\sin{\alpha}:\sin{(\theta - \alpha)} = u : V$.

To find $\alpha$ from this, develop the second term of the proportion and divide
the first two terms by $\sin{\alpha}$, which gives us
\begin{flalign*}
&&&1 : \sin{\theta}\cot{\alpha} - \cos{\theta} = u : V,&&\\
&\text{whence } &&u\sin{\theta}\cot{\alpha} = V + u\cos{\theta},&&\\
&\text{and } &&\cot{\alpha} = \frac{V + u\cos{\theta}}{u\sin{\theta}}&&\phantom{whence }
\intertext{\indent Taking the reciprocal of this we have }
&&  &\tan{\alpha} = \frac{u}{V + u\cos{\theta}}\sin{\theta}. &&
\intertext{The second term in the denominator is insensible, since $u$ is only about one
ten-thousandth of $V$, so that we may neglect it.\footnotemark\
This gives the formula in
the shape in which it ordinarily appears, viz.,}
&& &\tan{\alpha} = \frac{u}{V}\sin{\theta}.  &&
\end{flalign*}
  \footnotetext{The velocity of light, according to the latest determinations of Newcomb and
  Michelson, is 299860 kilometers${}\pm 30$ kilometers (which equals 186,330 miles${}\pm 20$
  miles). The mean velocity of the earth in its orbit, if we assume the solar
  parallax to be $8''.8$, is 29.77 kilometers, or 18.50 miles; this makes the constant
  of aberration $20''.478$, a little smaller than that given in the text.}
\end{fineprint}
\vspace*{-2ex} % end{flalign*} followed be end{fineprint} gives rather large space

The value of $\alpha$ (denoted by $\alpha_0$) which obtains when $\theta = 90°$ and
$\sin{\theta} =$ unity, is called the \textit{Constant of Aberration}.

\includegraphicsouter{illo072}{\textsc{Fig.~72.}\\
Aberrational Orbit of a Star.}

The latest, and probably the most accurate, determination of this
constant (derived from the Pulkowa Observations
by Nyrén in 1882) is $20''.492$.
Aberration was discovered and explained
by Bradley, the English Astronomer Royal,
in 1726.

\nbarticle{226.} \nbparatext{The Effect of Aberration upon the
Apparent Places of the Stars.}---As the earth
moves in an orbit nearly circular, and with
a velocity so nearly uniform that we may
for our present purpose disregard its variations,
it is clear that a star at the pole of
the ecliptic will be always displaced by the
same amount of $20''.5$, but in a direction
%% -----File: 154.png---Folio 143-------
continually changing. It must, therefore, appear to describe a
little circle $41''$ in diameter during the year, as shown in \figref{illo072}{Fig.~72}.
Now the direction of the earth's orbital motion is always in the plane
of the ecliptic, and towards the right hand as we stand facing the
sun. At the vernal equinox, therefore, we are moving toward the
point of the ecliptic, which is $90°$ \textit{west of the sun, i.e.}, towards
the winter solstitial point, and the star is then displaced in that
direction. Three months later the star will be displaced in a line
directed towards the vernal equinox, and so on. The earth, therefore,
so to speak, \textit{drives the star before it} in the aberrational orbit,
keeping it just a quarter of a revolution ahead of itself.

A star on the ecliptic simply appears to oscillate back and forth in
a straight line $41''$ long.

Generally, in any latitude whatever, the aberrational orbit is an
ellipse, having its major axis parallel to the ecliptic, and always
$41''$ long, while its minor axis is $41'' × \sin{\beta}$, $\beta$ being the star's latitude,
or distance from the ecliptic.

\begin{fineprint}
\nbarticle{226*.}
\nbparatext{Diurnal Aberration.}---The motion of an observer due to the
earth's \textit{rotation} also produces a slight effect known as the \textit{diurnal aberration}.
Its ``constant'' is only $0''.31$ for an observer situated at the equator; anywhere
else it is $0''.31\cos{\phi}$, $\phi$ being the latitude of the observer.

For any given star it is a maximum when the star is crossing the meridian,
and then its whole effect is slightly to \textit{increase the right ascension} by an
amount given by the formula
\[
\Delta\alpha = 0''.31\cos{\phi}\sec{\delta},
\]
$\delta$ being the star's declination.

\smallskip
See Chauvenet, ``Practical Astronomy,'' I. p.~638.

\end{fineprint}
\chelabel{CHAPTERVI}
%% -----File: 155.png---Folio 144-------

\Chapter{VII}{The Moon}
\nbchapterhang{\stretchyspace THE MOON: HER ORBITAL MOTION AND VARIOUS KINDS OF
MONTH.---DISTANCE AND DIMENSIONS, MASS, DENSITY AND
GRAVITY.---ROTATION AND LIBRATIONS.---PHASES.---LIGHT
AND HEAT.---PHYSICAL CONDITION AND INFLUENCES EXERTED
ON THE EARTH.---TELE\-SCOPIC ASPECT.---SURFACE
AND POSSIBLE CHANGES UPON IT.}

\nbarticle{227.}  \textsc{We} pass next to a consideration of our nearest neighbor in
the celestial spaces, the moon, which is a satellite of the earth and
accompanies us in our annual motion around the sun. She is much
smaller than the earth, and compared with most of the other heavenly
bodies, a very insignificant affair; but her proximity makes her
far more important to us than any of them except the sun. The
very beginnings of Astronomy seem to have originated in the study
of her motions and in the different phenomena which she causes,
such as the eclipses and tides; and in the development of modern
theoretical astronomy the lunar theory with the problems it raises
has been perhaps the most fertile field of invention and discovery.

\nbarticle{228.} \nbparatext{Apparent Motion of the Moon.}---Even superficial observation
shows that the moon moves eastward among the stars every
night, completing her revolution from \textit{star to star} again in about $27\frac{1}{4}$ days.
In other words, she revolves around the earth in that time; or,
more strictly speaking, they both revolve about their common centre
of gravity. But the moon is so much smaller than the earth that
this centre of gravity is situated within the ball of the earth on the
line joining the centres of the two bodies at a point about 1100
miles below its surface.

As the moon moves eastward so much faster than the sun, which
takes a year to complete its circuit, she every now and then, at the
time of the new moon, overtakes and passes the sun; and as the
phases of the moon depend upon her position with reference to the
sun, this interval from new moon to new moon is what we ordinarily
understand as the month.
%% -----File: 156.png---Folio 145-------

\sloppy
\nbarticle{229.} \nbparatext{Sidereal and Synodic Revolutions.}---\textit{The} \textsc{Sidereal} \textit{revolution
of the moon is the time occupied in passing from a star
to the same star again}, as the name implies. It is equal to
$27^\text{d}\ 7^\text{h}\ 43^\text{m}\ 11^\text{s}.545 \pm 0^\text{s}.01$, or $27^\text{d}.32166$. The moon's mean daily
motion among the stars equals $360°$ divided by this, which is
$13°\ 11'$ (nearly).

\fussy
\textit{The} \textsc{Synodic} \textit{revolution is the interval from new moon to new
moon again}, or from full to full. It varies somewhat on account
of the eccentricity of the moon's orbit and of that of the earth
around the sun, but its \textit{mean} value is $29^\text{d}\ 12^\text{h}\ 44^\text{m}\ 2^\text{s}.684\pm 0^\text{s}.01$,
or $29^\text{d}.53059$; and \textit{this is the ordinary month}. (The word synodic
is derived from the Greek \mytextgreek{s'un} and \mytextgreek{<od'os}, and has nothing to
do with the \textit{nodes} of the moon's orbit. The word is \textit{syn-odic}, not
\textit{sy-nodic}).

A synodical revolution is longer than the sidereal, because during
each sidereal month of 27.3 days the sun has advanced among the
stars, and must be caught up with.

\nbarticle{230.} \nbparatext{Elongation, Syzygy, etc.}---The angular distance of the moon
from the sun is called its \textit{Elongation}. At new moon it is zero, and
the moon is then said to be in ``\textit{Conjunction}.'' At full moon it is
$180°$, and the moon is then in ``\textit{Opposition}.'' In either case the
moon is said to be in ``\textit{Syzygy}'' (\mytextgreek{s'un zug'on}). When the elongation
is $90°$, as at the half-moon, the moon is in ``\textit{Quadrature}.''

\nbarticle{231.} \nbparatext{Determination of the Moon's Sidereal Period.}---This is
effected directly by observations of the moon's right ascension and
declination (with the meridian circle), kept up systematically for a
sufficient time.

If it were not for the so-called ``secular acceleration'' of the
moon's motion (Arts.\ \arnref{459}--\arnref{461}), an exceedingly accurate determination
of the moon's \textit{synodic} period could be obtained by comparing
ancient eclipses with modern.

The earliest authentically recorded eclipse is one that was observed
at Nineveh in the year 763~\textsc{b.c.}\ between 9 and 10~o'clock on the
morning of June 15th.

By comparing this eclipse with (say) the eclipse of August, 1887,
we have an interval of more than 35,000 months, and so an error of
ten hours even, in the observed time of the Nineveh eclipse, would
make only about one second in the length of the month. But the
month is a little shorter now than it was 2000 years ago.
%% -----File: 157.png---Folio 146-------

\sloppy
\nbarticle{232.} \nbparatext{Relation of Sidereal and Synodic Periods.}---The fraction of
a revolution described by the moon in one day equals $\frac{1}{M}$, $M$ being the
length of the \textit{sidereal} month. In the same way $\frac{1}{E}$ represents
the earth's daily motion in its orbit, $E$ being the length of the year.
The difference of these two equals the fraction of a revolution which
the moon \textit{gains} on the sun during one day. In a synodic month, $S$, it
gains one whole revolution, and therefore must gain each day $\frac{1}{S}$ of a
revolution; so that we have the equation
\[
\frac{1}{M} - \frac{1}{E} = \frac{1}{S};
\]
or, substituting the numerical values of $E$ and $S$,
\[
\frac{1}{M} - \frac{1}{365.25635} = \frac{1}{29.53059},
\]
whence we derive the value of $M$.

\fussy
\begin{fineprint}
Another way of looking at it is this: In a year there must be \textit{exactly one
more} sidereal revolution than there are synodic revolutions, because the sun
completes one entire circuit in that time. Now the number of synodic revolutions
in a year is given by the fraction
\[
  \cfrac{365\frac{1}{4}}{S} = 12.369\!+\!.
\]
There will therefore be 13.369 sidereal revolutions in the year, and the
length of one sidereal revolution equals $365\frac{1}{4}$ days divided by this number
13.369, which will be found to give the length of the sidereal revolution as
before.
\end{fineprint}

\nbarticle{233.} \nbparatext{Moon's Path among the Stars.}---By observing with the meridian
circle the right ascension and declination of the moon daily
during the month, just as in the case of the sun, we obtain the position
of the moon for each day, and joining the points thus found, we
can draw the path of the moon in the sky. It is found to be a great
circle inclined at a mean angle of $5°\, 8'$ to the ecliptic, which it cuts
in two points called the \textit{nodes} (from \textit{nodus}, a ``knot'').

We say the path is found to be a great circle. This must be taken
with some reservation, since at the end of the month the moon never
returns \textit{precisely} to the position it occupied at the beginning, owing
%% -----File: 158.png---Folio 147-------
to the regression of the nodes and other so-called ``perturbations,''
which will be discussed hereafter.

\begin{fineprint}
\nbarticle{234.} \nbparatext{Moon's Meridian Altitude.}---Since the moon's orbit is inclined
to the ecliptic $5°\, 8'$, its inclination to the equator varies from $28°\, 36'$ ($23°\, 28'
+ 5°\, 8'$), when the moon's ascending node is the vernal equinox, to $18°\, 20'$,
when, $9\frac{1}{2}$ years later, the same node is at the autumnal equinox. In the first
case the moon's declination will change during the month by $57°\, 12'$,
from $- 28°\, 36'$ to $+ 28°\, 36'$. In the other case it will change only by $36°\, 40'$,
so that at different times the difference in the behavior of the moon in
this respect is very striking.
\end{fineprint}

\nbarticle{235.} \nbparatext{Interval between Moon's Transits.}---On the average the moon
gains $12°\, 11'.4$ on the sun daily, so that it comes to the meridian about
51 minutes of solar time later each day.

To find the mean interval between the successive transits of the
moon we may use the proportion
\[
(360° - 12°\, 11'.4) : 360° = 24^\text{h} : x;\quad \text{whence } x = 24^\text{h}\, 50^\text{m}.6.
\]
The variations of the moon's motion in right ascension, which are
very considerable (much greater than in the case of the sun), cause
this interval to vary from $24^\text{h}\ 38^\text{m}$ to $25^\text{h}\ 06^\text{m}$.

\nbarticle{236.} \nbparatext{The Daily Retardation of the Moon's Rising and Setting.}---The
\textit{average} daily retardation of the moon's rising and setting is,
of course, the same as that of her passage across the meridian,
viz., $51^\text{m}$; but the actual retardation of rising is subject to very
much greater variations than these of the meridian passage, being
affected by the moon's changes in declination as well as by
the inequalities of her motion in right ascension. When the moon
is very far north, having her maximum declination of $28°\, 36'$, she
will rise in our latitudes much earlier than when she is farther
south.

In the latitude of New York the least possible daily retardation
of moon-rise is 23
minutes, and the greatest is 1~hour and 17~minutes.
In higher latitudes the variation is greater yet.

\begin{fineprint}
\nbarticle{237.} \nbparatext{Harvest and Hunter's Moons.}---The variations in the retardation
of the moon's rising attract most attention when they occur at the time
of the full moon. When the retardation is at its minimum, the moon rises
soon after sunset at nearly the same time for several successive evenings;
whereas, when the retardation is greatest, the moon appears to plunge nearly
%% -----File: 159.png---Folio 148-------
vertically below the horizon by her daily motion. When the full moon
occurs at the time of the autumnal equinox, the moon itself will be near the
first of Aries.

Now, as will be seen by reference to \figref{illo073}{Fig.~73}, the portion of the ecliptic
near the first of Aries makes a much smaller angle with the eastern horizon
than the equator.

[The line $HN$ is the horizon, $E$ being the east point---the \figref{illo073}{figure} being
drawn to represent a celestial globe, as if the observer were looking at the
eastern side of the celestial sphere \textit{from the outside}.]

$EQ$ is the equator. Now, when the autumnal equinoctial point or first of
Libra is on the horizon at $E$, the position of the ecliptic will be that represented
by $ED$; more steeply inclined to the horizon than $EQ$ is, by the
angle $QED$, $23\frac{1}{2}°$. But when the first of Aries is at $E$, the ecliptic will be in
the position $JJ'$. And if the ascending node of the moon's orbit happens
then to be near the first of Aries, the moon's path will be $MM'$.

\includegraphicsmid{illo073}{\textsc{Fig.~73.}---Explanation of the Harvest Moon.}

Accordingly, when the moon is in Aries, it, so to speak, coasts along the
eastern horizon from night to night, its time of rising not varying very
much; and this, when it occurs near the full of the moon, gives rise to the
phenomenon known as the harvest moon, the \textit{harvest moon being the full moon
nearest to the autumnal equinox}. The full moon next following is called the
\textit{hunter's} moon.

In Norway and Sweden, under these circumstances, the moon's orbit may
actually coincide with the horizon, so that she will rise at absolutely the same
time for a considerable number of successive evenings.
\end{fineprint}

\nbarticle{238.} \nbparatext{The Moon's Orbit.}---As in the case of the sun, the observation
of the moon's path in the sky gives no information as to the real
size of its orbit; but its \textit{form} may be found by measuring the apparent
diameter of the moon, which ranges from $33'\ 30''$ to $29'\ 21''$ at
different points. The orbit turns out to be an ellipse like the orbit
of the earth, but with an eccentricity more than three times as great---about
%% -----File: 160.png---Folio 149-------
$\frac{1}{18}$ on the average, but varying from $\frac{1}{14}$ to $\frac{1}{22}$ on account of
perturbations.

The extremities of the major axis of the moon's orbit are called
the \textit{perigee} and \textit{apogee} (from \mytextgreek{per\`i g\newtie h} and \mytextgreek{>ap\`o g\newtie h}).

The line of apsides, which passes through these two points, moves
around towards the east once in about nine years, also on account
of perturbations.

\nbarticle{239.} \nbparatext{Parallax and Distance of the Moon.}---These can be found in
several ways, of which the simplest is the following: At two observatories
$B$ and $C$ (\figref{illo074}{Fig.~74}) on, or
very nearly on, the same meridian
and very far apart (in the
northern and southern hemispheres
if possible; Greenwich
and the Cape of Good Hope, for
instance) let the moon's zenith
distance $ZBM$ and $Z'CM$ be observed
simultaneously with the
meridian circle. This gives in
the quadrilateral $BOCM$ the two
angles at $B$ and $C$, each of
which is the supplement of the geocentric zenith distance. The
angle at the centre of the earth, $BOC$, is the difference of the geocentric
latitudes and is known from the geographical positions of
the two observatories. Knowing the three angles in the quadrilateral,
the fourth at $M$ is of course known, since the sum of the four
must he four right angles. The sides $BO$ and $CO$ are known, being
radii of the earth; so that we can solve the whole quadrilateral by
a simple trigonometrical process.

\includegraphicsouter{illo074}{\textsc{Fig.~74.}---Determination of the Moon's Parallax.}

\begin{fineprint}
First find from the triangle $BOC$ the partial angles $OCB$ and $OBC$, and
the side $BC$. Then in the triangle $BCM$ we have $BC$ and the two angles
$CBM$ and $MCB$, from which we can find the two sides $BM$ and $CM$.
Finally, in the triangle $OBM$, we now know the sides $OB$ and $BM$ and the
included angle $OBM$, so that the side $OM$ can be computed, which is the
distance of the moon from the earth's centre. Knowing this, the horizontal
parallax $KMO$, or the semi-diameter of the earth as seen from the moon,
follows at once.

The moon's parallax can also be deduced from observations at a single
station on the earth, but not so simply. If she did not move among the
stars, it would be very easy, as all we should have to do would be to compare
her apparent right ascension and declination at different points in her diurnal
%% -----File: 161.png---Folio 150-------
circle. Near the eastern horizon the parallax (always depressing an object)
increases her right ascension; at setting, \textit{vice versa}. On the meridian
the declination only is affected. But the motion of the moon must be allowed
for, as the observations to be compared are necessarily separated by
considerable intervals of time, and this complicates the calculation.

A third, and a very accurate, method is by means of occultations of
stars, observed at widely separated points on the earth. These occultations
furnish the moon's place with great accuracy, and so determine the parallax
very precisely; but the calculation is not very simple, as the moon's
motion in this case also enters into it, since the observations cannot be
simultaneous.
\end{fineprint}

\nbarticle{240.} \nbparatext{The Distance of the Moon is continually changing} on account
of the eccentricity of its orbit, varying all the way, according to Neison,
between 252,972 and 221,614 miles; the \textit{mean} distance being
238,840 miles, or 60.27 times the equatorial radius of the earth.
The mean parallax of the moon is $57'\ 2''$, subject to a similar percentage
of change. This value of the parallax, it will be noted,
indicates that the earth, as seen from the moon, has a diameter of
nearly $2°$.

Knowing the size of the moon's orbit and the length of the month,
the velocity of her motion around the earth is easily calculated. It
comes out 2288 miles per hour, or about 3350 feet a second.

\includegraphicsmid{illo075}{\textsc{Fig.~75.}---Moon's Path with Reference to the Sun.}

\nbarticle{241.} \nbparatext{Form of the Moon's Orbit with Reference to the Sun.}---While
the moon moves in a small elliptical orbit around the earth, it
also moves around the sun in company with the earth. This \textit{common}
%% -----File: 162.png---Folio 151-------
motion of the moon and earth, of course, does not affect their \textit{relative}
motion; but to an observer outside the system the moon's motion
around the earth would only be a very small component of the moon's
movement as seen by \textit{him}.

The distance of the moon from the earth, 239,000 miles, is very
small compared with that of the earth from the sun, 93,000,000 miles---being
only about $\frac{1}{400}$ part. The speed of the earth in its orbit
around the sun is also more than thirty times faster than that of the
moon in its orbit around the earth, so that for the moon the resulting
path in space is one which is always \textit{concave towards the sun},
as shown in \figref{illo075}{Fig.~75}. It is \textit{not} like Figs.~76 and 77, as often represented.
If we represent the orbit of the earth by a circle with a
radius of 100~inches (8~feet 4~inches), the moon would only move
out and in a quarter of an inch, crossing the circumference twenty-five
times in going once around it.

\begin{center}
\begin{tabular}{cc}
  \includegfx{illo076}&
  \includegfx{illo077}\\
  \footnotesize\textsc{Fig.~76.} & \footnotesize\textsc{Fig.~77.}
\end{tabular}
\captionof*{figure}{False Representations of Moon's Motions.}
\end{center}

\nbarticle{242.} \nbparatext{Diameter of the Moon.}---The mean apparent diameter of the
moon is $31'\ 7''$. This gives it a real diameter of 2163 miles (plus or
minus one mile), which equals 0.273 of the earth's diameter. Since
the surfaces of globes are as the squares of their diameters, and their
volumes as their cubes, this makes the surface of the moon 0.0747 of
the earth's (between $\frac{1}{13}$ and $\frac{1}{14}$); and the volume 0.0204 of the earth's
volume (almost exactly $\frac{1}{49}$); that is, it would take 49 balls each as
large as the moon in bulk to make a ball of the size of the earth.

\nbarticle{243.} \nbparatext{Mass of the Moon.}---This is about $\frac{1}{80}$ of the earth's mass,
different authorities giving the value from $\frac{1}{75}$ to $\frac{1}{85}$. It is not easy
to determine it with accuracy. In fact, though the moon is the
nearest of all the heavenly bodies, it is more difficult to ``weigh'' her
than to weigh Neptune, although he is the most remote of the planets.

There are four ways of approaching the problem: (1)~(perhaps
easiest to understand) \textit{by finding the position of the common centre of
gravity of the earth and moon with reference to the centre of the earth}.
Since it is this \textit{common centre of gravity} of the two bodies which
describes around the sun the ellipse which we have called the
earth's orbit, and since the earth and moon revolve around this
common centre of gravity once a month, it follows that this monthly
motion of the earth causes an alternate eastward and westward
displacement of the sun in the sky, which can be measured.
At the time of the new and full moon this displacement is zero, the
centre of gravity being on the line which joins the earth and sun;
%% -----File: 163.png---Folio 152-------
but when the moon is at \textit{quadrature} (that is, $90°$ from the sun, as at
the time of half-moon), the sun
is apparently displaced in the sky
\textit{towards the moon}, as is evident
from \figref{illo078}{Fig.~78}. It will be about
$6''.3$ east of its mean place at the
first quarter of the moon, \figref{illo078}{Fig.~78}
($B$), and as much west at the time
of the last quarter, \figref{illo078}{Fig.~78} ($A$);
(\textit{i.e.}, when the angle $MGS$ is $90°$,
the angle $MCS$ is always \textit{less} than
$90°$ by $6''.3$, which is therefore the
value of the angle $CSG$). Now
since the parallax of the sun
(which is the earth's semi-diameter
seen from the sun---the
angle $CSK$) is about $8''.8$, it follows
that the distance of the centre
of gravity of the earth and
moon from the centre of the
earth is the fraction $\frac{6.3}{8.8}$ of the
earth radius, or, about 2830 miles.
This is just about $\frac{1}{80}$ of the distance
from the earth to the moon, whence we conclude that the mass
of the earth is 80 times that of the moon.

\includegraphicsouter[21]{illo078}{\textsc{Fig.~78.}\\
Apparent Displacement of Sun at First and
Third Quarters of the Month.}
\begin{fineprint}
\nbarticle{244.} (2) A second method is by comparing the moon's \textit{actual period} with
the \textit{computed period to which a single particle at the moon's distance from the earth
ought to have}, according to the known force of gravity of the earth, as determined
by pendulum experiments. The explanation of this method cannot
be given until we have further studied the motion of bodies under the law
of gravitation. It is not susceptible of great accuracy.

(3) Still another method is by comparing the \textit{tides produced by the moon
with these produced by the sun}. This gives us the mass of the moon as compared
with that of the sun; and the mass of the sun compared with that of
the earth being known, it gives us ultimately the mass of the moon compared
with that of the earth.

(4) The ratio of the moon's mass to the sun's can also be computed from
the \textit{nutation} of the earth's axis, (See \chapref{CHAPTERXIII}{Chap.~XIII.})

\nbarticle{245.}  No other satellite is nearly as large as the moon, in comparison
with its primary planet. The earth and moon together, as seen from a
%% -----File: 164.png---Folio 153-------
distant star, are really in many respects more like a \textit{double planet} than like
a planet and satellite, as ordinarily proportioned to each other. At a time,
for instance, when Venus happens to be near the earth, at a distance of
about twenty-five millions of miles, the earth to her would appear just about
as bright as Venus at her best does to us; and the moon would be about as
bright as Sirius, at a distance of about half a degree from the earth.
\end{fineprint}

\nbarticle{246.} \nbparatext{Density and Superficial Gravity of the Moon.}---Since the
density of a body is equal to $\dfrac{\text{mass}}{\text{volume}}$, the density of the moon as
compared with the earth is found from the fraction
\[
\frac{\frac{1}{80}}{\frac{1}{49}}, \text{ or }\frac{0.125}{0.0204}.
\]

This makes the moon's density 0.613 of the earth's density, or
about $3\frac{4}{10}$ the density of water---somewhat above the average density
of the rocks which compose the crust of the earth.

\begin{fineprint}
This small density of the moon is not surprising, nor at all inconsistent
with the belief that it once formed part of the same mass with the earth,
since if such were the case, the moon was probably formed by the separation
of the \textit{outer portions} of that mass, which would be likely to have a smaller
specific gravity than the rest.
\end{fineprint}

\nbarticle{247.} \nbparatext{The superficial gravity}, or the attraction of the moon for bodies
at its own surface, may be found by the equation
\[
g' = g × \frac{m}{r^{2}},
\]
in which $g'$ signifies the superficial gravity of the moon, $g$ is the force
of gravity on the earth, while $m$ and $r$ are the mass and radius of the
moon as compared with those of the earth. This gives us
\[
g' = g × \frac{0.0125}{0.0747},
\]
or (very approximately) $g'$ equals \textit{one-sixth} of $g$; that is, a body which
weighs six pounds on the earth's surface would at the surface of the
moon weigh only one (in a spring balance). A man on the moon
could jump six times as high as he could on earth and could throw
%% -----File: 165.png---Folio 154-------
a stone six times as far. This is a fact to be remembered in connection
with the enormous scale of the surface-structure of the moon.
Volcanic forces, for instance, upon the moon would throw the rejected
materials to a vastly greater distance there than on the earth.

\includegraphicsouter[9]{illo079}{\textsc{Fig.~79.}}

\nbarticle{248.} \nbparatext{Rotation of the Moon.}---The moon rotates on its axis once a
month, in precisely the same time as that occupied
by its revolution around the earth. In the
long run it therefore keeps the same side always
towards the earth: we see to-day precisely the
same face and aspect of the moon as Galileo did
when he first looked at it with his telescope, and
the same will continue to be the case for thousands
of years more, if not forever.

\begin{fineprint}
It is difficult for some to see why a motion of this
sort should be considered a \textit{rotation} of the moon, since
it is essentially like the motion of a ball carried on a
revolving crank. See \figref{illo079}{Fig.~79}. Such a ball, they say, ``\textit{revolves} around the
shaft, but does not \textit{rotate} on its own axis.'' It does rotate, however. The
shaft being vertical and the crank horizontal, suppose that a compass
needle be substituted for the ball, as in \figref{illo080}{Fig.~80}. The pivot turns underneath
it as the crank whirls, but the compass
needle does not rotate, maintaining always
its own direction with the marked end north.
On the other hand, if we mark one side of
the ball (in the preceding \figref{illo079}{figure}), we shall
find the marked side presented successively to
every point of the compass as the crank revolves,
so that the ball as really turns on its
own axis as if it were whirling upon a pin
fastened to a table. The ball has \textit{two} distinct
motions by virtue of its connection with the crank: \textit{first}, the motion
of translation, which carries its centre of gravity, like that of the compass
needle, in a circle around the axis of the shaft; \textit{secondly}, an additional
motion of rotation around a line drawn through its centre of gravity parallel
to the shaft.

\includegraphicsouter{illo080}{\textsc{Fig.~80.}}

\sloppy
\nbarticle{248*.}
\nbparatext{Definition of Rotation.}---A body ``\textit{rotates}'' whenever \textit{a line
drawn from its centre of gravity outward, through any point selected at random
in its mass, describes a circle in the heavens}. In every rotating body, one such
line can be so drawn that the circle described by it in the sky becomes infinitely
small, This is the \textit{axis} of the body. Another set of points can be
found such that lines drawn from the centre of gravity outward through
%% -----File: 166.png---Folio 155-------
\textit{them} describe a great circle in the sky $90°$ distant from the point pierced
by the axis, and these points constitute the \textit{equator} of the body.
\end{fineprint}

\fussy
\nbarticle{249.} \nbparatext{Librations of the Moon.}---1.~\textit{Libration in Latitude}. The axis
of revolution of the moon is not perpendicular to its orbit. It makes
a constant angle of about $88\frac{1}{2}°$ with the ecliptic, and the moon's equator
is so placed that it is always edge-wise to the earth when the moon is
at her node, being maintained in that position by an action of the earth,
which produces a precessional motion of the moon's axis. The angle
between the moon's equator and the plane of her orbit, therefore, is $1\frac{1}{2}°+$
the inclination of the moon's orbit, which together make up an angle
of a little more than $6\frac{1}{2}°$; but, as the inclination of the moon's orbit
to the ecliptic is constantly varying slightly, this inclination of the
moon's axis to her orbit also changes correspondingly. This inclination
of the moon's axis produces changes in the aspect of the moon
towards the earth similar to these produced by the inclination of the
earth's axis towards the ecliptic. At one time, just as the north pole
of the earth is turned towards the sun, so also the north pole of the
moon is tipped towards the earth at an angle of $6\frac{1}{2}°$, and in the opposite
half of the moon's orbit the south pole is similarly presented to
us. In consequence we alternately look over the northern and southern
portions of the moon's disc.

\begin{fineprint}
The period of this libration is the time of the moon's revolution from
node to node, called a \textit{nodical
revolution}. This is 27.21 days---about
2~hours and 38~minutes
shorter than the sidereal revolution
of the moon, since the
nodes always move westward,
completing the circuit in about
19~years.
\end{fineprint}

\nbarticle{250.} 2.~\textit{Libration in Longitude}.
The moon's orbit
being eccentric, she moves
faster when near perigee,
and slower when near apogee;
half-way between perigee
and apogee she is more
than $6°$ ahead of the position
she would have if she had moved with the \textit{mean} angular velocity.
Now the rotation is \textit{uniform}. A point, therefore, on the moon's
%% -----File: 167.png---Folio 156-------
surface which is directed toward the earth at perigee will not
have revolved far enough to keep it directed toward the earth when
she is half-way (\textit{in time}) between perigee and apogee, as is evident
from \figref{illo081}{Fig.~81}. For in the quarter-month next following the perigee,
the moon will travel to a point $M$, considerably more than half-way to
apogee. But the point $a$ will have made only one quarter-turn, which
is not enough to bring it to the line $ME$. We shall therefore see a little
around the \textit{western} edge. Similarly on the other side of the orbit,
half-way between apogee and perigee, we shall look around the \textit{eastern}
edge to the same extent. At perigee and apogee both, the libration
is, of course, zero. The amount of this libration is evidently at any
moment just the same as that of the so-called ``equation of the centre,''
which, it will be remembered, is the difference between the \textit{mean} and
\textit{true anomalies} of the moon at any moment. Its maximum possible
value is $7°\ 45'$.

\includegraphicsouter{illo081}{\textsc{Fig.~81.}---The Libration in Longitude.}

\begin{fineprint}
The period of this libration is the time it takes the moon to go around
from perigee to perigee---the so-called \textit{anomalistic revolution}, which is 27.555
days, about 5~hours and 36~minutes longer than the sidereal month, and 8
hours 14~minutes longer than the moon's \textit{nodical} revolution, which determines
the libration in latitude.

The cause of the increased length of the anomalistic revolution is of
course the fact that the line of apsides continually advances \textit{eastward}, making
one revolution every nine years.
\end{fineprint}

\nbarticle{251.} 3.~\textit{Diurnal Libration}. This is strictly a libration not of the
moon, but of the observer; still, as far as the aspect of the moon
goes, the effect is precisely the same as if it were a true lunar libration.
The moon's motions have reference to the earth's centre. We,
on the surface of the earth, look down over the western edge of the
moon when it is rising, and over the eastern when it is setting, by
an amount which is equal to the semi-diameter of the earth as seen
from the moon; that is, about one degree (the moon's parallax).

On the whole, taking all three librations into account, we see considerably
more than half the moon, the portion which never disappears
being about \textit{forty-one per cent} of the moon's surface, that never visible
also \textit{forty-one} per cent, while that which is alternately visible and
invisible is \textit{eighteen} per cent.

\nbarticle{252.} The agreement between the moon's time of rotation and of
her orbital revolution cannot be accidental. It is probably due to the
action of the earth on some slight protuberance on the moon's surface,
%% -----File: 168.png---Folio 157-------
analogous to a tidal wave. If the moon were ever plastic, such a
bulge must have been formed on the side of the moon next the earth,
and would serve as the handle by which the earth always keeps the
same face of the moon towards herself. This subject will be resumed
later.

\includegraphicsmid{illo082}{\textsc{Fig.~82.}---Explanation of the Phases of the Moon.}

\sloppy
\nbarticle{253.} \nbparatext{The Phases of the Moon.}---Since the moon is an opaque
globe, shining entirely by reflected light, we can see only that hemisphere
of her surface which happens to be illuminated, and of course
only that part of the illuminated hemisphere which is at the time turned
towards the earth. At new moon, when the moon is between the
earth and the sun, the dark side is towards us. A week later, at the
end of the first quarter, half of the illuminated hemisphere is seen,
and we have the half moon, just as we do a week after the full. Between
the new moon and the half moon, during the first and last
quarters of the lunation, we see less than half of the illuminated portion,
and then have the ``crescent'' phase. See \figref{illo082}{Fig.~82} (in which the
%% -----File: 169.png---Folio 158-------
light is supposed to come from a point far above the moon's orbit).
Between the half moon and the full, during the second and third
quarters of the lunation, we see more than half of the moon's illuminated
side, and have what is called the ``gibbous'' phase.

\fussy
Since the terminator or line which separates the dark portion of the
disc from the bright is always a \textit{semi-ellipse} (being a semi-circle viewed
obliquely), the illuminated surface is always a figure made up of a
\textit{semi-circle} plus or minus a \textit{semi-ellipse}, as shown in \figref{illo083}{Fig.~83}, $A$.

\includegraphicsouter{illo083}{\textsc{Fig.~83.}}

\begin{fineprint}
It is sometimes incorrectly attempted to represent the crescent form
by a construction like \figref{illo083}{Fig.~83}, $B$ (where a
smaller circle is cut by a larger one). It is
to be noticed that $ab$, the line which \DPtypo{joints}{joins}
the cusps, is always perpendicular to the line
directed to the sun, and \textit{the horns are always
turned away from the sun}; so that the precise
position in which they will stand at any time is
always predictable, and has nothing whatsoever
to do with the weather. Artists are sometimes careless in the manner in
which they introduce the moon into landscapes. One occasionally sees the
moon near the horizon with the \textit{horns turned downwards}, a piece of drawing
fit to go with Hogarth's barrel which shows both its heads at once.

\nbarticle{254.} \nbparatext{Earth-Shine on the Moon.}---Near the time of new moon the whole
disc of the satellite is easily visible, the portion on which sunlight does not fall
being illuminated by a pale ruddy light. This light is \textit{earth-shine}, the earth
as seen from the moon being then nearly full; for seen from the moon the earth
shows all the phases that the moon does, the earth's phase in every case being
exactly supplementary to that of the moon as seen by us.

As the earth has a diameter nearly four times that of the moon, the earth-shine
at any phase would be about thirteen times as strong as moonlight, if
the reflective power of the earth's surface were the same. Probably, taking
the clouds and snow into account, the earth's surface on the whole is rather
more brilliant than the moon's, so that near new moon the earth-shine, by
which the dark side of the moon is then illuminated, is from fifteen to
twenty times as strong as full moonlight. The ruddy color is due to the
fact that light sent to the moon from the earth has twice penetrated our
atmosphere and so has acquired the sunset tinge.
\end{fineprint}

\nbarticle{255.} \nbparatext{Physical Characteristics of the Moon.}---1.~\textit{Its Atmosphere}.
The moon's atmosphere, if it has any at all, is extremely rare, probably
not producing a barometric pressure to exceed $\frac{1}{25}$ of an inch
of mercury, or $\frac{1}{750}$ of the pressure at the earth's surface. The
evidence on this point is twofold.
%% -----File: 170.png---Folio 159-------

(\textit{a}) \textit{The telescopic appearance}. The parts of the moon near the
edge of the disc, which, if there were any atmosphere, would be
seen through its greatest possible depth, are seen without the least
distortion: there is no haze, and all shadows are perfectly black.
There is no sensible twilight at the cusps of the moon; no evidences
of clouds or storms, or anything like atmospheric phenomena.

(\textit{b}) \textit{The absence of refraction} when the moon intervenes between
us and any more distant object. For instance, at an eclipse of the
sun there is no distortion of the sun's limb where the moon cuts it,
nor any ring of light running out on the edge of the moon like that
which encircles the disc of Venus at the time of a transit. The most
striking evidence of this sort comes, however, from occultations of
the stars. When the moon hides a star from sight, the phenomenon,
if it occurs at the moon's dark edge, is an exceedingly striking
one. The star retains its full brightness in the field of the telescope
until all at once, without the least warning, it simply is not there,
the disappearance generally being absolutely instantaneous. Its reappearance
is of the same sort, and still more startling. Now if the
moon had any perceptible atmosphere (or the star any sensible diameter)
the disappearance would be gradual. The star would change
color, become distorted, and fade away more or less gradually.

The spectroscope adds its evidence in the same direction. There is
no modification of the spectrum of the star in any respect at the time
of its disappearance; and we may add that the spectrum of moonlight
is identical with that of sunlight pure and simple, there being no
traces of any effect whatever produced upon the sunlight by its reflection
from the moon, nor any signs of its having passed through
an atmosphere.

\nbarticle{256.} The time during which a star would be hidden behind the
moon would also be decreased by the refraction of any sensible
atmosphere, making the observed duration of an occultation less
than that computed from the known diameter of the moon and its rate
of motion. Certain Greenwich observations \textit{apparently show a difference},
amounting to about \textit{two seconds} of time. This may possibly be
due in some part to the action of a \textit{real}, but exceedingly rare, lunar
atmosphere; for if the whole phenomenon were due simply to atmospheric
action, it would indicate an atmosphere having a density about
$\frac{1}{2000}$ of our own,---far within the limits which were stated above.
But the difference may be, and very probably is, attributable, in part
at least, to a slight error in the measured diameter of the moon, due to
%% -----File: 171.png---Folio 160-------
\textit{irradiation}: the diameter of a bright object always appears a little
larger than it really is. An error of about $2''$ of this sort would
explain the whole discrepancy, without any need of help from an
atmosphere.

\begin{fineprint}
\nbarticle{257.} \nbparatext{What has become of the Moon's Atmosphere.}---if the moon
ever formed a part of the same mass as the earth, she must once have had
an atmosphere. There are a number of possible and more or less probable
hypotheses to account for its disappearance. It has been surmised (1)~that
there may be great cavities left within the moon's mass by volcanic eruptions,
and that the rocks themselves have been transformed into a sort of
pumice-stone structure, and that the air has retired into these internal
cavities.

(2) That the air has been absorbed by the inner lunar rocks in cooling. A
heated rock expels any gases that it may have absorbed; but if it afterwards
cools slowly, it reabsorbs them, and can take up a very great quantity. The
earth's core is supposed to be now too intensely heated to absorb much gas;
but if it goes on cooling, it will absorb more and more, and in time it may
rob the surface of the earth of all its air. There are still other hypotheses,
which we can not take space even to mention.
\end{fineprint}

\nbarticle{258.} \nbparatext{Water on the Moon's Surface.}---Of course without an atmosphere
there can be no water, since the water would immediately evaporate
and form an atmosphere of water vapor if there were no air
present. It is not impossible, however, or even improbable, that
\textit{solid} water, that is, ice and snow, may exist on the moon's surface
at a temperature too low for any sensible evaporation. There are
many things in the moon's appearance that seem to indicate the former
existence of seas and oceans on her surface, and the same
hypotheses have been suggested to account for their disappearance
that were suggested in the case of the moon's atmosphere. It may
be added also that many kinds of molten rock in crystallizing would
take up large quantities of water of crystallization, not merely absorbed
as a sponge absorbs water, but chemically united with the other
constituents of the rock. In whatever way, however, it may have
come about, it is certain that \textit{now} no substances that are gaseous, or
that can be evaporated at low temperatures, exist in any quantity on
the moon's surface---at least, not \textit{on our side} of the moon.

\begin{fineprint}
There have been speculations that on the other side---that celestial country
so near us and so absolutely concealed from us---there may be air and
water and abundant life; the idea being that our side of the moon is a great
table land many miles in elevation, while the other side is a corresponding
%% -----File: 172.png---Folio 161-------
depression, like the valley of the Caspian Sea, only vastly deeper. An insufficiently
grounded conclusion of Hansen's, that the centre of gravity of
the moon is some thirty miles farther from us than its centre of figure, for a
time gave color to the idea, but it is now practically abandoned, Hansen's
conclusion having been shown to be unwarranted by the facts.
\end{fineprint}

\nbarticle{259.} \nbparatext{The Moon's Light.}---As to \textit{quality} it is simple sunlight, showing
a spectrum which, as has been said, is identical in every detail
with that of light coming directly from the sun. Its \textit{brightness} as
compared with that of sunlight is difficult to measure accurately, and
different experimenters have found results for the ratio between full
moonlight and sunlight ranging all the way from $\frac{1}{300000}$ (Bouguer)
to $\frac{1}{800000}$ (Wollaston). The value now usually accepted is that
determined by Zöllner, viz., $\frac{1}{618000}$. According to this, if the whole
visible hemisphere were packed with full moons, we should receive
from it about one-eighth part of the light of the sun.

\begin{fineprint}
It is found, also, that the half moon does not give even nearly half as
much light as the full moon. The law which connects the phase of the
moon with the amount of light given at the time, is rather complicated, but
the gist of the matter is that at any time, except at the full, the visible surface
is more or less darkened by the shadows cast by the irregularities of the
surface. Zöllner has calculated that an average angle of $52°$ for these elevations
and depressions would account for the law of illuminations actually
observed.
\end{fineprint}

The average ``\textit{albedo},'' or reflecting power of the moon's surface,
Zöllner states as 0.174; that is, the moon's surface \textit{reflects a little
more than one-sixth} part of the light that falls upon it. This is about
the albedo of a rather light-colored sandstone, and agrees well with
the estimate of Sir John Herschel, who found the moon to be very
exactly of the same brightness as the rock of Table Mountain when
it was setting behind it, illuminated as were the rocks themselves by
the light of the rising sun. There are, however, great variations in
the brightness of different portions of the moon's surface. Some
spots are nearly as white as snow or salt, and others as dark as
slate.

\nbarticle{260.} \nbparatext{Heat of the Moon.}---For a long time it was impossible to
detect the moon's heat. It is too feeble to be detected by the most
delicate mercurial thermometer even when concentrated by a large
lens. The first sensible effect was obtained by Melloni, in 1846,
%% -----File: 173.png---Folio 162-------
with the then newly invented thermopile, by a series of observations
from the summit of Vesuvius. Since then several physicists have
worked upon the subject with more or less success, but the most recent
and reliable investigations are those of Lord Rosse and Professor
Langley. With modern apparatus there is no difficulty in detecting
the heat in the lunar radiations, but \textit{measurements} are extremely
difficult and liable to error. A considerable percentage of the lunar
heat seems to be heat simply \textit{reflected} (like light), while the rest,
perhaps three-fourths of the whole, is ``\textit{obscure heat}''; that is, heat
which has been first absorbed by the moon's surface and then \textit{radiated},
like the heat from a brick surface that has been warmed by sunshine.
This is shown by the fact that a comparatively thin plate of glass
cuts off some 86 per cent of the heat received from the moon in the
same way that it does the heat of a stove, while the heat of direct
sunlight, or of an electric arc, would pass through the same plate
with very little diminution. The same thing appears also from direct
measurements upon the \textit{heat-spectrum} of the moon made by Langley
with his bolometer, described further on. (\artref{Art.}{343}.)

\nbarticle{261.} As to the \textit{temperature of the moon's surface}, it is difficult to
affirm much with certainty. On one hand, the lunar rocks are exposed
to the sun's rays in a cloudless sky for fourteen days at a time, so
that if they were blanketed by air like our own rocks they would certainly
become intensely heated. A few years ago, Lord Rosse inferred
from his observations that the temperature of the lunar surface
rose at its maximum (about three days after full moon) far above
that of boiling water.

But his own later investigations and these of Langley throw great
doubt on this conclusion. There is no air-blanket at the moon's
surface to prevent it from losing heat as fast as it receives it; and
it now seems rather more probable that the temperature never rises
above the freezing-point of water, as is the case on the highest of
our mountains, where there is perpetual ice, and the temperature is
always low even at noon. So far as we can judge, the condition of
things on the moon's surface must correspond to an elevation many
times higher than any mountain on the earth; for no terrestrial mountain
is so high that the density of the air at its summit is even nearly
as low as that of the densest supposable lunar atmosphere.

This idea, that the temperature is low, is borne out, also, by the
fact that the bolometer shows the presence, in the lunar radiations,
of a considerable quantity of heat having a wave-length greater than
that of the heat radiated from a block of ice.
%% -----File: 174.png---Folio 163-------

At the end of the long lunar night of fourteen days the temperature
must fall appallingly low, \textit{certainly} $200°$ below zero.

The whole amount of heat sent by the full moon to the earth is
estimated by Rosse as \textit{about one eighty-thousandth part of that sent by
the sun}.

\nbarticle{262.} \nbparatext{Lunar Influences on the Earth.}---The moon's attraction cooperates
with that of the sun in producing tides, of which we shall
speak hereafter. There are also certain distinctly ascertained disturbances
of \DPtypo{terrestial}{terrestrial}
magnetism connected with the approach and
recession of the moon at perigee and apogee; and this ends the
chapter of \textit{ascertained} lunar influences.

\begin{fineprint}
The multitude of current beliefs as to the controlling influence of the
moon's phases and changes over the weather and the various conditions of
life are mostly unfounded, and in the strict sense of the word ``superstitions,''---mere
survivors from a past credulity.

It is quite certain that if there is any influence at all of the sort it is extremely
slight---so slight that it cannot be demonstrated with certainty,
although numerous investigations have been made expressly for the purpose
of detecting it. We have never been able to ascertain, for instance, with
certainty, whether it is \textit{warmer or not}, or \textit{less cloudy or not}, at the time of the
full moon. Different investigations have led to contradictory results.

The frequency of the moon's changes is so great that it is always easy to
find instances by which to verify a belief that changes of the moon control
conditions on the earth. A change of the moon necessarily occurs about
once a week, the interval from quarter to quarter being between seven and
eight days. \textit{All} changes, of the weather for instance, must therefore occur
within three and three-fourths days of a change of the moon, and fifty per
cent of them ought to occur within forty-six hours of a change, even if there
were no causal connection whatever.

Now it requires only a very slight prepossession in favor of a belief in
the effectiveness of the moon's changes to make one forget a few of the
weather changes that occur too far from the proper time. Coincidences
enough can easily be found to justify a preėxisting belief.
\end{fineprint}

\section*{THE MOON'S SURFACE.}

\nbarticle{263.} Even to the naked eye the moon is a beautiful object,
diversified with darker and lighter markings which have given rise to
numerous popular superstitions. With a powerful telescope these
naked-eye markings mostly vanish, and are replaced by a countless
multitude of smaller details, which are interesting in the highest
degree. The moon on the whole, on account of this diversity of
%% -----File: 175.png---Folio 164-------
detail, is the finest of all telescopic objects; especially to moderate-sized
instruments, say from six to ten inches in diameter, which
generally give a more pleasing view of our satellite than instruments
either larger or smaller.

\includegraphicsouter{illo084}{\textsc{Fig.~84.}---A Normal Lunar Crater (Nasmyth.)}

\nbarticle{264.} \nbparatext{How near the Telescope brings the Moon.}---An instrument
of this size, with magnifying powers between 250 and 500, brings
up the moon virtually to a distance ranging from 1000 miles to 500;
and since an object a mile in diameter on the moon subtends an
angle of about $0''.86$, with the higher powers of such an instrument
objects less than a mile in diameter become visible under favorable
atmospheric conditions. A long line or streak, even less than a
quarter of a mile across, could probably be seen. With larger telescopes
the power can now and then be carried at least twice as high,
and correspondingly smaller details made out. When everything is
at its best, the great Lick telescope of 30 inches aperture, with a
power of 2500 or so, may possibly reduce the virtual distance of our
satellite to about 100 miles for visual purposes. It is evident that
while with our telescopes we should be able to see such objects as
lakes, rivers, forests, and great cities, if they exist on the moon, it
will be hopeless to expect to distinguish single buildings, or any of
the ordinary operations and indications of life, if such there are.

\begin{fineprint}
There are a few mountains on the earth from which a range of 100 miles
is obtained in the landscape. Those who have seen such a landscape know
how little is to be made out with the naked eye at that distance. Still, the
comparison is not quite fair, because in looking at a terrestrial object a hundred
miles away the line of vision passes through a dense atmosphere, while
in looking upward towards the moon it penetrates a much less thickness
of air.
\end{fineprint}

\nbarticle{265.} \nbparatext{The Moon's Surface Structure.}---The moon's surface for the
most part is extremely uneven and broken, far more so than that of
the earth. The structure, however, is not like that of the earth's
surface. On the earth the mountains are mostly in long ranges, such
as the Alps, the Andes, and Himalayas. On the moon such mountain
ranges are few in number, though they exist; but the surface is
pitted all over with great craters, resembling very closely the volcanic
craters on the earth's surface, though on an immensely greater
scale. One of the largest craters upon the earth, if not the largest,
is the Aso San in Japan, about seven miles across. Many of those
on the moon are fifty and sixty miles in diameter, and some are
%% -----File: 176.png---Folio 165-------
over 100 miles across, while smaller ones from a half-mile to eight
or ten miles in diameter are counted by the thousand.

\includegraphicsmid{illo085}{\textsc{Fig.~85.}---Map of the Moon.}

The normal lunar crater is nearly circular, surrounded by an elevated
ring of mountains
which rise anywhere from
1000 to 20,000 feet above
the surrounding country.
Within the floor of the
crater the surface may be
either above or below the
outside level. Some craters
are deep, some filled
nearly to the brim. In
some cases the surrounding
mountain ring is entirely
absent, and the crater is a mere hole in the plain. In the
centre of the crater there usually rises a group of peaks, which attain
about the same elevation as the encircling ring, and these central
peaks often show little holes or craterlets in their summits.

\begin{fineprint}
In most cases the resemblance of these formations to terrestrial volcanic
structures, like those exemplified by Vesuvius and others in the surrounding
region, makes it natural to assume that they had a similar origin.
This, however, is not absolutely certain, for there are considerable difficulties
in the way, especially in the case of the great ``Bulwark Plains,'' so called,
which are so extensive that a person standing in the centre could not see
the summit of the surrounding ring at any point; and yet no line of demarcation
can be drawn between them and the smaller craters. The series
is continuous. Moreover, on the earth, volcanoes necessarily require the
action of air and water, which do not now exist on the moon. It is obvious,
therefore, that if these lunar craters are the result of true volcanic eruptions,
they must be fossil formations; for it is quite certain that no evidence of
existing volcanic activity has ever been found. The moon's surface appears
to be absolutely quiescent---still in death.
\end{fineprint}

On some portions of the moon these craters stand very thickly.
Older craters have been encroached upon, or more or less completely
obliterated by the newer, and the whole surface is a chaos, of which
the counterpart is hardly to be found on the earth, even in the roughest
portions of the Alps. This is especially the case near the moon's
south pole. It is noticeable that, as on the earth the newest mountains
are generally the highest, so on the moon the more newly
formed craters are generally deeper and more precipitous than the
older ones.
%% -----File: 177.png---Folio 166-------

{\stretchyspace \nbarticle{266.} \nbparatext{Lunar Nomenclature.}---The great plains were called by
Galileo} oceans or seas (\textit{Maria}), and some of the smaller ones
marshes (\textit{Paludes}) and lakes, for he supposed that the grayish surfaces
visible to the naked eye, and conspicuous in a small telescope,
were covered with water. Thus we have the ``Oceanus Procellarum,''
the ``Mare Imbrium,'' and a number of other ``seas,'' of which
``Mare Fecunditatis,'' ``Mare Serenitatis,'' and ``Mare Tranquilitatis,''
are the most conspicuous. There are twelve of them in all, and
eight or nine Paludes, Lacus, and Sinus.

\sloppy
The ten mountain ranges on the moon are mostly named after
terrestrial mountains, as Caucasus, Alps, Apennines, though two or
three bear {\stretchyspace the names of astronomers, like Leibnitz,} Dörfel, etc.
%% -----File: 178.png---Folio 167-------
The conspicuous craters bear the names of the more eminent ancient
and medięval astronomers and philosophers, as Plato, Archimedes,
Tycho, Copernicus, Kepler, and Gassendi; while hundreds of smaller
and less conspicuous formations bear the names of more modern or
less noted astronomers.

\fussy
\begin{fineprint}
The \textit{system} seems to have originated with Riccioli in 1650, but most of
the names have been more recently assigned by the later map-makers, the
most eminent of whom have been the German astronomers Beer and Maedler
(who published their map in 1837), and Schmidt of Athens, whose great
map of the moon, on a scale seven feet in diameter, was published by the
Prussian government a few years ago. It is not at all too much to say that
our maps of the earth's surface do not, on the whole, compare in fulness and
accuracy with our maps of the moon. Of course this is not true of such
countries as France and England, or others that have been trigonometrically
surveyed; but there are no such \textit{lucunę} in our maps of the moon as exist in
our maps of Asia and Africa, for instance.
\end{fineprint}

\includegraphicsouter[16]{illo086}{\textsc{Fig.~86.}---Archimedes and the Apennines (Nasmyth).}

\sloppy
\nbarticle{267.} \nbparatext{Other Lunar Formations.}---The craters and mountains are
not the only interesting formations
on the moon's surface.
There are many deep,
narrow, crooked valleys that
go by the name of ``rills''
(German \textit{Rillen}), some of
which may once have been
watercourses. Then there
are numerous ``clefts,'' half
a mile or so wide and of unknown
depth, running in
some cases several hundred
miles, straight through mountain
and valley, without any
apparent regard for the accidents
of the surface.
They seem to be deep
cracks in the crust of our
satellite. Several of them
are shown in \figref{illo086}{Fig.~86}. Most
curious and interesting of all are the light-colored \textit{streaks} or ``\textit{rays}''
which radiate from certain of the craters, extending in some cases
a distance of several hundred miles. They are usually from five to
ten miles wide, and neither elevated nor depressed to any extent with
%% -----File: 179.png---Folio 168-------
reference to the general surface. They pass across mountain and
valley, and sometimes through craters without any change in width
or color. We do not know whether they are like the so-called ``trap-dykes''
on the earth,---fissures which have been filled up from below
with some light-colored material,---or whether they are mere surface
markings. No satisfactory explanation has ever been given.

\fussy
The most remarkable system of ``rays'' of this kind is the one
connected with the great crater Tycho, not very far from the moon's
south pole. They are not very conspicuous until within a few days
of full moon, but at that time they, and the crater from which they
radiate, constitute by far the most striking feature of the whole lunar
landscape.

\includegraphicsouter{illo087}{\textsc{Fig.~87.}---Gassendi (Nasmyth).}

\nbarticle{268.} \hspace{-2.1pt}\nbparatext{Changes on the Moon.}\\---It is certain that there are no \textit{conspicuous}
changes. The observer
has before him no
such ever-varying vision
as he would have in looking
toward the earth,---no
flying clouds, no alternations
of seasons with the
transformation of the snowy
wastes to green fields, nor
any considerable apparent
movement of objects on the
disc. The sun rises on them
slowly as they come one
after the other to the terminator,
and sets as slowly.
At the same time it is confidently
maintained by many
observers that here and there
changes are still going on in
the details of the surface.
Others as stoutly dispute it.

\begin{fineprint}
\nbarticle{269.} Probably the most notable and best advocated instance of such a
change is that of the little crater Linné, in the Mare Serenitatis. It was observed
by Schroeter very early in the century, and is figured and described by
Beer and Maedler as being about five and a half or six miles in diameter, quite
deep and very bright. In 1866 Schmidt, who had several times observed it
before, announced that it had disappeared. A few months later it was
%% -----File: 180.png---Folio 169-------
visible again, and there were many reported changes in its appearance
during the next year or two. There is no question that it does not now at
all agree in conspicuousness and size with the representation of Beer and
Maedler, for it is at present, and has been for several years, only a minute
dark spot, with a whitish spot surrounding it. Astronomers would feel
more confident that this was a case of real change were it not that Schroeter's
earlier picture much more resembles the present appearance than does
that of Beer and Maedler. As the latter observers worked with rather a
small telescope, and had no reason for taking any special pains in the delineation
of this particular object, the evidence
is less conclusive than it might
seem at first. The change, however, if
real, was certainly as great as in the
instance of \DPtypo{Krakatoa}{Krakatćo}, the great volcano
whose eruption in 1883 filled the earth's
atmosphere with smoke and vapor for
more than two years, and caused the
``twilight conflagrations'' of the sky.
The phenomenon in the case of Linné,
if real, was probably a falling in of the
walls of the crater, exposing fresh unweathered
surfaces.

The reason why it is so difficult to
be sure of changes lies in the great
variations in the appearance of a lunar object under the varying illumination.
To insure certainty in such delicate observations, comparisons
must be made between the appearance seen at precisely the same phase of
the moon, with telescopes (and eyes too) of equal power; and under substantially
the same conditions otherwise, such as the height of the moon
above the horizon, the clearness and steadiness of the air, etc. It is of
course very difficult to secure such identity of conditions.
\end{fineprint}

\includegraphicsouter{illo088}{\textsc{Fig.~88.}}

\nbarticle{270.} \nbparatext{Measurements of Heights of Lunar Mountains.}---When the
terminator approaches a lunar mountain, the top of the mountain
catches the sunlight first, and appears as a star entirely detached
from the rest of the illuminated portion, like the little bright spots
opposite $a$ and $b$ in \figref{illo088}{Fig.~88}. As time passes, the bright spot becomes
larger as the light extends lower down the mountain side, until
the terminator reaches and passes it.

If now we measure the apparent distance, $AD$ or $a$, \figref{illo089}{Fig.~89}, from the
peak to the terminator at the moment when it first appears like a star,
it is easy to compute $AB$, and from this, the height of the mountain.

\begin{fineprint}

In \figref{illo089}{Fig.~89} the angle $S'CN$ very approximately the moon's ``elongation''
at the time of observation, since the line from the \textit{earth} to the sun is nearly
%% -----File: 181.png---Folio 170-------
parallel to $S'C$ (the moon's distance being only about $\frac{1}{400}$ of the sun's). Now
the angle $BAD = GBE'' = 90° - S'CN$, so that $AB$, or $b$, $= AD$ (or $a$)$÷\sin{S'CN}$.
Knowing $b$, and the radius of the moon $r$, we get
\[
(r + h)^{2} = r^{2} + b^{2},
\]
in which $h$ is the height of the mountain, and $b$ is the distance $AB$. From
this equation we find
\begin{flalign*}
&&  h &= \sqrt{r^{2} + b^{2}} - r,    &&\phantom{\text{or (very nearly) }}\\
&\text{or (very nearly) }&
    h &= \tfrac{1}{2}\frac{b^{2}}{r}. &&
\end{flalign*}

If, for instance, $b$ were 60 miles,
\[
h\text{(in miles)} = \frac{3600}{2× 1081} \text{ or } 1.664 \text{ miles } = 8845 \text{ feet. }
\]
\includegraphicsouter{illo089}{\textsc{Fig.~89.}\\
Measurement of the Height of a Lunar Mountain.}

Since the terminator is very ragged, it is sometimes best to measure from
the mountain top clear across
to the edge of the moon, as
indicated by the little arrows
in \figref{illo088}{Fig.~88}. The position of
the theoretical terminator
(the terminator as it would
be if the moon were a smooth
sphere) is known from the
moon's age, so that $AB$ can
be deduced by measuring
from the limb as well as
from the terminator.
\end{fineprint}

The height of a mountain
can also be ascertained
by measuring the
length of its shadow in
cases where the shadow
falls on a reasonably level surface. A few of the lunar mountains
reach the height of 22,000 or 23,000 feet, but there are none which
attain the elevation of the very highest terrestrial mountains. Heights
ranging from 10,000 to 20,000 feet are common.

\begin{fineprint}
\nbarticle{271.} \nbparatext{The Best Time to look at the Moon with a Telescope.}---The
moon when full is not so satisfactory an object as when near the half, because
at the full moon there are no shadows, so that at that time the ``relief''
of the surface structure is entirely lost. Certain features, however, as has
been before mentioned, are then best seen, as, for instance, the streaks or
%% -----File: 182.png---Folio 171-------
rays. Generally, any particular mountain, crater, rill, or cleft, is best studied
when it is just on or very near the terminator, that is, at the time when the
sun is rising or setting near it, because then the shadows are longest. The
best general view of the moon is that obtained a few days after the half
moon, when Copernicus and Tycho are both near the terminator, and Plato
is still near enough to it to show very well.

\nbarticle{272.} \nbparatext{Photographs of the Moon.}---A great deal of attention has been
paid to this subject, and some fine results have been reached. The earliest
success was that of Bond in 1850, with the old daguerreotype process; then
followed the work of De la Rue in England, and of Dr.~Henry Draper, and
especially of Mr.~Rutherfurd in this country. Rutherfurd's pictures have
remained absolutely unrivalled until very recently.

To these older experimenters the moon's motion offered a great difficulty,
but now that with the sensitive plates at present used, a fraction of a second
is a sufficient exposure, that difficulty has disappeared, and the plates which
have recently been taken at Cambridge, Mass., are far in advance even of
Rutherfurd's, showing such craters as Copernicus or Ptolemy with a diameter
of two inches, on a scale larger than that of Schmidt's map. The Lick
Observatory has also taken up the work, and is making admirable pictures.
\end{fineprint}
\chelabel{CHAPTERVII}
%% -----File: 183.png---Folio 172-------

\Chapter{VIII}{The Sun}
\nbchapterhang{\stretchyspace
THE SUN: DISTANCE AND DIMENSIONS.---MASS AND DEN\-SITY.---ROTATION.---STUDY
OF THE SURFACE: GENERAL VIEWS AS
TO THE SUN'S CONSTITUTION.---SUN SPOTS: THEIR APPEARANCE,
NATURE, DISTRIBUTION, AND PERIODICITY.---THE
SPECTROSCOPE AND THE SOLAR SPECTRUM.---CHEMICAL
ELEMENTS RECOGNIZED IN THE SUN.---THE CHROMOSPHERE
AND PROMINENCES.---THE CORONA.}

\nbarticle{273.} \textsc{The} SUN is simply a \textit{star}; a hot, self-luminous globe of
enormous magnitude as compared with the earth and the moon,
though probably only of medium size among its stellar compeers.
But to the earth and the other planets which circle around it, it is
the grandest of all physical objects. Its attraction confines its
planets to their orbits and controls their motions, and its rays supply
the energy which maintains every form of activity upon their
surfaces and makes them habitable.

\nbarticle{274.} \nbparatext{Its Distance and Dimensions.}---Its distance is determined by
finding its horizontal parallax; that is, the semi-diameter of the
earth as seen from the sun. The mean value of this parallax is
probably very near $8''.8$,\footnote
  {In the American Ephemeris the value deduced by Newcomb in 1867 is used, viz.,
$8''.85$. The British ``Nautical Almanack'' uses the same value, and the French the
value deduced by Leverrier a little earlier, $8''.86$; but more recent observations
seem to show that this value is a little too large, and that the number stated,
$8''.8$, is more probably correct. The difference is of no importance for \textit{almanac
  purposes}.}
plus or minus $0''.03$.

We reserve to a separate \chapref{CHAPTERXVI}{chapter} the discussion of the methods by
which this most fundamental and important of all astronomical data
has been ascertained, merely remarking here that the problem is one
of extreme \textit{practical} difficulty, though the principles involved are
simple enough.

Assuming the parallax at $8''.8$, the mean distance of the sun (putting
$r$ for the earth's radius) equals
\[
r÷\sin{8''.8} = 23,\!439× r.
\]

With Clarke's value of $r$ (\artref{Art.}{145}), this gives 149,500000 kilometers,
%% -----File: 184.png---Folio 173-------
or 92,897,000 miles; which, however, is uncertain by at least 200,000
miles, and is \textit{variable}, also, to the extent of about three million
miles on account of the eccentricity of the earth's orbit, the earth
being nearer the sun in December than in June.

\nbarticle{275.} This distance is so much greater than any with which we
have to do on the earth that it is possible to reach a conception of it
only by illustrations of some sort. Perhaps the simplest is that
drawn from the motion of a railway train. Such a train going 1000
miles a day (nearly forty-two miles an hour, and faster than the
Chicago Limited on the Pennsylvania Railroad) would take $254\frac{1}{3}$
years to make the journey.

If sound were transmitted through interplanetary space, and at the
same rate as through our own atmosphere, it would make the passage
in about fourteen years; \textit{i.e.}, an explosion on the sun would be
heard by us fourteen years after it actually occurred. A cannon-ball
moving unretarded, at the rate of 1700 feet per second, would
travel the distance in nine years. Light does it in 499 seconds.

\sloppy
\nbarticle{276.} \nbparatext{Diameter.}---The sun's mean apparent diameter is $32'\ 04''\pm2''$.
Since at the sun one second equals 450.36 miles, its diameter
equals 866,500 miles, or $109\frac{1}{2}$ times the diameter of the earth. It
is quite possible that this diameter is variable to the extent of a few
%% -----File: 185.png---Folio 174-------
hundred miles, since, as will appear hereafter, the sun (at least the
surface which we see) is not solid.

\fussy
Representing the sun by a globe two feet in diameter, the earth
would be $\frac{22}{100}$ of an inch in diameter,---the size of a very small pea,
or a ``22-calibre'' round pellet. Its distance from the sun on that
scale would be just about 220 feet, and the \textit{nearest star} (still on the
same scale) \textit{would be eight thousand miles away, at the antipodes}.

If we were to place the earth in the centre of the sun, supposing
it to be hollowed out, the sun's surface would be 433,000 miles away
from us. Since the distance of the moon is only about 239,000
miles, it would be only a little more than half-way out from the
earth to the inner surface of the hollow globe, which would thus
form a very good background for the study of the lunar motions.

\begin{fineprint}
It is perhaps worth noticing, as a help to memory, that the sun's diameter
exceeds the earth's just about as many times as it is itself exceeded by the
radius of the earth's orbit; or, in other words, the sun's diameter is \textit{nearly} a
mean proportional between the earth's distance from the sun and the earth's
diameter, 110 being the common ratio.
\end{fineprint}

\includegraphicsmid{illo090}{\textsc{Fig.~90.}---Dimensions of the Sun compared with the Moon's Orbit.}

\nbarticle{277.} \nbparatext{Surface and Volume.}---Since the \textit{surfaces} of globes are proportional
to the \textit{squares} of their radii, the surface of the sun exceeds
that of the earth in the ratio of $(109.5)^2$ to 1; that is, its surface is
about 12,000 times the surface of the earth.

The \textit{volumes} of spheres are proportional to the \textit{cubes} of their radii;
hence the \textit{sun's volume} is $(109.5)^3$, or 1,300000 times that of the earth.

\nbarticle{278.} \nbparatext{The Sun's Mass.}---The mass of the sun is \textit{very nearly three
hundred and thirty-two thousand times that of the earth}, subject to a
probable error of at least one per cent. There are various ways of
getting at this result. For our purpose here, perhaps the most convenient
is by comparing the earth's attraction for bodies at her surface
(as determined by pendulum experiments) with the attraction of the
sun for the earth,---the central force which keeps her in her orbit.
Put $f$ for this force (measured, like gravity, by the velocity it generates
in one second), $g$ for the force of gravity (32~feet 2~inches
per second), $r$ the earth's radius, $R$ the sun's distance, and let $E$
and $S$ be the masses of the earth and sun respectively. Then, by
the law of gravitation, we have the proportion
\begin{flalign*}
            &&&f : g :: \frac{S}{R^2} : \frac{E}{r^2}, \text{ or } S = E\left(\frac{f}{g}\right)\left(\frac{R}{r}\right)^2. &&&(a)&\\
&\text{Now, }&&\frac{R}{r} = 23,440 \text{ (nearly)}.&&&\phantom{Now, }&
\end{flalign*}
%% -----File: 186.png---Folio 175-------
Its square equals 549,433,600. $g = 386$ inches. To find $f$ we have
from Mechanics (Physics, p.~62),
\[
  f = \frac{V^2}{R}, \tag{\textit{b}}
\]
this being the expression for the ``central force'' in the case of a
body revolving in a circle. (We may neglect the eccentricity of the
earth's orbit in a merely approximate treatment of the problem.)
$V$ is the orbital velocity of the earth, which is found by dividing the
circumference of the orbit, $2\pi R$, by $T$, the number of seconds in a
sidereal year. This velocity comes out 18.495 miles per second.
Putting this into formula (\textit{b}), we get $f = 0.2333$ inches,
\begin{flalign*}
&\text{so that }&  &\frac{f}{g} = 0.0006044 = \frac{1}{1654} \text{ (nearly)};&&\\
&\text{whence }&  &S = E× \frac{1}{1654}× 519,\!433,\!600; \text{ or }S \text{ equals 332,000.}&&
\end{flalign*}
We may note in passing that half of $f$ expresses the distance by
which the earth \textit{falls towards the sun} every second, just as half $g$ is
the distance a body at the earth's surface falls in a second. This
quantity (0.116 inch), a trifle more than a ninth of an inch, is the
amount by which the earth's orbit deviates from a straight line in a
second. In travelling \textit{eighteen and one-half miles} the deflection is
only \textit{one-ninth of an inch}.

\begin{fineprint}
\nbarticle{278*.} By substituting $\dfrac{2\pi R}{T}$ for $V$ in equation (\textit{b}), we get
\begin{flalign*}
&&&f=\frac{4{\pi}^2 R}{T^2};&&
\intertext{and putting this value $f$ into equation (\textit{a}) and reducing, we obtain}
&&&S = E\left[ \left( \frac{4{\pi}^2}{T^2} \right)
               \left( \frac{r}{g}\right)
               \left( \frac{R}{r}\right)^3 \right],  &(\textit{c})&\\
&\text{or, since }&  &\frac{R}{r}=\frac{1}{\sin p}  &\phantom{or\ since }&
\intertext{($p$ being the sun's horizontal parallax), we have finally}
&&&S = E\left[ \left( \frac{4{\pi}^2}{T^2} \right)
               \left( \frac{r}{g} \right) \frac{1}{\sin^3p} \right].   &(\textit{d})&
\end{flalign*}

It will be noticed that in this expression the \textit{cube} of the parallax appears,
and this is the reason why an uncertainty of one per cent in $p$ involves an
uncertainty of three per cent in $S$.
%% -----File: 187.png---Folio 176-------

In obtaining the mass of the sun it will be seen that we require as
data, $T$, the length of the sidereal year in seconds; the value of
gravity, $g$ (which is derived from pendulum experiments); the radius
of the earth, $r$ (deduced from geodetic surveys); and finally (and
most difficult to get), the sun's parallax, $p$, or else, what comes to
the same thing, the ratio $\dfrac{r}{R}$.
\end{fineprint}

\nbarticle{279.} \nbparatext{The Sun's Density.}---This density\footnote
  {The determination of the sun's density does not \textit{necessarily} involve its parallax.
Put $\rho$ for the sun's radius, and $Ds$ for its density: also let $De$ be the earth's mean
density. Substitute in equation ($c$), and we have $\frac{4}{3}\pi\rho^{3}Ds = \frac{4}{3}\pi r^{3} De \left(\dfrac{r}{g}\right) \left(\dfrac{R}{r}\right)^{3}$,
whence $Ds = De \left(\dfrac{r}{g}\right)  \left(\dfrac{R}{\rho}\right)^{3}$. But $\left(\dfrac{\rho}{R}\right) = \sin{\Sigma}$, $\Sigma$ being the sun's (angular) semi-diameter.
   Hence, finally, $Ds = De \left(\dfrac{r}{g}\right) \dfrac{1}{\sin^{3}{\Sigma}}$.}
as compared with that of
the earth is found by simply dividing its mass by its volume (both as
compared with the earth); that is, it equals the fraction
\[
\frac{332\, 000}{1300\, 000} = 0.255,
\]
a little more than \textit{a quarter} of the earth's density. To get its
``\textit{specific gravity}'' (\textit{i.e.}, density as compared with \textit{water}), we must
multiply this by 5.58, the earth's mean specific gravity. This gives
1.41; that is, \textit{the sun's mean density is not $1\frac{1}{2}$ times that of water},---a
most significant result as bearing on its physical condition.

\nbarticle{280.} \nbparatext{Superficial Gravity.}---This is found by dividing its mass by
the square of its radius; that is,
\[
\frac{332\, 000}{(109\frac{1}{2})^{2}},
\]
which equals 27.6. A body weighing one pound on the earth's surface
would there weigh 27.6 lbs. A body would fall 444 feet in a
second, instead of sixteen feet, as here.

\nbarticle{281.} \nbparatext{The Sun's Rotation.}---The sun's surface often shows spots
upon it, which pass across the disc from east to west. These are
evidently attached to its surface, and not bodies circling around the
sun at a distance above it, as was imagined by some early astronomers,
because, as Galileo early demonstrated, they continue in sight
just as long as the time during which they are invisible; which would
not be the case if they were at any considerable elevation.
%% -----File: 188.png---Folio 177-------

\textbf{Period of Rotation.}---The average time occupied by a spot in passing
around the sun and returning to the same position again is 27.25
days,---\textit{average} because different spots show considerable differences
in this respect. This interval, however, is not the \textit{true time} of solar
rotation, but the \textit{synodic}, since the earth advances in the interval of
a revolution so that the sun has to turn on its axis a little farther each
time to bring the spot again into conjunction with the earth. The
equation by which the true period is deduced from the synodic is the
same as in the case of the moon (\artref{Art.}{232}), viz.:
\begin{flalign*}
&&  &\frac{1}{T} - \frac{1}{E} = \frac{1}{S},  &&
\intertext{$T$ being the true period of the sun's rotation, $E$ the length of the
year, and $S$ the observed \textit{synodic} rotation;}
&\text{whence }&  &\frac{1}{T} = \frac{1}{27.25} + \frac{1}{365.25},
&&\phantom{whence }
\end{flalign*}
which gives $T = 25^{\text{d}}.35$. Different observers get slightly different
results. Carrington finds $25^{\text{d}}.38$; Spoerer, $25^{\text{d}}.23$.

\nbarticle{282.} \nbparatext{Position of the Sun's Axis.}---On watching the spots with
care as they cross the disc, it appears that they usually describe paths
more or less oval, showing that the sun's axis is inclined to the
ecliptic. Twice a year, however, the paths become straight, at the
times when the earth is in the plane of the sun's rotation. These
dates are about June~3 and Dec.~5.

\begin{fineprint}
The ascending node of the sun's equator is in celestial longitude $73°\, 40'$
(Carrington), and the inclination of its equator to the plane of the ecliptic
is $7°\, 15'$. Its inclination to the plane of the terrestrial equator is $26°\, 25'$.
The position of the point in the sky towards which the sun's pole is directed
is in right ascension $18^{\text{h}}\ 44^{\text{m}}$, declination $+64°$, almost exactly half-way
between the bright star $\alpha$~Lyrę and the Pole Star.
\end{fineprint}

\nbarticle{283.} \nbparatext{Peculiar Law of the Sun's Rotation.}---\textit{Equatorial Acceleration}.
The earth rotates \textit{as a whole}, every point on its surface making
its diurnal revolution in the same time; so also with the moon and
with the planet Mars. Of course it is necessarily so with any \textit{solid}
globe. But this is not the case with the sun. It was noticed quite
early that the different spots give different results for the rotation
period, but the researches of Carrington about thirty years ago first
%% -----File: 189.png---Folio 178-------
brought out the fact that the differences follow a regular law, showing
that at the solar equator the time of rotation is less than on either
side of it. Thus spots near the sun's equator give $T = 25$ days; at
solar latitude $20°$, $T= 25.75$ days; at solar latitude $30°$, $T = 26.5$
days; at solar latitude $40°$, $T = 27$ days. The time of rotation in
latitude $40°$ is fully two days longer than at the solar equator; but we
are unable to follow the law further towards the poles, because the
spots are rarely found beyond the parallels of $45°$ on each side of the
equator, and there are no well-defined markings between this point
and the poles by which we can accurately determine the motion.

\begin{fineprint}
\nbarticle{284.} Various formulę have been proposed to represent this law of rotation.
Carrington gives for the daily motion of a spot $X = 865' - 165'× \sin^{\frac{7}{4}}{l}$,
$l$ being the solar latitude of the spot. Faye, from the same observations, considering
that the exponent $\frac{7}{4}$ could have no physical justification, deduced
$X = 862' - 186'× \sin^{2}{l}$, which agrees almost as well with the observations.
Still other formulę have been deduced by Spoerer, Zöllner, and Tisserand,
all giving substantially the same results.
\end{fineprint}

The law, in any case, is simply \textit{empirical}; that is, it is deduced from
the observations, without being based upon any satisfactory physical
explanation, for no such explanation of this strange equatorial acceleration
has yet been found. Probably it has its origin somehow in
the effects produced by the outpour of heat from the sun's surface;
still, just how such a result should follow in the case of a cooling
globe, of which the particles are free to move among each other, is
not yet evident.

\begin{fineprint}
\nbarticle{285.} It is possible that the spots move on the surface of the sun, changing
their places just as do clouds or railroad trains upon the surface of the
earth, so that their motion does not represent the sun's true rotation. This
however, as we shall see later, is hardly probable, and if it were the case, it
would still be no less difficult to account for this systematic difference in
their behavior at the solar equator and in the higher latitudes.

\includegraphicsouter{illo091}{\textsc{Fig.~91.}---Telescope and Screen.}

It has been suggested that the spots may be due to the fall of matter from
a considerable elevation above the sun's surface, matter which has remained
at that elevation for some time, and acquired a corresponding velocity of
rotation due to that elevation. It can be shown that if the matter forming
the spots had thus fallen from an elevation of about 20,000 miles, it would
account for their apparent acceleration. Matter so falling would have an
apparent eastward motion, just as do bodies on the earth when falling from
the summit of a tower (\artref{Art.}{138}). From this point of view it is very interesting
to inquire whether the minuter markings upon the sun's surface, such
as the ``granules,'' to be spoken of very soon, do, or do not, possess the same
%% -----File: 190.png---Folio 179-------
rate of motion as the spots. There is no decisive observational\footnote
  {Mr.~Crew has recently made at Baltimore, under the direction of Professor
Rowland, an extensive series of observations upon the displacement of the lines
of the spectrum at the eastern and western limbs of the sun. This displacement,
which is very slight, is due, according to Doppler's principle (\artref{Art.}{321}), to the
rotation of the sun; and Mr.~Crew's results, so far as they can be considered
decisive, go to show that the absorbing layer of gases by which the Fraunhofer
lines are formed \textit{does not} behave like the sun spots, but is slightly \textit{retarded} at the
sun's equator. The observations are so delicate, however, that the conclusion,
though made very probable, can hardly be considered to be absolutely proved
  beyond question.}
evidence at
present that they do not. But the subject is an extremely difficult one; and
yet important, because the solution of the problem of the sun's equatorial
acceleration will probably throw much light upon its real constitution.
\end{fineprint}

\nbarticle{286.} \nbparatext{The Phenomena of the Sun's Surface.}---In order to study
the sun with the telescope it is necessary to be provided with some
special forms of apparatus. Its heat
and light are so intense that it is
impossible to look directly at it, as
we do at the moon. A very convenient
method of exhibiting the sun
to a number of persons at once is simply
to attach to the telescope a frame
carrying a screen of white paper at a
distance of a foot or more from the
eye-piece, as shown in \figref{illo091}{Fig.~91}.  On
pointing the instrument to the sun and
properly adjusting the focus, a distinct image is formed on the
screen, which shows the main features very fairly. It is, however,
much more satisfactory to look at it directly, with a proper eye-piece.
With a small telescope, not more than two and a half or
three inches in diameter, a mere dark glass between the eye-piece and
the eye can be used, but this dark glass soon becomes very hot, and
is apt to crack. With larger instruments, it is necessary to use
eye-pieces especially designed for the purpose and known as \textit{solar
eye-pieces} or \textit{helioscopes}.

\begin{fineprint}
\includegraphicsouter{illo092}{\textsc{Fig.~92.}---Herschel Eye-piece.}

The simplest of them, and a very good one for ordinary purposes, is one
known as Herschel's, in which the sun's rays are reflected at right angles by a
plane of unsilvered glass (\figref{illo092}{Fig.~92}). This reflector is made either of a prismatic
form or concave, in order that the reflection from the back surface may not
%% -----File: 191.png---Folio 180-------
interfere with that from the front. About nine-tenths of the light passes
through this reflector, and is allowed to pass
out uselessly through the open end of the
tube. The remaining tenth is sent through
the eye-piece, and though still too intense
for the eye to endure, it requires only a
comparatively thin shade of neutral-tinted
glass to reduce it sufficiently, and in this
case the shade does not become uncomfortably
heated. It is well to have the shade-glass
made wedge-shaped,---thinner at one
end than at the other---so that one can
choose the particular thickness which is
best adapted to the magnifying power
employed.

\nbarticle{287.} The polarizing eye-pieces are still better when well made. In
these the light is reflected twice at plane surfaces of glass at the ``angle of
polarization'' (Physics, p.~480), and is then received on a second pair
of reflectors of black glass. When
the upper pair of reflectors is in either
of the two positions shown in \figref{illo093}{Fig.~93},
a strong beam of light is received at
$C$,---too strong for the eye to bear,
although more than ninety per cent
of it has already been rejected; but
by simply turning the box which
carries the upper reflectors one-quarter
of a revolution around the line
$BB'$ as an axis, the light may be
wholly extinguished; and any desired
gradation may be obtained by setting
it at the proper angle, without the
use of a shade-glass.
\end{fineprint}

\nbarticle{288.} It may be asked why it
will not answer merely to ``cap''
the object-glass, and so cut off
part of the light, instead of rejecting
it after it has once been
allowed to enter the telescope. It
is because of the fact, mentioned in \artref{Article}{43}, that the smaller the
object-lens of the telescope, the larger the image it makes of a luminous
point, or the wider its image of a sharp line. To cut down the
%% -----File: 192.png---Folio 181-------
aperture, therefore, is to sacrifice the definition of delicate details.
With a low power there is no objection to reducing the amount of
heat admitted into the telescope tube in that way, but with the higher
powers the whole aperture should always be used.

\includegraphicsouter{illo093}{\textsc{Fig.~93.}---Polarizing Helioscope.}

\nbarticle{289.} \hspace{-3.2pt}\nbparatext{Photography.}---In the study of the sun's surface photography
is for some purposes very advantageous and much used. The
instrument must have a special object-glass (\artref{Article}{42}), with an apparatus
for the quick exposure of plates. Such instruments are called
photo-heliographs, and with them photographs of the sun are made daily
at numerous observatories. The necessary exposure varies from $\frac{1}{500}$
to $\frac{1}{10}$ of a second, in different cases. The pictures made by these
instruments are usually from two inches up to eight or ten inches in
diameter, and some of Janssen's, made at Meudon, bear enlarging
up to forty inches in diameter. Photographs have the advantage of
freedom from prejudice and prepossession on the part of the observer;
but they take no advantage of the instants of fine seeing.
They represent the surface as it happened to be at the moment
when the plate was uncovered.

\begin{fineprint}
\nbarticle{290.} The study of the sun has become so important from a scientific
point of view that several observatories have recently been established
mainly for that purpose, though most of them connect with it that of other
topics in astronomical physics. The two most important of these solar or
astro-physical observatories, are the observatory at Meudon and the so-called
``Sonnenwarte'' at Potsdam. There ought to be one in this country.
\end{fineprint}

\nbarticle{291.} \nbparatext{General Views.}---Before passing to a discussion of the
details of the different solar phenomena, it will be well to give a
very brief summary of the objects and topics to be considered.

1. \textit{The photosphere; i.e.}, the luminous surface of the sun directly
visible to our telescopes. It is probably a sheet of \textit{luminous clouds}
formed by condensation into little drops and crystals (like the water-drops
and ice-crystals in our terrestrial clouds) of certain substances
which within the central mass of the sun exist in a gaseous form,
but are cooled at its surface below the temperature necessary for
their condensation; perhaps such substances as carbon, boron, and
silicon. The granules, faculę, and spots are all phenomena in this
photosphere.

2. The so-called ``\textit{reversing layer}'' is a stratum of unknown thickness,
but probably shallow, just above the photosphere, containing
the vapors of many of the familiar terrestrial elements; of which
%% -----File: 193.png---Folio 182-------
the presence, and to some extent their physical condition, can be investigated
by means of the spectroscope.

\includegraphicsouter{illo094}{\textsc{Fig.~94.}\\
Constitution of the Sun. From ``The Sun,'' by permission
of the Publishers.}

3. Above the photosphere, interpenetrating the atmosphere of
vapors just spoken of,
and perhaps indistinguishable
from it, is an envelope
of \textit{permanent} gases; that
is, gases which, under the
solar conditions, cannot
be condensed into clouds
of solid or liquid particles.
Among them hydrogen
is most conspicuous.
This envelope is the so-called
\textit{Chromosphere}; and
from it the \textit{prominences}
of various kinds rise,
sometimes to the height
of hundreds of thousands
of miles. These beautiful
objects are best seen
at total eclipses of the
sun, but to a certain extent
they can also be studied
at any time by the
help of a spectroscope.

4. Higher yet rises the
mysterious \textit{Corona}, of material
still less dense, and so
far observable only during
total eclipses of the sun.

Fig.~94 shows the relative positions of these different elements of
the solar constitution.

5. A fifth subject deals with the \textit{measurement of the sun's light}
and the relative brightness of different parts of the solar surface.

6. Another most interesting and important topic relates to the
amount of \textit{heat} radiated by the sun,---the sun's \textit{probable temperature}
and the mechanism by which its heat-supply is maintained.

\includegraphicsmid{illo095}{\textsc{Fig.~95.}\\
The Great Sun Spot of September, 1870, and the Structure of the Photosphere. From a Drawing
by Professor Langley. From ``The New Astronomy,'' by permission of the Publishers.}

\nbarticle{292.} \nbparatext{The Photosphere.}---The sun's visible surface is called the
\textit{photosphere}, and when studied under favorable atmospheric conditions,
%% -----File: 194.png---Folio 183-------
with rather a low magnifying power, it looks like rough drawing-paper.
With higher powers it is seen to be, as shown in
Fig.~95, made up of a comparatively darkish background sprinkled
over with grains, or ``nodules,'' as Herschel called them, of something
much more brilliant,---like \DPtypo{snowflakes}{snow-flakes} on gray cloth, according
to Langley. These are from 400 to 600 miles across, and in the
finest seeing are themselves resolved into more minute ``granules.''
For the most part, these nodules are about as broad as they are long,
though of irregular form; but here and there, especially in the
neighborhood of the spots, they are drawn out into long streaks.
Nasmyth seems first to have observed this structure, and called the
filaments ``willow leaves.'' Secchi called them ``rice grains.''
According to Huggins they were ``dots''; and there was for a long
time a pretty lively controversy as to their true form. Their shape,
however, unquestionably varies very much in different parts of the
%% -----File: 195.png---Folio 184-------
surface and under different circumstances. They are probably luminous
clouds floating in a less luminous atmosphere.

Near the edge the photosphere appears generally much less brilliant;
but certain bright streaks called ``faculę'' (from \textit{fax}, a
torch), which though visible are not very obvious at points further
from the limb, become there conspicuous. These faculę are elevations,---masses
of the same material as the rest of the photosphere
but elevated above the general level and intensified in brightness.
When one of them passes off the edge of the sun, it is sometimes
seen as a little projection. They are most abundant near the sun-spots,
and they are more conspicuous near the edge of the disc, as
shown in \figref{illo096}{Fig.~96}, because the sun's surface is overlaid by a gaseous
atmosphere which absorbs more of the light there than it does near
the centre, and these faculę push up through it like mountains.

\includegraphicsmid{illo096}{\textsc{Fig.~96.}---Faculę at Edge of the Sun. (De La Rue.)}

\nbarticle{293.} \nbparatext{The Sun Spots.}---Whenever these are present upon the sun's
surface, they are the most conspicuous objects to be seen upon it.
The appearance of a normal sun spot, \figref{illo097}{Fig.~97}, fully formed and not
yet beginning to break up, is that of a dark central ``\textit{umbra},'' more
or less nearly circular, with a fringing ``\textit{penumbra},'' composed of
filaments directed radially. The umbra itself is not uniformly dark
throughout, but is overlaid with filmy clouds which require a good
telescope and helioscope to make them visible. Usually, also, in the
%% -----File: 196.png---Folio 185-------
umbra there are several round and very black spots, which are sometimes
called ``\textit{nucleoli},'' but are often referred to us ``Dawes'
holes,'' after the name of their first discoverer. But while this is the
appearance of what may be taken as a normal spot, very few are
strictly normal. Most of them are more or less irregular in form.
They are often gathered in groups with a common penumbra, and
partly covered by brilliant ``\textit{bridges}'' extending across from the outside
photosphere. Often the umbra is out of the centre of the penumbra,
or has a penumbra only on one side, and the penumbral
filaments, instead of being strictly radial, are frequently distorted in
every conceivable way. In fact, the normal spots form a very small
proportion of the whole number.

\includegraphicsmid{illo097}{\textsc{Fig.~97.}---A Normal Sun Spot. (Secchi; modified.)}

The darkest portions of the umbra are dark only by contrast.
Photometric observations (by Langley) show that even the nucleus
gives at least one per cent as much light as a corresponding area
of the photosphere; that is to say, as we shall see hereafter, the
darkest portion of a sun spot is brighter than a calcium light.

\nbarticle{294.} The spots are unquestionably \textit{cavities} or \textit{depressions} in the
photosphere, filled with gases and vapors which are cooler than the
surrounding portions, and therefore absorb a considerable proportion
of light. The fact that they are \textit{cavities} is shown by the change
%% -----File: 197.png---Folio 186-------
in the appearance of a spot as it approaches the edge of the disc.
When a normal spot is near the centre of the disc, the nucleus is
nearly central. As it approaches the edge, the penumbra becomes
wider on the outer edge and narrower on the inner, and just before
the spot disappears around the limb of the sun, the penumbra on the
inner edge entirely disappears,---the appearance being precisely
such as would be shown by a saucer-shaped cavity in the surface
of a globe, the bottom of the cavity being painted black to represent
the umbra, and the sloping sides gray for the penumbra. \figref{illo098}{Fig.~98}
represents the phenomena in a schematic way. Observations
upon a single spot would hardly be sufficient to substantiate this,
because the spots are so irregular in their form; but by observing
the behavior of several hundred of them the truth comes out quite
clearly. Occasionally, when a very large spot passes off the sun's
limb, the depression can be seen with the telescope.

\includegraphicsmid{illo098}{\textsc{Fig.~98.}---Sun Spots as Cavities.}

The fact was first discovered by Wilson of Glasgow something
more than a hundred years ago. Previously it had very commonly
been supposed that the spots were \textit{elevated} above the general surface
of the sun, and the idea still survives in certain quarters, though certainly
incorrect.

\nbarticle{295.} The \textit{penumbra} is usually composed of ``thatch-straws,'' or
long drawn-out granules of photospheric matter, which, as has been
said, converge in a general way towards the centre of the spot. At
the inner edge the penumbra, from the convergence of these filaments,
is usually brighter than the outer. The inner ends of the filaments
are generally club-formed; but sometimes they are drawn out into
fine points, which seem to curve downward into the umbra like the
rushes over a pool of water. The outer edge of the penumbra is
usually pretty definite, and the penumbra there is darker. Around
%% -----File: 198.png---Folio 187-------
the spot the photosphere is much disturbed and elevated into faculę,
which sometimes radiate outward from the spot like streams of
lava from a crater, though, of course, they are really nothing of the
sort.

\nbarticle{296.} \nbparatext{Dimensions of Sun Spots.}---The diameter of the umbra of
a sun spot ranges all the way from 500 to 1000 miles in the case of
a very small one, to 50,000 or 60,000 miles in the case of the larger
ones. The penumbra surrounding a group of spots is sometimes
150,000 miles across, though that would be rather an exceptional
size. Not infrequently sun spots are large enough to be seen by the
naked eye, and they have been often so seen at sunset or through a
fog. The depth by which the umbra is depressed below the general
surface of the photosphere is very difficult to determine, but according
to Faye, Carrington, and others, it seldom exceeds 2500 miles,
and more often is between 500 and 1500.

\nbarticle{297.} \nbparatext{Development and Changes of Form.}---Generally the origin
of a sun spot fails to be observed. It begins from an insensible
point, and rapidly grows larger, the penumbra usually appearing
only \textit{after the nucleus is fairly developed}.

If the disturbance which causes the spot is violent, the spot usually
breaks up into several fragments, and these again into others which
tend to separate from each other. At each new disturbance the forward
portions of the group show a tendency to advance eastward
on the sun's surface, leaving behind them a trail of smaller spots.

\begin{fineprint}
\nbarticle{298.} The ``segmentation'' of a spot, as Faye calls it, is usually effected
by the formation of a ``bridge,'' or streak of brilliant light, which projects
itself across the penumbra and umbra from the outside photosphere. These
bridges are mere extensions of the surrounding faculę, and are often intensely
bright.

Occasionally a spot shows a distinct cyclonic motion, the filaments being
drawn inward spirally; and in different members of the same group of spots
the cyclonic motions are not seldom in opposite directions.

When a spot at last vanishes it is usually by the rapid encroachment of
the photospheric matter, which, as Secchi expresses it, appears to ``fall pell-mell
into the cavity,'' completely burying it and leaving its place covered
by a group of faculę. Figs.~99--104 (see page~\figpage{99-104}) show the changes which
took place in the great spot of September, 1870. They are from photographs
by Mr.~Rutherfurd of New York, and are borrowed from ``The New Astronomy''
of Professor Langley, through the courtesy of his publishers.
\end{fineprint}
%% -----File: 199.png---Folio 188-------

\nbenlargepage
\nbarticle{299.} Spots within $15°$ or $20°$ of the sun's equator generally, on
the whole, drift a little \textit{towards} it, while these in higher latitudes
drift \textit{away from} it; but the motion is slight, and exceptions are frequent.

In and around the spot itself the motion is usually \textit{inward towards
the centre}, and \textit{downward at} the centre. Not infrequently the fragments
at the inner end of the penumbral filaments appear to draw
off, move towards the centre of the spot, and then descend. Occasionally,
though seldom, the motion is vigorous enough to be detected
by the displacement of lines in the spectrum.

\begin{includegraphicspage}{99-104}
  \begin{tabular}{c@{\qquad\qquad}c}
    \includegfx{illo099}           & \includegfx{illo100}\\
    \footnotesize\textsc{Fig.~99.}---Sept.~19.  & \footnotesize\textsc{Fig.~100.}---Sept.~20.\\[4ex]
    \includegfx{illo101}           & \includegfx{illo102}\\
    \footnotesize\textsc{Fig.~101.}---Sept.~21. & \footnotesize\textsc{Fig.~102.}---Sept.~22.\\[4ex]
    \includegfx{illo103}           & \includegfx{illo104}\\
    \footnotesize\textsc{Fig.~103.}---Sept.~23. & \footnotesize\textsc{Fig.~104.}---Sept.~26.\\[4ex]
  \end{tabular}
  \captionof*{figure}{The Great Sun Spot of 1870.}
\end{includegraphicspage}

\nbarticle{300.} \nbparatext{Duration.}---The duration of the spots is very various, but,
astronomically speaking, they are always short-lived phenomena,
sometimes lasting for only a few days, more frequently, perhaps, for
a month or two. In a single instance, a spot has been observed
through as many as eighteen successive revolutions of the sun.

\nbarticle{301.} \nbparatext{Distribution.}---It is a significant fact that the spots are confined
mostly to two zones of the sun's surface between $5°$ and $40°$ of
latitude north and south. A few are found near the equator, none
beyond the latitude of $45°$. \figref{illo105}{Fig.~105} shows the distribution of several
thousand spots as observed by Carrington and Spörer.

\includegraphicsmid{illo105}{\textsc{Fig.~105.}---Distribution of Sun Spots in Latitude.}

Occasionally, what Trouvelot calls ``veiled spots'' are seen beyond
the $45°$ limits---grayish patches surrounded by faculę, which look as
if a dark mass were submerged below the surface and dimly seen
through a semi-transparent medium.
%% -----File: 200.png---Folio 189-------

\nbarticle{302.} \nbparatext{Theories as to the Nature of the Spots.}---We first mention
(\textit{a})~the theory of Sir William Herschel, because it still finds place
in certain text-books, though certainly incorrect. His belief was
that the spots were openings through two luminous strata, which he
supposed to surround the central globe of the sun. This globe he
supposed to be \textit{dark} (\textit{and even habitable!}). The outer stratum, the
photosphere, was the brighter of the two, and the opening in it the
larger, while the inner shell between it and the solid globe was of less
luminous substance, and formed the penumbra. He thought the opening
through these might be caused by volcanoes on the globe beneath.

\nbarticle{303.} (\textit{b}) Another theory, now abandoned, was proposed independently
both by Secchi and Faye about 1868. They supposed that
the spots were openings in the photosphere caused by the bursting
outward of the imprisoned gases underneath it; the photosphere at
that time being supposed to be \textit{liquid}.

\begin{fineprint}
They explained the darkness of the centre of the spot by the fact that a
heated gas at a given temperature has a lower radiating power and sends
out much less light than a \textit{liquid surface}, or than \textit{clouds} formed by the condensation
of the same material at even a lower temperature. This is true
of gases at low pressure, but not of gases under great compression, such as
must be the case within the body of the sun. Besides, if the gases possessed
the small radiating power necessary to this theory, they would also possess
small \textit{absorbing} power, and therefore would be transparent; the inner side of
the photosphere on the opposite side of the sun would therefore be visible
through the opening, so that the centre of such an eruption would not be
\textit{dark}, but, if anything, brighter than the general solar surface. Moreover, as
we now know from the spectroscopic evidence, the motion at the centre of a
spot is \textit{inward}, not \textit{outward}.
\end{fineprint}

\nbarticle{304.} (\textit{c}) Faye more recently has proposed and now maintains a
theory which has numerous good points about it, and is accepted by
many, viz.: that the spots are analogous to storms on the earth, being
\textit{cyclones}, due to the fact that the portions of the sun's surface near
the equator make their revolution in a shorter time than these in
higher latitudes. This causes a relative drift in adjacent portions of
the photosphere, and according to him gives rise to \textit{vortices} or \textit{whirlpools}
like these in swiftly running water. The theory explains the
distribution of the spots (which abound precisely in the regions
where this relative drift is at the maximum) and many other facts,
such as their ``segmentation.'' According to it, however, \textit{all} spots
should be cyclonic, and the spiral motion of all the spots in the
%% -----File: 201.png---Folio 190-------
northern hemisphere should be \textit{clock-wise}, while in the southern hemisphere
they should be \textit{counter-clock-wise}. Now, as a matter of fact,
only a very few of the spots show such spiral motions, and there is
no such agreement in the general direction of the motion as the
theory requires.

\begin{fineprint}
Faye attempts to account for this by saying that we do not see the vortex
itself, but only the cloud of cooler materials which is drawn together by
the down-rushing vortex, itself hidden beneath this cloud. Still, it would
seem that in such a case the cloud itself should gyrate. Moreover, the
relative drift of the adjacent portions of the photosphere is too small to
account for the phenomena satisfactorily. In the solar latitude of $20°$ two
points separated by $1'$ of the sun's surface (123 miles) have a relative daily
drift of only about four and one-sixth miles, insufficient to produce any sensible
whirling.
\end{fineprint}

\nbarticle{305.} (\textit{d}) Secchi's later theory. He supposed the spots to be due
to eruptions from the inner portions of the sun's surface, not \textit{in} the
spot, however, but only \textit{near} it; the spot itself being formed by
the settling down upon the photosphere of materials thrown out by
the eruption and cooled by their expansion and their motion through
the upper regions. We have, however, in fact, as a usual thing, not
a single eruption, but a ring of eruptions all around every large spot,
all of them converging their bombardment, so to speak, upon the
same centre,---a fact very difficult to explain if the spot originates
in the eruption, but not difficult to understand if the eruptions
are the result of the spot.

Perhaps the true explanation may be that when an eruption occurs
at any spot, the \textit{photosphere somewhere in the neighborhood settles
down in consequence of the diminution of the pressure beneath}, thus
forming a ``\textit{sink},'' so to speak, which is of course covered by a
greater depth of cooler vapors above, and so looks dark.

\nbarticle{306.} (\textit{e}) Mr. Lockyer, in his recent work on the chemistry of the
sun, revives an old theory, first suggested by Sir John Herschel and
accepted by the late Professor Peirce, that the spots are not formed
by any action from within, but by \textit{cool matter descending from above},---matter
very likely of meteoric origin; but it is difficult to see how
the distribution of the spots with reference to the sun's equator can
be accounted for in this way.

On the whole it is impossible to say that the problem of the origin
of sun spots is yet satisfactorily solved. There is no question that
%% -----File: 202.png---Folio 191-------
sun spots are closely associated with eruptions from beneath; but
which is cause and which effect, or whether both are due to some
external action, remains undetermined.

\nbarticle{307.} \nbparatext{Periodicity of Sun Spots.}---In 1851 Schwabe of Dessau, by
the comparison of an extensive series of observations running over
nearly thirty years, showed that the sun spots are \textit{periodic}, being at
times vastly more numerous than at others, with a roughly regular
recurrence every ten or eleven years. This had been surmised
by Horrebow more than a century before, though not proved.

\sloppy
Subsequent study fully confirms this remarkable result of Schwabe.
Wolf of Zurich has collected all the observations discoverable and
finds a pretty complete record back to 1610. From these records is
constructed the annexed diagram, \figref{illo106}{Fig.~106}. The ordinates of the
curve represent what Wolf calls his ``relative numbers,''\footnote
  {This ``relative number'' is formed in rather an arbitrary manner from the
observations which Wolf hunted up as the basis of his investigation. The
formula is, $r$ (the relative number)${}- k (10g + f)$, in which $g$ is the number
of groups and isolated spots observed, $f$ the total number of spots which can
be counted in these groups and singly, while $k$ is a coefficient which depends
upon the observer and the size of his telescope: it is large for a small telescope
  and not very persistent observer, and approaches unity the more likely the
  observer may be supposed to have noted every sun spot that appeared during
  the time covered by his observations.}
which he
has adopted as representing the spottedness.
%% -----File: 203.png---Folio 192-------

\fussy
\includegraphicsmid{illo106}{\textsc{Fig.~106.}---Wolf's Sun-Spot Numbers.}

The average period is eleven and one-tenth years, but, as the \figref{illo106}{figure}
shows, the spot maxima are quite irregular, both in time and as to
the extent of spottedness. The last spot maximum occurred in
1883--84 (a year or two behind time), and we are now (1888) approaching
a minimum. During a maximum the surface of the sun
is never free from spots, from twenty-five to fifty being frequently
visible at once. During a minimum, on the other hand, weeks often
pass without the appearance of a single one.

\begin{fineprint}
\nbarticle{308.} \nbparatext{Possible Cause of the Periodicity.}---The cause of this periodicity
is not known. It has been attempted to connect it with planetary action.
Some things in the Kew statistics of the sun spots look as if Venus, Mercury,
and the Earth had something to do with it, the sun's surface being
more spotted when these planets approach nearer; but the evidence is
insufficient, or at least needs to be supplemented by further comparisons.
Jupiter also has been suspected. His period is 11.86 years, which is not
very different from the mean sun-spot period; but an examination of the
different spot maxima show that some of them have occurred when he was
near perihelion, and others when he was near aphelion; and on the whole
there is very little reason for supposing that he has any considerable influence
in the matter.

Sir John Herschel suggested that it might be due to streams of meteors
moving in an oval orbit with a period of about eleven years, and approaching
so near at perihelion that numerous members of the meteoric group actually
fall into the sun; but, as has been said before, the distribution of the
spots would seem to contradict the idea.
\end{fineprint}

\nbarticle{309.} \nbparatext{Terrestrial Influence of the Sun Spots.}---One influence of
the sun spots upon the earth is perfectly demonstrated. When the
spots are numerous, magnetic disturbances (the so-called magnetic
storms) are most numerous and violent upon the earth, a fact not to
be wondered at since violent disturbances upon the sun's surface
have been in many individual cases immediately followed by magnetic
storms, with a brilliant exhibition of the Aurora Borealis. The
nature and mechanism of the connection is as yet unknown, but of
the fact there can be no question. The dotted lines in the figure of
the sun-spot periodicity (\figref{illo106}{Fig.~106}) represent the magnetic storminess
of the earth at the indicated dates; and the correspondence between
these curves and the curve of spottedness makes it impossible to
doubt the connection.
%% -----File: 204.png---Folio 193-------

\nbarticle{310.} It has been attempted, also, to show that greater or less
disturbance of the sun's surface, as indicated by the greater frequency
of the sun's spots, is accompanied by \textit{effects upon the meteorology
of the earth}, upon its temperature, barometric pressure, storminess,
and the amount of rain-fall. The researches of Mr.~Meldrum
of Mauritius with respect to the cyclones in the Indian Ocean appear
to bear out the conclusion that there may be some such connection in
that case, but the general results are by no means decisive. In some
parts of the earth the rain-fall seems to be greater during a spot
maximum; in others, less.

As to the temperature, it is still uncertain whether it is higher or
lower at the time of a spot maximum. The \textit{spots themselves are
cooler} (as Henry, Secchi, and Langley have shown) than the general
surface of the photosphere; but their extent is never sufficient to
reduce the amount of heat radiated from the sun by us much as $\frac{1}{1000}$
part. On the other hand, when the spots are most numerous, the
generally disturbed condition of the photosphere would, as Langley
has shown, necessarily be accompanied by an increased radiation.

Dr.~Gould considers that the meteorological records in the Argentine
Republic between 1875 and 1885 show an indubitable connection
between the \textit{wind currents} and the number of sun spots. But the
\textit{demonstration} of such a relation really requires observations running
through several spot periods. On the whole, it is now quite certain
that whatever influence the sun spots exert upon terrestrial meteorology
is very slight, if it exists at all.

\section*{THE SOLAR SPECTRUM AND ITS REVELATIONS.}

\nbarticle{311.}  About 1860 the spectroscope appeared in the field as a new
and powerful instrument of astronomical research, at once resolving
many problems as to the nature and constitution of the heavenly bodies
which before had not seemed to be even open to investigation.

The essential part of the apparatus is either a prism or train of
prisms, or else a diffraction grating,\footnote
  {The grating is merely a piece of glass or speculum metal, ruled with many
thousand straight, equidistant lines, from 5000 to 20,000 in the inch. Usually the
surface before ruling is accurately plane, but for some purposes the \textit{concave} gratings,
  originated by Professor Rowland, are preferable.}
which is capable of performing
the same office of dispersing---that is, of spreading and sending in
different directions---the light rays of different wave-lengths. If,
with such a ``\textit{dispersion piece},'' as it may be called (either prism or
grating), one looks at a distant point of light, as a star, he will see
%% -----File: 205.png---Folio 194-------
instead of a point a long streak of light, red at one end and violet
at the other. If the object looked at be not a point, but a \textit{line of
light} parallel to the edge of the prism or to the lines of the grating,
then, instead of a mere colored streak without width, one gets a
\textit{spectrum}, a colored band of light, which may show markings that
will give the observer most valuable information. (Physics, pp.~458--460.)
For convenience' sake it is usual to form this line
of light by admitting the light through a narrow ``\textit{slit},'' which is
at one end of a tube having at the other end an achromatic object-glass
at such a distance that the slit is in its principal focus.
This tube with slit and lens constitutes the ``\textit{collimator},'' so called because
it is precisely the same as the instrument used in connection with
the transit instrument to adjust its line of collimation (\artref{Article}{60}).

Instead of looking at the spectrum with the naked eye, however,
it is better in most cases to use a small telescope; called the ``\textit{view-telescope},''
to distinguish it from the large telescope, to which the
spectroscope is often attached.

\nbarticle{312.} \nbparatext{Construction of the Spectroscope.}---The instrument, therefore,
as usually constructed, and shown in \figref{illo107}{Fig.~107}, consists of three
parts,---collimator, dispersion-piece, and view-telescope; but in the
direct-vision spectroscope, shown in the \figref{illo107}{figure}, the view-telescope is
omitted. If the slit, $S$, be illuminated by strictly homogeneous light
%% -----File: 206.png---Folio 195-------
all of one wave-length, say yellow, the ``real image'' of the slit
will be found at $Y$. If at the same time light of a different
wave-length be also admitted, say red, a second image will be
formed at $R$, and the observer will see a spectrum with two ``bright
lines,'' the lines being really nothing more than \textit{images of the slit}. If
light from a candle be admitted, there will be an infinite number of
these slit-images, close packed, like the pickets in a fence, without
interval or break, and we then get a continuous spectrum; but if we
look at sunlight or moonlight, we shall find a spectrum continuous in
the main, but crossed by numerous dark lines, as if some of the
``pickets'' had been knocked off, leaving gaps.

\includegraphicsmid{illo107}{\textsc{Fig.~107.}---Different Forms of Spectroscope.}

\sloppy
\nbarticle{313.} \nbparatext{Integrating and Analyzing Spectroscope.}---If we simply
direct the collimator of a spectroscope towards a distant luminous
object, every part of the slit receives light from every part of the
object, so that in this case every elementary streak of the spectrum
%% -----File: 207.png---Folio 196-------
is a spectrum of the entire body, without distinction of parts. A
spectroscope used in this way is said to be an \textit{integrating} instrument.

\fussy
If, however, we interpose a lens (the object-glass of a telescope)
between the luminous object and the slit, so as to have in the plane
of the slit a distinct, real image of the object, then the top of the
slit, for instance, will be illuminated wholly by light from one part
of the object, the middle of it by light from another point, and the
bottom by light from still a third. The spectrum formed by the top
of the slit belongs, then, to the light from that particular point of the
object whose image falls upon that part of the slit; and so of the
rest. We thus separate the spectra of the different parts of the
object, and so \textit{optically analyze} it. An instrument thus used is
spoken of as an ``\textit{analyzing spectroscope}.'' The combined instrument
formed by attaching a spectroscope to a large telescope for
the spectroscopic observation of the heavenly bodies has been called
by Mr.~Lockyer a ``telespectroscope.'' \figref{illo108}{Fig.~108} shows the apparatus
used by the writer for some years at Dartmouth College.

For solar purposes a grating spectroscope is generally better than a
prismatic, being less complicated and more compact for a given power.

\nbenlargepage
\includegraphicsmid{illo108}{\textsc{Fig.~108.}---The Telespectroscope.}

\nbarticle{314.} \hspace{-1.7pt}\nbparatext{Principles upon which Spectrum Analysis depends.}---These,
substantially as announced by Kirchoff in 1858, are the three
following:---

1st, A \textit{continuous spectrum} is given by every incandescent body,
the molecules of which so interfere with each other as to prevent
their free, independent, luminous vibration; that is, by bodies which
are either \textit{solid} or \textit{liquid}, or, if gaseous, are \textit{under high pressure}.

2d, The spectrum of a gaseous element, \textit{under low pressure}, is
discontinuous, made up of \textit{bright lines}, and these lines are characteristic;
that is, the same substance under similar conditions always
gives the same set of lines, and generally does so even under widely
different conditions.

\includegraphicsmid{illo109}{\textsc{Fig.~109.}---Reversal of the Spectrum.}

\sloppy
3d, A gaseous substance \textit{absorbs} from white light passing through
it \textit{precisely these rays of which its own spectrum consists}. The spectrum
of white light which has been transmitted through it then exhibits
a ``\textit{reversed}'' spectrum of the gas; that is, one which shows dark
lines instead of the characteristic bright lines.

\fussy
Fig.~109 illustrates this principle. Suppose that in front of the slit
of the spectroscope we place a spirit lamp with a little carbonate of
soda and some salt of thallium upon the wick. We shall then get a
spectrum showing the two yellow lines of sodium and the green line
%% -----File: 208.png---Folio 197-------
of thallium, \textit{bright}. If now the lime-light be started right behind the
lamp flame, we shall at once get the effect shown in the lower \figref{illo109}{figure},---a
continuous spectrum crossed by \textit{black} lines just where the bright
lines were before. Insert a screen between the lamp flame and the
lime, and the dark lines instantly show bright again.

\includegraphicsouter{illo110}{\textsc{Fig.~110.}---The Comparison Prism.}

\nbarticle{315.} \nbparatext{Chemical Constituents of the Sun.}---By taking advantage
of these principles we can detect the presence of a large number of
well-known terrestrial elements in the sun. The solar spectrum is
crossed by dark lines, which, with
an instrument of high dispersion,
number several thousand, and by
proper arrangements it is possible
to identify among these lines many
which are due to the presence in
the sun's lower atmosphere of
known terrestrial elements in the
state of vapor.  To effect the comparison
necessary for this purpose,
the spectroscope must be so arranged that the observer can have before
him, side by side, the spectrum of sunlight and that of the substance
to be tested. In order to do this, half of the slit is fitted with a little
``\textit{comparison prism},'' so-called (\figref{illo110}{Fig.~110}), which reflects into it the
light from the sun, while the other half of the slit receives directly
%% -----File: 209.png---Folio 198-------
the light of some flame or electric spark. On looking into the eye-piece
of the spectroscope, the observer will then see a spectrum, the
\textit{lower} half of which, for instance, is made by sunlight, while the
\textit{upper} half is made by light coming from an electric spark between
two metal points, say of iron.

\includegraphicsmid{illo111}{\textsc{Fig.~111.}\\
Comparison of the Solar Spectrum with that of Iron. From a Negative by Prof.~Trowbridge.}

Photography may also be most effectively used in these comparisons
instead of the eye. \figref{illo111}{Fig.~111} is a rather unsatisfactory reproduction,
on a reduced scale, of a negative recently made by Professor
Trowbridge at Cambridge. The lower half is the violet portion
of the spectrum of the sun, and the upper half that of the vapor
of iron in an electric arc. The reader can see for himself with what
absolute certainty such a photograph indicates the presence of iron
in the solar atmosphere. A few of the lines in the photograph
which do not show corresponding lines in the solar spectrum are due
to impurities in the carbon, and not to iron.

\nbarticle{316.} As the result of such comparisons we have the following list
of twelve elements, which are \textit{certainly} known to exist in the sun,
viz.:---\\
\begin{tabular}{llll}
\hspace{.2\textwidth}
&hydrogen,      &&cobalt,\\
&iron,          &&chromium,\\
&titanium,      &&barium,\\
&calcium,       &&sodium,\\
&manganese,     &&magnesium,\\
&nickel,        &&platinum.\\[1ex]
\multicolumn{4}{@{}p{\textwidth}@{}}{There are evidences, perhaps not quite conclusive, of the presence of
nearly as many more, viz.:---}\\
&copper,        &&uranium,\\
&palladium,     &&aluminium,\\
&vanadium,      &&cadmium,\\
&molybdenum,    &&carbon,\\
&&lead.
\end{tabular}
%% -----File: 210.png---Folio 199-------

As to carbon, however, the spectrum is so peculiar, consisting of
bands rather than lines, that it is very difficult to be sure, but the
tendency of the latest investigations (of Rowland and Hutchins) is to
establish its right to a place on the list.

\begin{fineprint}
The more recent researches have thrown much doubt on the presence of
several substances which, a few years ago, were usually included in the list,
as, for instance, strontium and cerium. It has been generally admitted also
that the photographs of Dr.~Henry Draper had demonstrated the presence
of oxygen in the sun, represented in the solar spectrum, not by dark lines
like other elements, but by certain \textit{wide, bright bands}. The latest work,
while it does not absolutely refute Dr.~Draper's conclusion, appears however
to turn the balance of evidence the other way.
\end{fineprint}

\nbarticle{317.} It will be noticed that all the bodies named in the list, carbon
alone excepted, are \textit{metals} (chemically hydrogen is a metal), and that
a multitude of the most important terrestrial elements fail to appear;
oxygen~(?), nitrogen, chlorine, bromine, iodine, sulphur, phosphorus,
silicon, and boron are all missing. We must be cautious, however,
as to \textit{negative} conclusions. It is quite conceivable that the spectra of
these bodies under solar conditions may be so different from their spectra
as presented in our laboratories that we cannot recognize them; for
it is now quite certain that some substances, nitrogen, for instance,
under different conditions, give two or more widely different spectra.

Among the many thousand lines of the solar spectrum only a few
hundred are so far identified.

\sloppy
\nbarticle{318.} \nbparatext{Mr.~Lockyer's Views.}---Mr.~Lockyer thinks it more probable
that the missing substances are not truly elementary, but are
decomposed or ``dissociated'' on the sun by the intense heat, and so
do not exist there, but are replaced by their components; he believes,
in fact, that none of our so-called elements are really elementary, but
that all are decomposable, and, to some extent actually decomposed
in the sun and stars, and some of them by the electric spark in our
own laboratories. Granting this, a crowd of interesting and remarkable
spectroscopic facts find easy explanation. At the same time the
hypothesis is encumbered with great difficulties and has not yet been
finally accepted by physicists and chemists. For a full statement
of his views the reader is referred to his ``Chemistry of the Sun.''

\fussy
\nbarticle{319.} \nbparatext{The Reversing Layer.}---According to Kirchoff's theory the
dark lines are formed by the passing of light from the minute solid
and liquid particles of which the photospheric clouds are supposed to
%% -----File: 211.png---Folio 200-------
be formed, through vapors containing the substances which we recognize
in the solar spectrum. If this be so, the spectrum of the gaseous
envelope, which by its absorption forms the \textit{dark} lines, should by itself
show a spectrum of corresponding bright lines. The opportunities are
of course rare when it is possible to obtain the spectrum of this gas-stratum
alone by itself; but at the time of a total eclipse, at the
moment when the sun's disc has just been obscured by the moon,
and the sun's atmosphere is still visible beyond the moon's limb,
if the slit of the spectroscope be carefully adjusted to the proper
point, the observer ought to see this bright-line spectrum. The
author succeeded in making this very observation at the Spanish
eclipse of 1870. The lines of the solar spectrum, which up to the
final obscuration of the sun had remained dark as usual (with the
exception of a few belonging to the spectrum of the chromosphere),
were suddenly ``reversed,'' and the whole length of the spectrum was
filled with brilliant-colored lines, which flashed out quickly and then
gradually faded away, disappearing in about two seconds,---a most
beautiful thing to see. Substantially the same thing has since then
been several times observed.

\begin{fineprint}
\nbarticle{320.}  The natural interpretation of this phenomenon is, that \textit{the formation
of the dark lines in the solar spectrum is mainly, at least, produced by a very
thin layer close down to the photosphere}, since the moon's motion in two seconds
would cover a thickness of only about 500 miles. It was not possible, however,
to be certain that \textit{all} the dark lines were reversed, and in this uncertainty
lies the possibility of a different interpretation. Mr.~Lockyer doubts
the existence of any such \textit{thin stratum}. According to his views the solar
atmosphere is very extensive, and these lines of iron, which correspond to
the more complex combinations of its constituents, are formed only in the
regions of lower temperature, \textit{high up} in the sun's atmosphere. They should
appear \textit{early} at the time of an eclipse and \textit{last long}, but not be very bright.
Those due to the constituents of iron which are found only close down to
the solar surface should be short and bright; and he thinks that the
numerous bright lines observed under the conditions stated are due to such
substances only. Observation needs to be directed to the special point to
determine whether \textit{all} of the dark lines are reversed at the edge of the sun,
or \textit{only a few};
and if so, what ones.
\end{fineprint}

\includegraphicsouter{illo112}{\textsc{Fig.~112.}\\
The C line in the Spectrum of a Sun Spot,
Sept.~22, 1870.}

\sloppy
\nbarticle{321.} \hspace{0.8em}\nbparatext{Sun-Spot Spectrum.}---\hspace{0pt}
This is like the general solar spectrum,
except that certain lines are much widened, while certain others are
thinned, and sometimes the lines of hydrogen become bright. It is
to be noticed that by far the larger proportion of the dark lines of any
%% -----File: 212.png---Folio 201-------
%% -----File: 213.png---Folio 202-------
given substance are not affected at all in the spot spectrum, but only
a certain few of them, a point
which Mr.~Lockyer considers very
important. Not infrequently it
happens that certain lines of the
spectrum are crooked and broken
in connection with sun spots, as
shown by \figref{illo112}{Fig.~112}. Such phenomena
are caused, according to
\textit{Doppler's principle},\footnote
  {Doppler's principle is this: that when we are approaching, or approached by,
a body which is emitting regular vibrations, then the \textit{number} of waves received
by us in a second is \textit{increased}, and their wave-length correspondingly \textit{diminished};
and \textit{vice versa} when the distance of the vibrating body is increasing. Thus the
\textit{pitch} of a musical tone rises while we are approaching the sounding body, and
falls as we recede; in just the same way the ``refrangibility'' of the rays, say of
hydrogen, emanating from the sun is increased (the wave-length being shortened)
whenever we are approaching it with a speed which bears a sensible ratio to the
velocity of light. Calling $\lambda$ the wave-length of the ray when the observer and
the luminous object are relatively at rest, and $\lambda'$ the wave-length as affected by
their relative motion; putting $V$ also for the velocity of light (about 186,330 miles
per second), and $s$ for the speed with which the observer and source of light
\textit{approach} each other, we have
\[
\lambda' = \lambda \left(\frac{V}{V + s}\right).
\]
[If the distance is increasing instead of diminishing, the denominator will be
$(V - s)$.] With the most powerful spectroscopes motions of from one to two miles
  per second along the line of sight can thus be detected.}
by the swift
motion of matter towards or from
the observer. In the particular case shown in the \figref{illo112}{figure}, hydrogen is
the substance, and the greatest motion indicated was towards the
observer at the rate of about 300~miles a second---an unusual velocity.
These effects are most noticeable, not \textit{in} the spots, but near them,
usually just at the outer edge of the penumbra.

\fussy
\begin{fineprint}
The dark and apparently continuous spectrum which is due to the nucleus
of a sun spot is not truly continuous, but under high dispersion is resolved
into a range of extremely fine, close-packed, dark lines, separated by narrow
spaces. At least this is so in the green and blue portions of the spectrum; it
is more difficult to make out this structure in the yellow and red. It appears
to indicate that the absorbing medium which fills the hollow of a sun spot
is \textit{gaseous}, and not composed of precipitated particles like smoke, as has
been suggested.
\end{fineprint}
%% -----File: 214.png---Folio 203-------

\nbarticle{322.} \nbparatext{The Chromosphere.}---The \textit{chromosphere} is a region of the
sun's gaseous envelope which lies close above the photosphere, the
``\textit{reversing layer},'' if it exists at all, being only the most dense and
hottest part of it. The chromosphere is so called, because as seen
for an instant, during a total solar eclipse, it is of a bright scarlet
color, the color being due to the hydrogen which is its main constituent.
It is from 5000 to 10,000 miles in thickness, and in structure
is very like a sheet of scarlet flame, not being composed of horizontal
sheets, but of (approximately) upright filaments. Its appearance
has been compared very accurately to that of ``a prairie on fire'';
but the student must carefully guard against the idea that there is
any real ``burning'' in the case; \textit{i.e.}, any \textit{process of combination}
between hydrogen and some other substance. The temperature is
altogether too high for any formation of hydrogen compounds at the
sun's surface.

\nbarticle{323.} \nbparatext{The Prominences.}---At a total eclipse, after the totality has
fairly set in, there are usually to be seen at the edge of the moon's
disc a number of scarlet, star-like objects, which in the telescope
appear as beautiful, fiery clouds of various form and size. These are
the so-called ``\textit{prominences},'' which very non-committal name was
given while it was still doubtful whether they were solar or lunar.
Photography, in 1860, proved that they really belong to the sun,
for the photographs taken during the totality showed that the moon
obviously moves over them, covering those upon the eastern limb,
and uncovering those upon the western.

In 1868, during the observation of the eclipse which occurred
that year in India, the spectroscope showed them to be gaseous,
and that their main constituent is hydrogen. Their spectrum contains
also a conspicuous yellow line known as the $D_{3}$ line, because
it is very near the two ``$D$'' lines of sodium. This line
is not due to hydrogen, and is not identified with any known element;
but an element has been assumed for it and called provisionally
``\textit{helium}.''

In connection with this eclipse, Janssen, who observed it in India,
found that the lines of the prominence spectrum were so bright
that he was able to observe them the next day after the eclipse in
full sunlight; and he also found that by a proper management of
his instrument he could study the form and behavior of the prominences
nearly as well without an eclipse as during one. Lockyer,
in England, some time earlier had come to similar conclusions from
%% -----File: 215.png---Folio 204-------
theoretical grounds, and he practically perfected his discovery a few
weeks later than Janssen, although without knowledge of what he
had done. By a remarkable but accidental coincidence their discoveries
were communicated to the French Academy on the same day;
and in their honor the French have struck a medal bearing their united
effigies.

\sloppy
\nbarticle{324.} \nbparatext{How the Spectroscope makes the Prominences Visible.}---The
only reason we cannot see the prominences at any time is on
account of the bright illumination of our own atmosphere. We can
screen off the direct light of the sun; but we cannot screen off the
reflected sunlight coming from the air which is directly between us
and the prominences themselves; a light
so brilliant that the prominences cannot
be seen through it without some kind of
aid.

\fussy
\includegraphicsouter{illo113}{\textsc{Fig.~113.}\\
Spectroscope Slit adjusted for
Observation of the Prominences.}

The spectrum of this air-light is, of course,
just the same as that of the sun---a continuous
spectrum with the same dark lines
upon it. When, therefore, we arrange the
apparatus as indicated in \figref{illo113}{Fig.~113}, pointing
the telescope so that the image of the sun's
limb just touches the slit of the spectroscope,
then, if there is a prominence at that point, we shall have in
our spectroscope two spectra superposed upon each other; namely,
the spectrum of the air-illumination and that of the prominence.
The latter is a spectrum of \textit{bright lines}, or, if the slit is opened
a little, of bright \textit{images} of whatever part of the prominence may
fall within the edges of the slit. Now, the brightness of these
images is not affected by any increase of dispersion in the spectroscope.
Increase\footnote
  {Too high dispersion injures the definition, however, because the lines in the
  spectrum of hydrogen are rather broad and hazy.}
of dispersion merely sets these images farther
apart, without making them fainter. The spectrum of the aerial illumination,
on the other hand, is made very faint by its extension; and,
moreover, it \textit{presents dark lines} (or \textit{spaces} when the slit is opened)
precisely at the points where the bright images of the prominences
fall.

\includegraphicsouter[14]{illo114}{\textsc{Fig.~114.}\\
The Chromosphere and Prominences seen in the
Spectroscope.}

\hspace{-1.1pt}A spectroscope of dispersive power sufficient to divide the two $E$
lines, attached to a telescope of four or five inches aperture, gives a
%% -----File: 216.png---Folio 205-------
very satisfactory view of these beautiful objects; the \textit{red} image corresponds
to the $C$ line, and is by far the best for such observations,
though the $D_{3}$ line or the $F$ line can also be used. When the instrument
is properly adjusted, the slit opened a little, and the image of
the sun's limb brought exactly to the edge of the slit, the observer
at the eye-piece of the spectroscope will see things about as we have
attempted to represent them in \figref{illo114}{Fig.~114}; as if he were looking
at the clouds in an evening sky through a slightly opened window-blind.

\sloppy
\nbarticle{325.} \nbparatext{Different Kinds of Prominences; Their Forms and Motions.}---The
prominences may be broadly divided into two classes,---the
\textit{quiescent} or diffused, and the \textit{eruptive} or ``metallic,'' as Secchi
calls them, because they show
in their spectrum the lines of
many metals besides hydrogen.
The former, illustrated by \figref{illo115a}{Fig.~115}
(see p.~\figpage{115}), are immense
clouds, often 60,000 miles in
height, and of corresponding
horizontal dimensions, either
resting upon the chromosphere
or connected with it by slender
stems like great banyan-trees.
They are not very brilliant, and
are composed almost entirely
of hydrogen and ``helium.''
They often remain nearly unchanged
for days together as
they pass over the sun's limb.
They are found on all portions of the disc, at the poles and equator
as well as in the spot zones. Some of them are clouds floating
entirely detached from the sun's surface.

\fussy
Usually these clouds are simply the remnants of prominences which
appear to have been thrown up from below, but in some cases they
actually form and grow larger without any visible connection with
the chromosphere---a fact of considerable importance, as showing in
these regions the presence of hydrogen, invisible to our spectroscopes
until somehow or other it is made to give out the rays of its familiar
spectrum. All the forms and motions of the prominences, it
may be said further, seem to indicate the same thing---that they
%% -----File: 217.png---Folio 206-------
%% -----File: 218.png---Folio 207-------
%% -----File: 219.png---Folio 208-------
exist and move, not \textit{in a vacuum}, but in a medium of density comparable
with their own, as clouds do in our own atmosphere.

\nbenlargepage
\nbarticle{326.}  The \textit{eruptive} prominences, on the other hand, are brilliant
and active, not \textit{usually} so large as the quiescent, but at times
enormous, reaching elevations of 100,000, 200,000, or even 400,000
miles. They are illustrated by \figref{illo116a}{Fig.~116}. Most frequently they are
in the form of spikes or flames; but they present also a great variety
of other fantastic shapes, and are sometimes so brilliant as to be
visible with the spectroscope on the surface of the sun itself, and not
merely at the limb. Generally prominences of this class are associated
with active sun spots, while both classes appear to be connected
with the faculę. The \figref{illo115a}{figures} given are from drawings of
individual prominences that have been observed by the author at
different times.

\begin{includegraphicspage}{115}
  \begin{tabular}{c@{\qquad\qquad}c}
    \includegfx{illo115a}           & \includegfx{illo115b}\\[1ex]
    \footnotesize Clouds.      & \footnotesize Diffuse.\\[4ex]
    \includegfx{illo115c}           & \includegfx{illo115d}\\[1ex]
    \footnotesize Filamentary. & \footnotesize Stemmed.\\[4ex]
    \includegfx{illo115e}           & \includegfx{illo115f}\\[1ex]
    \footnotesize Plumes.      &  \footnotesize Horns.\\[4ex]
  \end{tabular}
  \captionof*{figure}{\textsc{Fig}.~115.\\
  Quiescent Prominences. Scale 75,000 Miles to the Inch. From ``The Sun,''\\
  by Permission of D. Appleton \& Co.}
\end{includegraphicspage}

\begin{includegraphicspage}{116}
  \begin{tabular}{c@{\qquad\qquad}c}
    \includegfx{illo116a}           & \includegfx{illo116b}\\[1ex]
    \footnotesize Vertical Filaments.      & \footnotesize Prominences Sept. 7, 1871, 12.30 \textsc{p.m.}\\[4ex]
    \includegfx{illo116c}           & \includegfx{illo116d}\\[1ex]
    \footnotesize Cyclone. & \footnotesize Same at 1.15 \textsc{p.m.}\\[4ex]
    \includegfx{illo116e}           & \includegfx{illo116f}\\[1ex]
    \footnotesize Flame.      &  \footnotesize Jets near Sun's Limb, Oct. 5, 1871.\\[4ex]
  \end{tabular}
  \captionof*{figure}{\textsc{Fig}.~116.\\
  Eruptive Prominences. From ``The Sun.'' By Permission of D. Appleton \& Co.}
\end{includegraphicspage}

These solar clouds are most fascinating objects to watch, on account
of the beauty of their forms, and the rapidity of their changes.
In the case of the eruptive prominences the swiftness of the changes
is sometimes wonderful---portions can be actually seen to move, and
this implies a real velocity of at least 250 miles a second, so that it
is no exaggeration to speak of such phenomena as veritable ``explosions'':
of course, in such cases the lines in the spectrum are greatly
broken and distorted, and frequently a ``magnetic storm'' follows
upon the earth, with a brilliant Aurora Borealis.

The number visible at a single time is variable, but it is not very
unusual to find as many as twenty on the sun's limb at once.

\nbarticle{327.} \nbparatext{The Corona.}---This is a halo, or ``glory,'' of light which surrounds
the sun at the time of the total eclipse. From the remotest
times it has been well known, and described with enthusiasm, as being
certainly one of the most beautiful of natural phenomena.

The portion of the corona nearest the sun is almost dazzlingly bright,
with a greenish, pearly tinge which contrasts finely with the scarlet
blaze of the prominences. It is made up of streaks and filaments
which on the whole radiate outwards from the sun's disc, though
they are in many places strangely curved and intertwined. Usually
these filaments are longest in the sun-spot zones, thus giving the
corona a more or less quadrangular figure. At the very poles of
the sun, however, there are often tufts of sharply defined threads.

For the most part the streamers have a length not much exceeding
the sun's radius, but some of them at almost every eclipse go
%% -----File: 220.png---Folio 209-------
far beyond this limit. In the clear air of Colorado during the eclipse
of 1878, two of them could be traced for five or six degrees,---a
distance of at least 9,000000 miles from the sun. A most striking
feature of the corona usually consists of certain dark rifts which
reach straight out from the moon's limb, clear to the extremest limit
of the corona.

The corona varies much in brightness at different eclipses, and of
course the details are never twice the same. Its total light under
ordinary circumstances is at least two or three times as great as that
of the full moon.

\nbarticle{328.} \nbparatext{Photographs of the Corona.}---While the eye can perhaps
grasp some of its details more satisfactorily than the photographic
plate can do, it is found that drawings of the corona are hardly to be
trusted. At any rate, it seldom happens that the representations of
two artists agree sufficiently to justify any confidence in their scientific
accuracy. Photographs, on the other hand, may be trusted as far as
they go, though they may fail to bring out some things which are
conspicuous to the eye. \figref{illo117}{Fig.~117} is from the photograph of the Egyptian
eclipse of 1882, when a little comet was found close to the sun.

\includegraphicsmid{illo117}{\textsc{Fig.~117.}---Corona of the Egyptian Eclipse, 1882.}

\begin{fineprint}
Of course, as in the case of the prominences, the only reason we cannot
see the corona without an eclipsed sun is the illumination of the earth's
atmosphere. If we could ascend above our atmosphere, and manage to exist
and to observe there, we could see it by simply screening off the sun's disc.
%% -----File: 221.png---Folio 210-------
So long, however, as the brightness of the illuminated air is more than about
sixty times that of the corona, it must remain invisible to the eye. Dr.~Huggins
has thought that it might be possible by means of photographs to
detect differences of illumination less than $\frac{1}{60}$ (the limit of the eye's perception),
and so to obtain pictures of the corona at any time; especially as it
appears that the coronal light is far richer in ultra-violet rays (the photographic
rays) than the general sunlight with which the air is illuminated.
His attempts so far, however, have yielded only doubtful success.
\end{fineprint}

\nbarticle{329.} \nbparatext{Spectrum of the Corona.}---This was first definitely observed
in 1869 during the eclipse which passed over the western part of the
United States in that year. It was then found that its most remarkable
characteristic is a bright line in the green, which the writer identified
as coinciding with the dark line at 1474 on the scale of Kirchoff's
map ($\lambda = 5316$). This line was also observed by Harkness.

\begin{fineprint}
This result was for a time very puzzling, since the dark line in question
is given by \AA ngstrom and other authorities as due to the spectrum of \textit{iron}.
The mystery has since been removed, however, by the discovery that under
high dispersion the line is double, and that the corona line coincides with
the more refrangible of the two components, while the other one is the line
due to iron. We have as yet been unable to identify with any terrestrial
element the substance to which this line is due, but the provisional name
``\textit{coronium}'' has been proposed for it. The recent researches of Grünewald
make it somewhat probable that both coronium and helium are components
of hydrogen, which (in line with Mr.~Lockyer's speculations) is supposed to
be partially decomposed under solar conditions.
\end{fineprint}

Besides this conspicuous green line, the hydrogen lines are also
faintly visible in the spectrum of the corona; and by means of a photographic
camera used during the Egyptian eclipse of 1882, it was
found that the upper or violet portion of the spectrum is very rich in
lines, among which $H$ and $K$ are specially conspicuous. There is
also, through the whole spectrum, a faint continuous background,
which, however, according to Mr.~Lockyer's statements, is not of
uniform brightness, but ``\textit{banded}.'' In it some observers have reported
the presence of a few of the more conspicuous dark lines of
the ordinary solar spectrum, but the evidence on this point is rather
conflicting.

If during the totality we look at the eclipsed sun with a diffraction
grating, or through a prism of high dispersive power, we see three
rings which are really images of the corona. One of them, the brightest
and the largest, is the green ring due to the 1474 line; the others
%% -----File: 222.png---Folio 211-------
are a red ring due to $C$, and a blue one due to the $F$ line of
hydrogen.

\nbarticle{330.} \nbparatext{Nature of the Corona.}---It is evident that the corona is a
truly solar and not merely an optical or atmospheric phenomenon, from
two facts: first, \textit{the identity of detail in photographs made at widely
separate stations}. In 1871, for instance, photographs were obtained at
the Indian station of Bekul, in Ceylon, and in Java, three stations separated
by many hundreds of miles; but, excepting minute differences
of detail, such as might be expected to have resulted from the changes
that would naturally go on in the corona, during the half-hour while
the moon's shadow was travelling from Bekul to Java, all the photographs
agree exactly, which of course would not be the case if the
corona depended in any way upon the atmospheric conditions at the
observer's station.

Second (but \textit{first} historically), \textit{the presence of bright lines in the
spectrum of the corona} proves that it cannot be a terrestrial or lunar
phenomenon, by demonstrating the presence in the corona of a \textit{self-luminous
gas}, which observation fails to find either near to the moon
or in our own atmosphere. It must, therefore, be at the sun.

But while it is thus certain that the corona contains luminous gas, it
also is very likely that finely divided solid or liquid matter may be present
in the corona; that is, fog or dust of some kind.

\nbarticle{331.}  The corona cannot be a true ``solar \textit{atmosphere}'' in any strict
sense of the word. No gaseous envelope in any way analogous to the
earth's atmosphere could possibly exist there in gravitational equilibrium
under the solar conditions of pressure and temperature. The
corona is probably a phenomenon due somehow to the intense activity
of the forces there at work; meteoric matter, cometic matter, matter
ejected from within the sun, are all concerned.

That this matter is inconceivably rare is evident from the fact that
in several cases comets have passed directly through the corona without
experiencing the least perceptible disturbance of their motions. It is
altogether probable that at a very few thousand miles above the sun's
surface, its density becomes far less than that of the best vacuum we
can make in an electric lamp.
\chelabel{CHAPTERVIII}
%% -----File: 223.png---Folio 212-------

\Chapter{IX}{The Sun's Light and Heat}
\nbchapterhang{\stretchyspace
THE SUN'S LIGHT AND HEAT: COMPARISON OF SUNLIGHT WITH
ARTIFICIAL LIGHTS.---MEASUREMENT OF THE SUN'S HEAT,
AND DETERMINATION OF THE ``SOLAR CONSTANT.''---PYRHELIOMETER,
ACTINOMETER, AND BOLOMETER.---THE SUN'S
TEMPERATURE.---THEORIES AS TO THE MAINTENANCE OF
THE SUN'S RADIATION, AND CONCLUSIONS AS TO THE SUN'S
POSSIBLE AGE AND FUTURE DURATION.}

\nbarticle{332.} \nbparatext{The Sun's Light.}---\textit{The Quantity of Sunlight}. It is very easy
to compare (approximately) sunlight with the light of a standard\footnote
  {A standard candle is a sperm candle weighing one-sixth of a pound and burning
120 grains an hour. The French ``Carcel burner,'' used as a standard in their
photometry, gives just ten times the quantity of light given by this standard
candle. An ordinary gas-burner consuming five feet of gas hourly gives a light
  equivalent to from twelve to fifteen standard candles.}
candle; and the result is, that when the sun is in the zenith, it illuminates
a white surface about 60,000 times as strongly as a standard
candle at a distance of one metre. If we allow for the atmospheric absorption,
the number would he fully 70,000. If we then multiply 70,000
by the square of 150,000 million (roughly the number of metres in
the sun's distance from the earth), we shall get what a gas engineer
would call the sun's ``\textit{candle power}.'' The number comes out 1575
billions of billions (English); \textit{i.e.}, 1575 with twenty-four ciphers
following.

\begin{fineprint}
\nbarticle{333.} One way of making the comparison is the following: Arrange matters
as in \figref{illo118}{Fig.~118}. The sunlight is brought into a darkened room by a
mirror $M$, which reflects the rays through a lens $L$ of perhaps half an inch in
diameter. After the rays pass the focus they diverge and form on the
screen $S$ a disc of light, the size of which may be varied by changing the
distance of the screen. Suppose it so placed that the illuminated circle is
just ten feet in diameter; that is, 240 times the diameter of the lens. The
illumination of the disc will then be less than that of direct sunlight in
the ratio of $240^2$ (or 57,600) to 1 (neglecting the loss of light produced by
%% -----File: 224.png---Folio 213-------
the mirror and the lens, a loss which of course must he allowed for). Now
place a little rod like a pencil near the screen, as at $P$, light a standard
candle, and move the candle back and forth until the two shadows of the
pencil, one formed by the candle, and the other by the light from the lens,
are equally dark. It will be found that the candle has to be put at a distance
of about one metre from the screen; though the results would vary a
good deal from day to day with the clearness of the air.
\end{fineprint}

\includegraphicsmid{illo118}{\textsc{Fig.~118.}---Comparison of Sunlight with a Standard Candle.}

\nbarticle{334.} When the sun's light is compared with that of the full moon
and of various stars, we find, as stated (\artref{Art.}{259}), that it is about
600,000 times that of the full moon. It is 7,000,000000 times as
great as the light received from Sirius, and about 40,000,000000
times that from Vega or Arcturus.

\nbarticle{335.} \nbparatext{The Intensity of the Sun's Luminosity.}---This is a very
different thing from the total quantity of its light, as expressed by
its ``candle power'' (a surface of comparatively feeble luminosity
can give a great quantity of light if large enough). It is the \textit{amount
of light per square inch of luminous surface} which determines the
intensity. Making the necessary computations from the best data
obtainable (only roughish approximations being possible), it appears
that the sun's surface is about 190,000 times as bright as that of a
%% -----File: 225.png---Folio 214-------
candle flame, and about 150 times as bright as the lime of the calcium
light. \textit{Even the darkest part of a solar spot outshines the lime}. The
intensely brilliant spot in the so-called ``crater'' of an electric arc
comes nearer sunlight than anything else known, being from one-half
to one-fourth as bright as the surface of the sun itself. But either
the electric arc or the calcium light, when interposed between the
eye and the sun looks like a dark spot on the disc.

\nbarticle{336.}  \nbparatext{Comparative Brightness of Different Portions of the Sun's
Surface.}---By forming a large image of the sun, say a foot in diameter,
upon a screen, we can compare with each other the rays
coming from different parts of the sun's disc. It thus appears that
there is a great diminution of light at the edge, the light there, according
to Professor Pickering's experiments, being just about one-third
as strong as at the centre. There is also an obvious difference of
color, the light from the edge of the disc being brownish red as compared
with that from the centre. The reason is, that the red and
yellow rays of the spectrum lose much less of their brightness at the
limb than do the blue and violet. According to Vogel, the latter rays
are affected nearly twice as much as the former. For this reason,
photographs of the sun exhibit the darkening of the limb much more
strongly than one usually sees it in the telescope.

\includegraphicsouter{illo119}{\textsc{Fig.~119.}\\Cause of the Darkening of the Sun's Limb.}

\nbarticle{337.} \nbparatext{Cause of the Darkening of the Limb.}---It is due unquestionably
to the general absorption of the sun's rays by the lower portion
of the overlying atmosphere.
The reason is obvious from the
figure (\figref{illo119}{Fig.~119}). The \textit{thinner} this
atmosphere, other things  being
equal, the \textit{greater the} ratio \textit{between
the percentage of absorption at the
centre and edge of the disc, and
the more obvious the darkening of
the limb}.

Attempts have been made to
determine from the observed differences
between the brightness of centre and limb the total
percentage of the sun's light thus absorbed. Unfortunately we have
to supplement the observed data with some very uncertain assumptions
in order to solve the problem; and it can only be said that
it is \textit{probable} that the amount of light, absorbed by the sun's atmosphere
%% -----File: 226.png---Folio 215-------
lies between fifty and eighty per cent; \textit{i.e.}, the sun deprived of
its gaseous envelope would probably shine from two to five times as
brightly as now. It is noticeable also, as Langley long ago pointed
out, that thus stripped, the ``complexion'' of the sun would be
markedly changed from yellowish white to a good full \textit{blue}, since the
blue and violet rays are much more powerfully absorbed than these
at the lower end of the spectrum.

\section*{THE SUN'S HEAT.}

\nbarticle{338.} \nbparatext{Its Quantity; the ``Solar Constant.''} By the ``\textit{quantity of
heat}'' received by the earth from the sun we mean the number of
heat-units received in each unit of time by a square unit of surface
when the sun is in the zenith. The heat-unit most employed by
engineers is the \textit{calorie}, which is the quantity of heat required to
raise the temperature of one kilogram of water one degree centigrade.
It is found by observation that each square metre of surface exposed
perpendicularly to the sun's rays receives from the sun each minute
from twenty-five to thirty of these calories; or rather it \textit{would do so} if
a considerable portion of the sun's heat were not stopped by the earth's
atmosphere, which absorbs some thirty per cent of the whole, even
when the sun is vertical, and a much larger proportion when the sun
is near the horizon. This quantity, \textit{twenty-five calories\footnote
  {For many scientific purposes the engineering calorie is inconveniently
large, and a smaller one is employed, which replaces the kilogram of water
by the \textit{gram} heated one degree---the smaller calorie being thus only $\frac{1}{1000}$ of
the engineering unit. As stated by many writers (Langley, for instance), the
solar constant is the number of these \textit{small} calories received per square \textit{centimetre}
of surface in a minute. This would make the number 2.5 instead of 25. It
would perhaps be better to bring the whole down to the ``c.g.s.\ system'' by substituting
the \textit{second} for the minute; and this would give us for the solar constant,
  on the ``c.g.s.\ system,'' 0.0417 (\textit{small}) \textit{calories per square centimetre per second}.}
per square
metre per minute} (using the smaller of the values mentioned, which
\textit{certainly} is not too large), is known as the ``\textit{Solar Constant}.''

\nbarticle{339.} \nbparatext{Method of determining the ``Solar Constant.''}---The method
by which the solar constant is determined is simple enough in principle,
though complicated with serious practical difficulties which
affect its accuracy. It is done by allowing \textit{a beam of sunlight of
known cross-section to shine upon a known weight of water} (\textit{or other
substance of known specific heat}) \textit{for a known length of time, and}
%% -----File: 227.png---Folio 216-------
\textit{measuring the rise of temperature}. It is necessary, however, to determine
and allow for the heat received from other sources during the
experiment, and for that lost by radiation. Above all, the absorbing
effect of our own atmosphere is to be taken into account, and
this is the most difficult and uncertain part of the work, since the
atmospheric absorption is continually changing with every change of
the transparency of the air, or of the sun's altitude.

\begin{fineprint}
\includegraphicsouter{illo120}{\textsc{Fig.~120.}---Pouillet's Pyrheliometer.}

\sloppy
\nbarticle{340.} \nbparatext{Pyrheliometers and Actinometers.}---The instruments with
which these measurements are made, are known as ``pyrheliometers'' and
``actinometers.'' \figref{illo120}{Fig.~120} represents the pyrheliometer
of Pouillet, with which in 1838 he
made his determination of the solar constant,
at the same time that Sir John Herschel was
experimenting at the Cape of Good Hope in
practically the same way. They were the
first apparently to understand and attack the
problem in a reasonable manner. The pyrheliometer
consists essentially of a little cylindrical
box $ab$, like a snuff-box, made of thin
silver plate, with a diameter of one decimetre
and such a thickness that it holds 100 grams
of water. The upper surface is carefully
blackened, while the rest is polished as brilliantly
as possible. In the water is inserted
the bulb of a delicate thermometer, and the
whole is so mounted that it can be turned in
any direction so as to point it directly towards
the sun. It is used by first holding a screen
between it and the sun for (say) five minutes,
and watching the rise or fall of the mercury in
the thermometer at $m$. There will usually be
some slight change due to the radiation of
surrounding bodies. The screen is then removed,
and the sun is allowed to shine upon
the blackened surface for five minutes, the
instrument being continually turned upon
the thermometer as an axis, in order to keep the water in the calorimeter
box well stirred. At the end of the five minutes the screen is replaced
and the rise of the temperature noted. The difference between this and
the change of the thermometer during the first five minutes will give us the
amount by which a beam of sunlight one decimetre in diameter has raised
the temperature of 100 grams of water in five minutes, and were it not for
the troublesome corrections which must be made, would furnish directly the
value of the solar constant.
%% -----File: 228.png---Folio 217-------

\fussy
\includegraphicsouter{illo121}{\textsc{Fig.~121.}---Violle's Actinometer.}

\nbarticle{341.} The second apparatus, \figref{illo121}{Fig.~121}, is the actinometer of Violle, which
consists of two concentric metal spheres, the inner of which is blackened on
the inside, while the outer one is brightly polished, the space between the
two being filled with water at a known temperature, kept circulating by a
pump of some kind. The thermoscopic
body in this case, instead of being a box
filled with water, is the blackened bulb
of the thermometer $T$; and the observations
may be made either in the same
way as with the pyrheliometer, or simply
by noting the difference between the
temperature finally attained by the thermometer
$T$ after it has ceased to rise in
the sun's rays, and the temperature of
the water circulating in the shell.

\nbarticle{342.} \nbparatext{Correction for Atmospheric
Absorption.}---The correction for atmospheric
absorption is determined by
making observations at various altitudes
of the sun between zenith and horizon.
If the rays were \textit{homogeneous} (that is,
all of one wave-length), it would be
comparatively easy to deduce the true correction and the true value of
the solar constant. In fact, however, the \textit{visible} solar spectrum is but a
small portion of the whole spectrum of the sun's radiance, and, as Langley
has shown, it is necessary to determine the coefficient of absorption separately
for all the rays of different wave-length.

\sloppy
\nbarticle{343.} \nbparatext{The Bolometer.}---This he has done by means of his ``Bolometer,''
an instrument which is capable of indicating exceedingly minute changes in
the amount of radiation received by an extremely thin strip of metal. This
strip is so arranged that the least change in its electrical resistance due to
any change of temperature will disturb a delicate galvanometer. The
instrument is far more sensitive than any thermometer or even thermopile,
and has the especial advantage of being extremely quick in its response
to any change of radiation. \figref{illo122}{Fig.~122} shows it so connected with
a spectroscope that the observer can bring to the bolometer, $B$, rays of
any wave-length he chooses. The rays enter through the collimator lens
$L$, and are then refracted by the prism $P$ to the reflector $M$, whence they
are sent back to $B$.

\fussy
Langley has shown that the corrections for atmospheric absorption deduced
by earlier observers are all considerably too small, and has raised the received
value of the solar constant, from 20 or 25, which was the value
accepted a few years ago, to 30. We have, however, provisionally retained
%% -----File: 229.png---Folio 218-------
the 25, as his new results, though almost certainly correct, have not yet
been universally accepted, and perhaps need verification.
\end{fineprint}

\includegraphicsouter{illo122}{\textsc{Fig.~122.}\\
Langley's Spectro-Bolometer, as used for Mapping
the Energy of the Prismatic Spectrum.}

\nbarticle{344.} A less technical statement of the solar radiation may be
made in terms of thickness of
the quantity of \textit{ice} which would
be melted by it in a given time.
Since it requires about eighty
calories of heat to melt a kilogram
of ice, it follows that
twenty-five calories per minute
per square metre would liquefy
in an \textit{hour} a sheet of ice \textit{one
metre square and about nineteen
millimetres thick}. According
to this the sun's heat
would melt about 174 feet of
ice annually on the earth's
equator; or $136\frac{1}{2}$ feet yearly all
over the surface of the earth,
if the heat annually received
were equally distributed in all
latitudes. (See note at end of
the chapter, page~\pageref{pg:227}.)

\nbenlargepage
\nbarticle{345.} \nbparatext{Solar Heat expressed
as Energy.}---Since according to
the known value of the ``mechanical equivalent of heat'' (Physics, p.~159)
a horse-power corresponds to about $10\frac{7}{10}$ calories per minute, it
follows that \textit{each square metre of surface} (neglecting the air-absorption)
\textit{would receive, when the sun is overhead}, about \textit{two and one-third
horse-power continuously}. Atmospheric absorption cuts this
down to about one and one-half horse-power, of which about one-eighth
can be actually utilized by properly constructed machinery,
as, for instance, the solar engines of Ericsson and Mouchot (see
Langley's ``New Astronomy''). In Ericsson's apparatus the reflector,
about 11~feet by 16~feet, collected heat enough to work a
three-horse-power engine very well. Taking the earth's surface as a
whole, the energy received during a year aggregates about sixty mile-tons
for every square foot. That is to say, the \textit{heat annually received
on each square foot of the earth's surface, if employed in a}
%% -----File: 230.png---Folio 219-------
\textit{perfect heat engine, would be able to hoist sixty tons to the height
of a mile}.

\nbarticle{346.} \nbparatext{Solar Radiation at the Sun's Surface.}---If, now, we estimate
the amount of radiation at the sun's surface itself, we come to
results which are simply amazing and beyond comprehension.  It is
necessary to multiply the solar constant observed at the earth (which
is at a distance of 93,000000 miles from the sun) by the square of
the ratio between 93,000000 and 433,250, the radius of the sun. This
square is about 46,000; in other words, the amount of heat emitted
in a minute by a square metre of the sun's surface is about 46,000
times as great as that received by a square metre at the earth. Carrying
out the calculations, we find that this heat radiation at the surface
of the sun amounts to \textit{over a million calories per square metre
per minute}; that it is over 100,000 horse-power per square metre
continuously acting; that \textit{if the sun were frozen over completely to
a depth of fifty feet, the heat emitted is sufficient to melt this whole
shell in one minute of time}; that if an ice bridge could be formed
from the earth to the sun by a column of ice two and one-fourth
miles square at the base and extending across the whole 93,000000
of miles, and if by some means the whole of the solar radiation
could be concentrated upon this column, it would be melted in one
second of time, and in between seven and eight seconds more would
be dissipated in vapor. To maintain such a development of heat \textit{by
combustion} would require the \textit{hourly burning of a layer of the best
anthracite coal from sixteen to twenty feet thick} over the sun's entire
surface,---a ton for every square foot of surface,---at least nine
times as much as the consumption of the most powerful blast furnace
in existence. At that rate the sun, if made of solid coal, would
not last 6000 years.

\begin{fineprint}
\nbarticle{347.} \nbparatext{Waste Of Solar Heat.}---Those estimates are of course based on
the assumption that the sun radiates heat equally in all directions, and there
is no assignable reason why it should not do so. On this assumption, however,
\textit{so far as we can see}, only a minute fraction of the whole radiation ever
reaches a resting-place. The earth receives about $\frac{1}{2200,000000}$ of the whole, and
the other planets of the solar system, with the comets and the meteors, get
also their shares; all of them together, perhaps ten or twenty times as much
as the earth. Something like $\frac{1}{100,000000}$ of the whole seems to be utilized within
the limits of the solar system. As for the rest, science cannot yet tell what
becomes of it. A part, of course, reaches distant stars and other objects in
interstellar space; but by far the larger portion seems to be ``wasted,'' according
to our human ideas of waste.
\end{fineprint}
%% -----File: 231.png---Folio 220-------

\sloppy
\nbarticle{348.} Experiments with the thermopile, first conducted by Henry
at Princeton in 1845, show that the heat from the edges of the sun's
disc, like the light, is less than that from the centre---according
to Langley's measurements about half as much. The explanation
evidently lies in its absorption by the solar atmosphere.

\fussy
\includegraphicsouter{illo123}{\textsc{Fig.~123.}}

\nbarticle{349.} \nbparatext{The Sun's Temperature.}---While we can measure with some
accuracy the \textit{quantity} of heat sent us by the sun, it is different with
its \textit{temperature} in respect to which we can only say that it must be
very high---much higher than any temperature attainable by known
methods on the surface of the earth.

This is shown by a
number of facts, for instance,
by the great \textit{abundance
of the violet and
ultra-violet rays} in the
sunlight.

Again, by the \textit{penetrating}
power of sunlight;
a large percentage of the heat from a common fire, for instance,
being stopped by a plate of glass, while nearly the whole of the solar
radiation passes through.

The most impressive demonstration, however, follows from this
fact; viz., that at the focus of a powerful burning-lens all known
substances melt and vaporize, as in an electric arc. Now at the
focus of the lens the \textit{limit} of the temperature is that which would
be produced by the sun's direct radiation at a point where the sun's
angular diameter equals that of the burning-lens itself seen from the
focus, as represented in \figref{illo123}{Fig.~123}. An object at $F$ would theoretically
(that is, if there was no loss of heat conducted away by surrounding
bodies and by the atmosphere) reach the same temperature
as if carried to a point where the sun's angular diameter equals the
angle $LFL'$. In the most powerful burning-lenses yet constructed
a body at the focus is thus virtually carried up to within about
240,000 miles of the sun's surface, where its apparent diameter
would be about $80°$. Here, as has been said, the most refractory
substances are immediately subdued. If the earth were to approach
the sun as near as the moon is to us, she would melt and be
vaporized.

\begin{fineprint}
\nbarticle{350.}  Ericsson in 1872 made an exceedingly ingenious and interesting
experiment illustrating the intensity of the solar heat. He floated a calorimeter,
%% -----File: 232.png---Folio 221-------
containing about ten pounds of water, upon the surface of a large
mass of molten iron by means of a raft of fire-brick, and found that the
radiation of the metal was a trifle over 250 calories per minute for each
square foot of surface; which is only $\frac{1}{400}$ part of the amount emitted by the
same area of the sun's surface. He estimated the temperature of the metal
at $3000°$~F. or $1649°$~C.
\end{fineprint}

\sloppy
\nbarticle{351.} \nbparatext{Effective Temperature.}---The question of the sun's temperature
is embarrassed by the fact that it has no \textit{one} temperature; the
temperature at different parts of the solar photosphere and chromosphere
must be very different. We evade this difficulty to some
extent by substituting for the \textit{actual} temperature, as the object of
inquiry, what has been called the sun's ``\textit{effective temperature}''; that
is, the temperature which a sheet of \textit{lampblack} must have in order
to radiate the amount of heat actually thrown off by the sun. (Physicists
have taken the radiating power of lampblack as \textit{unity}.) If we
could depend upon the laws\footnote
  {A number of such laws have been formulated; for instance, the well-known
law of Dulong and Petit (Physics, p.~470). The French physicists Pouillet and
Vicaire, using this formula, have deduced values for the sun's effective temperature
running from $1500°$ to $2500°$~C\@. Ericsson and Secchi, using Newton's law
of radiation (which, however, is certainly inapplicable under the circumstances),
put the figure among the millions. Zöllner, Spörer, and Lane give values ranging
  from $25,000°$ to $50,000°$~C.}
deduced from laboratory experiments,
by which it has been sought to connect the temperature of the body
with its rate of radiation, the matter would then be comparatively
simple: from the known radiated \textit{quantity of heat} (in calories) we
could compute the \textit{effective temperature} in degrees. But at present it
is only by a very unsatisfactory process of extrapolation that we can
reach conclusions. The sun's temperature is so much higher than
any which we can manage in our laboratories, that there is not yet
much certainty to be obtained in the matter. Rosetti, the most
recent investigator, whose results seem to be on the whole the most
probable, obtains $10,000°$~C. or $18,000°$~F. for the effective temperature.

\fussy
\nbarticle{352.} \nbparatext{Constancy of the Sun's Heat.}---It is an interesting and thus
far unsolved problem, whether the total amount of the sun's radiation
varies perceptibly at different times.  It is only certain that the
variations, if real, are too small to be detected by our present means
of observation.  Possibly, at some time in the future, observations
on a mountain summit above the main body of our atmosphere may
decide the question.
%% -----File: 233.png---Folio 222-------

It is not unlikely that changes in the earth's climate such as
have given rise to glacial and carboniferous periods may ultimately
be traced to the condition of the sun itself, especially to changes in
the thickness of the absorbing atmosphere, which, as Langley has
pointed out, must have a great influence in the matter. Since the
Christian era, however, it is certain that the amount of heat annually
received from the sun has remained practically unchanged. This is
inferred from the distribution of plants and animals, which is still substantially
the same as in the days of Pliny.

\nbarticle{353.} \nbparatext{Maintenance of the Solar Heat.}---The question at once
arises, if the sun is sending off such an enormous quantity of heat
annually, how is it that it does not grow cold?

(\textit{a}) The sun's heat cannot be kept up by \textit{combustion}. As has
been said before, it would have burned out long ago, even if made
of solid coal burning in oxygen.

(\textit{b}) Nor can it be simply a \textit{heated body cooling down}. Huge as it
is, an easy calculation shows that its temperature must have fallen
greatly within the last 2000 years by such a loss of heat, even if it
had a specific heat higher than that of any known substance.

As matters stand at present, the available theories seem to be
reduced to two,---that of Mayer, which ascribes the solar heat to
the energy of meteoric matter falling on the sun; and that of Helmholtz,
who finds the cause in a slow contraction of the sun's diameter.

\nbarticle{354.} \nbparatext{Meteoric Theory of Sun's Heat.}---The first is based on the
fact that when a moving body is stopped, its mass-energy becomes
molecular energy, and appears mainly as heat. The amount of heat
developed in such a case is given by the formula
\[
Q = \frac{MV^{2}}{8339},
\]
in which $Q$ is the number of calories of heat produced, $M$ the mass
of the moving body in kilograms, and $V$ its velocity in metres
per second; the denominator is the ``mechanical equivalent of heat''
\DPtypo{}{(}Physics, p.~159) multiplied by $2g$ expressed in metres; \textit{i.e.}, $425 × 2 × 9.81$.

Now, the velocity of a body coming from any considerable distance
and falling into the sun can be shown to be about 380 miles per
second, or more than 610 kilometres. A body weighing one kilogram
%% -----File: 234.png---Folio 223-------
would therefore, on striking the sun with this velocity, produce about
45,000000 calories of heat,
\[
\left[\frac{(610000)^{2}}{8339}\right].
\]
This is 6000 times more than could be produced by \textit{burning} it, even
if it were coal or solidified hydrogen burning in pure oxygen.

Now, as meteoric matter is continually falling upon the earth, it
must be also falling upon the sun, and in vastly greater quantities,
and an easy calculation shows that a quantity of meteoric matter
equal to $\frac{1}{100}$ of the earth's mass striking the sun's surface annually
with the velocity of 600 kilometres per second would account for its
whole radiation.

\nbarticle{355.} \nbparatext{Objections to Meteoric Theory of Sun's Heat.}---There can be
no question that a certain fraction of the sun's heat is obtained in this
way, but it is very improbable that this fraction is a large one;
indeed, it is hardly possible that it can be as much as \textit{one per cent} of
the whole.

\begin{fineprint}
(1) The annual fall on the sun's surface of such a quantity of meteoric
matter implies the presence \textit{near} the sun of a vastly greater mass; for, as we
shall see hereafter, only a few of the meteors that approach the sun from
outer space would strike the surface: most of them would act like the
comets and swing around it without touching. Now, if there were any
considerable quantity of such matter near the sun, there would result disturbances
in the motions of the planets Mercury and Venus, such as observation
does not reveal.

(2) Professor Peirce has shown further that if the heat of the sun were
produced in this way, the earth ought to receive from the meteors that strike
her surface about half as much heat as she gets from the sun. Now the
quantity of meteoric matter which would have to fall upon the earth to furnish
us daily half as much heat as we receive from the sun, would amount to
nearly fifty tons for each square mile. It is not likely that we actually get
$\frac{1}{10,000000}$ of that amount. It is difficult to determine the amount of heat which
the earth actually does receive from meteors, but all observations indicate
that the quantity is extremely small. The writer has estimated it, from
the best data attainable, as less in a \textit{year} than we get from the sun in
a \textit{second}.
\end{fineprint}

\sloppy
\nbarticle{356.} \nbparatext{Helmholtz's Theory of Solar Contraction.}---We seem to be
shut up to the theory of Helmholtz, now almost universally accepted:
namely, that the heat necessary to maintain the sun's radiation is
principally supplied \textit{by the slow contraction of its bulk}, aided, however,
%% -----File: 235.png---Folio 224-------
by the accompanying liquefaction and solidification of portions of its
gaseous mass. When a body falls through a certain distance, \textit{gradually},
against resistance, and then comes to rest, the same total amount
of heat is produced as if it had fallen \textit{freely, and been stopped instantly}.
If, then, the sun does contract, heat is necessarily produced by the
process, and that in enormous quantity, since the attracting force at
the solar surface is more than twenty-seven times as great as terrestrial
gravity, and the contracting mass is immense. In this process
of contraction each particle at the surface moves inward by an
amount equal to the diminution of the sun's radius: a particle below
the surface moves less and under a diminished gravitating force; but
every particle in the whole mass, excepting only that at the exact
centre of the globe, contributes something to the evolution of heat.
In order to calculate the precise amount of heat evolved by a given
shrinkage it would be necessary to know the law of increase of the
sun's density from the surface to the centre; but Helmholtz has
shown that under the most unfavorable conditions \textit{a contraction in the
sun's diameter of about two hundred and fifty feet a year} (125 feet in
the sun's \textit{radius}) \textit{would account for the whole annual output of heat}.
This contraction is so slow that it would be quite imperceptible to
observation. It would require more than 9000 years to reduce the
sun's diameter by a single second of arc; and nothing much less
would be certainly detectible by our measurements. \textit{If the contraction
is more rapid than this}, the mean temperature of the sun
must be actually \textit{rising}, notwithstanding the amount of heat it is
losing. Long observation alone can determine whether this is really
the case or not.

\fussy
\begin{fineprint}
\nbarticle{357.} \nbparatext{Lane's Law.}---It is a remarkable fact, first demonstrated by
Lane of Washington, in 1870, that a gaseous sphere, losing heat by radiation
and contracting under its own gravity, \textit{must rise in temperature and actually
grow hotter}, until it ceases to be a ``perfect gas,'' either by beginning to
liquefy, or by reaching a density at which the laws of perfect gases no longer
hold. The kinetic energy developed by the shrinkage of a gaseous mass
is more than sufficient to replace the loss of heat which caused the shrinkage.
In the case of a \textit{solid or liquid} mass this is not so. The shrinkage
of such a mass contracting under its own gravity on account of the loss
of heat is never sufficient to make good the loss; but the temperature falls
and the body cools. At present it appears that in the sun the relative
proportions of true gases and liquids are such as to keep the temperature
nearly stationary, the liquid portions of the sun being of course the little
drops which are supposed to constitute the clouds of the photosphere.
\end{fineprint}
%% -----File: 236.png---Folio 225-------

\nbarticle{358.} \nbparatext{Future Duration of the Sun.}---If this shrinkage theory of the
solar heat is correct (and there is every reason to accept it), it follows
that in time the sun's heat must come to an end, and, looking backwards,
we see that there must have been a beginning.

We have not sufficient data to enable us to calculate the future
duration of the sun with exactness, though an approximate estimate
can he made. According to Newcomb, if the sun maintains its
present radiation, it will have shrunk to half its present diameter in
about 5,000000 years at the longest. Since when reduced to this
size it must be about eight times as dense as now, it can hardly
then continue to be mainly gaseous, and its temperature must begin
to fall. Newcomb's conclusion, therefore, is that it is not likely
that the sun can continue to give sufficient heat to support such life
on the earth as we are now acquainted with, for 10,000000 years
from the present time.

\nbarticle{359.} \nbparatext{Age of the Sun.}---As to the past of the solar history on this
hypothesis, we can be a little more definite. It is only necessary to
know the present amount of radiation, and the mass of the sun, to compute
how long the solar fire can have been maintained at its present
intensity by the processes of condensation. No conclusion of geometry
is more certain than this,---that the contraction of the sun to its
present size, from a diameter even many times greater than Neptune's
orbit, would have furnished about 18,000000 times as much
heat as the sun now supplies in a year, and therefore that the sun
cannot have been emitting heat \textit{at the present rate} for more than
18,000000 years, \textit{if its heat has really been generated in this manner}.

\begin{fineprint}
But of course this conclusion as to the possible past duration of the solar
system rests upon the assumption that the sun has derived its heat \textit{solely in
this way}; and moreover, that it radiates heat equally in all directions in
space,---assumptions which possibly further investigations may not confirm.
\end{fineprint}

\nbarticle{360.} \nbparatext{Constitution of the Sun.}---(\textit{a}) As to the nature of the main
body or nucleus of the sun, we cannot be said to have certain knowledge.
It is probably \textit{gaseous}, this being indicated by its low mean
density and its high temperature---enormously high even at the surface,
where it is coolest. At the same time the gaseous matter at the
nucleus must be in a very different state from gases as we commonly
know them in our laboratories, on account of the intense heat and the
extreme condensation by the enormous force of solar gravity. The
central mass, while still strictly gaseous, because observing the three
%% -----File: 237.png---Folio 226-------
physical laws of Boyle, Dalton, and Gay Lussac, which characterize
gases, would be denser than water, and viscous; probably something
like tar or pitch in consistency.\footnote
  {The law of Dalton (Physics, p.~181) is, that any number of different gases and
vapors tend to \textit{distribute themselves throughout the space which they occupy in common,
each as if the others were absent}. The law of Boyle or Mariotte (Physics, p.~110)
is, \textit{that at any given temperature the volume of any given amount of gas varies inversely
with the pressure: i.e., $pv = p'v'$.} The law of Gay Lussac (Physics, p.~185) is,
that a gas \textit{under constant pressure expands in volume uniformly under uniform
increment of temperature}, so that $V_{t} = V_{0} (1 + at)$. This is not true of \textit{vapors} in
presence of the liquids from which they have been evaporated; for instance, of
steam in a boiler.}

While this doctrine of the gaseous constitution of the sun is generally
assented to, there are still some who are disposed to consider
the great mass of the sun as liquid.

\nbarticle{361.} (\textit{b}) The \textit{photosphere} is probably a shell of \textit{incandescent
clouds}, formed by the condensation of the vapors which are exposed
to the cold of space.

\nbarticle{362.} (\textit{c}) The photospheric clouds float in an atmosphere containing,
still uncondensed, a considerable quantity \textit{of the same vapors
out of which they themselves have been formed}, just as in our own
atmosphere the air around a cloud is still saturated with water vapor.
This vapor-laden atmosphere, probably comparatively shallow, constitutes
the \textit{reversing layer}, and by its selective absorption produces
the dark lines of the solar spectrum, while by its general absorption
it probably produces the darkening at the limb of the sun.

\begin{fineprint}
But it will be remembered that Mr.~Lockyer and others are disposed to
question the existence of any such shallow absorbing stratum, considering
that the absorption takes place in all regions of the solar atmosphere even
to a great elevation.
\end{fineprint}

\nbarticle{363.} (\textit{d}) The \textit{chromosphere and prominences} are composed of
the \textit{permanent gases}, mainly hydrogen and helium, which are mingled
with the vapors of the reversing stratum in the region near the
photosphere, but usually rise to far greater elevations than do the
vapors. The appearances are for the most part as if the chromosphere
was formed of jets of heated hydrogen ascending through the
interspaces between the photospheric clouds, like flames playing over
a coal fire.
%% -----File: 238.png---Folio 227-------

\nbarticle{364.} (\textit{e}) The \textit{corona} also rests on the photosphere, and the peculiar
green line of its spectrum (\artref{Art.}{329}) is brightest just at the surface
of the photosphere, in the reversing stratum and in the chromosphere
itself; but the corona extends to a far greater elevation than even
the prominences ever reach, and seems to be not wholly gaseous,
but to contain, besides the hydrogen and the mysterious ``coronium,''
dust and fog of some sort, perhaps meteoric. Many of its phenomena
are as yet unexplained, and since it can only be observed during
the brief moments of total solar eclipses, progress in its study is
necessarily slow.

\begin{fineprint}
\nbarticle{364*.} \nblabel{pg:227}\textit{Note to }\artref{\textit{\DPtypo{article}{Article}}}{344}. The total heat received by the earth from the
sun in any given time is that intercepted by its diametrical cross-section,
\textit{i.e.}, by the area of one of its great circles kept always perpendicular to the
sun's rays. The quantity of ice which would be melted annually on this
circular plane by the solar rays would be a sheet having a thickness of 166.5
metres or 546 feet ($19^{\text{mm}} × 24 × 365\frac{1}{4} = 166.5^{\text{met}}$).

The thickness of the ice which could be melted in a year on a narrow
equatorial belt would be $\frac{546^{\text{ft}}}{\pi}$, or $174^{\text{ft}}$, since such a belt intercepts the rays
that would otherwise fall on a diametrical strip of the same width upon the
circular plane.

If the sun's heat were \textit{uniformly} distributed over the earth's whole surface,
which equals four great circles, ($4\pi R^{2}$), it could melt a shell having a thickness
of $\frac{546^{\text{ft}}}{4}$, or $136\frac{1}{2}^{\text{ft}}$.

It is true that at the sea-level the solar-constant is much diminished by
atmospheric absorption; and probably does not exceed fifteen calories per
minute \textit{directly} received from the sun's rays. But a large part of the solar
heat absorbed by the atmosphere reaches the earth's surface \textit{indirectly}, so
that it must not be considered as lost to the earth, because not directly
measurable by the actinometer.
\end{fineprint}
\chelabel{CHAPTERIX}
%% -----File: 239.png---Folio 228-------

\Chapter{X}{Eclipses}
\nbchapterhang{\stretchyspace
ECLIPSES: FORM AND DIMENSIONS OF SHADOWS.---LUNAR
ECLIPSES.---SOLAR ECLIPSES.---TOTAL, ANNULAR, AND PARTIAL.---ECLIPTIC
LIMITS AND NUMBER OF ECLIPSES IN A
YEAR.---THE SAROS.---OCCULTATIONS.}

\nbarticle{365.} \textsc{The} word eclipse (Greek \mytextgreek{>'ekleiyic}) is strictly a medical term,
meaning a \textit{faint} or \textit{swoon}. Astronomically it is applied to the darkening
of a heavenly body, especially of the sun or moon, though
some of the satellites of other planets besides the earth are also
``eclipsed'' from time to time. An eclipse of the \textit{moon} is caused
by its passage through the shadow of the earth; an eclipse of the
\textit{sun}, by the interposition of the moon between the sun and the observer,
or, what comes to the same thing, by the passage of the
moon's shadow over the observer.

\nbarticle{366.} \nbparatext{Shadows.}---If interplanetary space were slightly dusty, we
should see, accompanying the earth and moon and each of the
planets, a long black shadow projecting behind it and travelling
with it. Geometrically speaking, this shadow of a body, the earth
for instance, is a \textit{solid---not a surface}. It is the space from which
sunlight is excluded. If we regard the sun and other heavenly
bodies as truly spherical, these shadows are \textit{cones} with their axes
in the line joining the centres of the sun and the shadow-casting
body, the point being always directed away from the sun, because
the sun is always the larger of the two.

\nbarticle{367.} \nbparatext{Dimensions of the Earth's Shadow.}---The length of the
shadow is easily found. In \figref{illo124}{Fig.~124} we have from the similar
triangles $OED$ and $ECa$, $OD:Ea::OE:EC$ or $l$. $OD$ is the difference
between the radii of the sun and the earth, $= R-r$.
$Ea=r$, and $OE$ is the distance of the earth from the sun $= \Delta$.
\begin{flalign*}
&\indent\text{Hence }&&
  l = \Delta× \left( \frac{r}{R - r}\right)
    = \frac{1}{108.5}\Delta.  &&\phantom{\indent Hence }
\end{flalign*}

(The fractional factor is constant, since the radii of the sun and
%% -----File: 240.png---Folio 229-------
earth are fixed quantities. Substituting the values of the radii, we
find it to be $\frac{1}{108.5}$.) This gives 857,200 miles for the length of the
earth's shadow when $\Delta$ has its mean value of 93,000000 miles, regarding
the earth as a perfect sphere and taking its mean radius.
This length varies about 14,000 miles on each side of the mean as
the earth changes its distance from the sun.

\includegraphicsmid{illo124}{\textsc{Fig.~124.}---Dimensions of the Earth's Shadow.}

\begin{fineprint}
The semi-angle of the cone (the angle $ECb$, or $ECB$ in the \figref{illo124}{figure}) is
found as follows. Since $OEB$ is exterior to the triangle $BEC$,
\begin{flalign*}
&&            OEB &= EBC + BCE,  &&\phantom{or }  \\
&\text{or }&  BCE &= OEB - EBC.  &&
\end{flalign*}
Now, $OEB$ is the \textit{sun's apparent semi-diameter} as seen from the earth, and
$EBC$ is the earth's semi-diameter as seen from the sun, which is the same
thing as \textit{the sun's horizontal parallax} (\artref{Art.}{83}).
\end{fineprint}

Putting $S$ for the sun's semi-diameter, and $p$ for its parallax, we
have---
\[
\text{Semi-angle at } C = S - p.\footnote
  {Also, $l = \dfrac{r}{\sin{(S - p)}}$, an expression sometimes more convenient than the one
  given above.}
\]

From the cone $aCb$ all sunlight is excluded, or would be were it
not for the fact that the atmosphere of the earth by its refraction
bends some of the rays into this shadow. The effect is to make the
shadow a little larger in diameter, but less perfectly dark.

\nbarticle{368.} \nbparatext{Penumbra.}---If we draw the lines $Ba$ and $Ab$, crossing at
$C'$ between the earth and the sun, they will bound the \textit{penumbra}.
Within this space a part, but not the whole, of the sunlight is cut off:
an observer outside of the shadow, but within this cone-frustum,
%% -----File: 241.png---Folio 230-------
which tapers \textit{towards} the sun, would see the earth as a black body
encroaching on the sun's disc. The semi-angle of the penumbra $EC'a$
is easily shown to be $S + p$.

\nbarticle{369.} Although \textit{geometrically} the boundaries of the shadow and
penumbra are perfectly definite, they are not so optically. If a screen
were placed at $M$ (\figref{illo124}{Fig.~124}) perpendicular to the axis of the shadow,
no sharply defined lines would mark the boundaries of either shadow
or penumbra; near the edge of the shadow, the penumbra would
be very nearly as dark as the shadow itself, only a mere speck of the
sun being visible there; and at the outer limit of the penumbra the
shading would be still more gradual.

\nbarticle{370.} \nbparatext{Eclipses of the Moon.}---The axis of the earth's shadow is
always directed to a point exactly opposite the sun. If, then, at the
time of full moon, the moon happens to be near the ecliptic (that is,
\textit{not far from one of the nodes of her orbit}), she will pass into the
shadow and be eclipsed. Since, however, the moon's orbit is inclined
about five and one-fourth degrees to the plane of the ecliptic, this
does not happen very often (seldom more than twice a year). Ordinarily
the moon passes north or south of the shadow without touching
it.

Lunar eclipses are of two kinds,---partial and total: total when
she passes into the shadow completely; partial when she only partly
enters it, going so far to the north or south of the centre of the
shadow that only a portion of her disc is obscured.

\begin{fineprint}
We may also have a ``penumbral eclipse'' when she passes merely through
the penumbra, without touching the shadow. In this case, however, the loss
of light is so gradual and so slight, unless she almost grazes the shadow,
that an observer would notice nothing unusual.
\end{fineprint}

\nbarticle{371.} \nbparatext{Size of the Earth's Shadow at the Point where the Moon
crosses it.}---Since $EC$ in \figref{illo125}{Fig.~125} is 857,000 miles, and the distance
of the moon from the earth is on the average about 239,000 miles,
$CM$ must be 618,000 miles, and $MN$, the semi-diameter of the shadow
at this point, will be $\frac{618}{857}$ of the earth's radius. This gives $MN = 2854$
miles, and makes the whole diameter of the shadow a little over 5700
miles, about two and two-thirds times the diameter of the moon. But
this quantity varies considerably. The shadow is sometimes more
than three times as large as the moon, sometimes hardly more than
twice its size.
%% -----File: 242.png---Folio 231-------

\begin{fineprint}
\nbarticle{372.} We may reach the same result in another way. Considering the
triangle $ECN$, \figref{illo125}{Fig.~125}, we have the angular semi-diameter of the cross-section
of the shadow where the moon passes through it, as seen from the
earth, represented by $MEN$.
\begin{flalign*}
&\indent\text{But }&&  ENa = MEN + ECN;  &&\phantom{whence }\\
&\text{whence }    &&  MEN = ENa - ECN.
\end{flalign*}

Now $ENa$ is the semi-diameter of the earth as seen from the moon; that
is; it is the moon's \textit{horizontal parallax}, for which we write $P$. Hence, substituting
for $ECN$ its value $S - p$, we get
\[
MEN = P + p - S.
\]
$MEN$ is called ``the radius of the shadow.'' The mean value of $P$ is $57'\, 2''$;
of $p$, $8''.8$; and of $S$, $16'\, 2''$, which makes the mean value of $MEN = 41'\, 9''$.
The mean value of the moon's apparent semi-diameter is $15'\, 40''$, the ratio
between the semi-diameter of the moon and the radius of the shadow being
about $2\frac{2}{3}$, as before.

\includegraphicsmid{illo125}{\textsc{Fig.~125.}---Diameter of Earth's Shadow where the Moon crosses it.}

In computing a lunar eclipse, this angular value for the ``radius of the
shadow,'' as it is called, is more convenient than its value in miles. It is
customary to increase it by about $\frac{1}{60}$ part in order to allow for the effect of
the earth's atmosphere, the value ordinarily used being $\frac{61}{60}(P + p - S)$.
Some computers, however, use $\frac{51}{50}$, and others $\frac{76}{75}$. On account of the indistinctness
of the edge of the shadow it is not easy to determine what precise
value ought to be employed.
\end{fineprint}

\nbarticle{373.} \nbparatext{Duration of a Lunar Eclipse.}---When central, a total eclipse
of the moon may, all things favoring, continue total for about two
hours, the interval from the first contact to the last being about two
hours more. This depends upon the fact that the moon's hourly
motion is nearly equal to its own diameter. The whole interval from
first contact to last is the time occupied by the moon in moving from
%% -----File: 243.png---Folio 232-------
$a$ to $d$ (\figref{illo126}{Fig.~126}). The totality lasts while it moves from $b$ to $c$.
The duration of a non-central eclipse varies, of course, according to
the part of the shadow through which the moon passes.

\includegraphicsmid{illo126}{\textsc{Fig.~126.}---Duration of a Lunar Eclipse.}

\nbarticle{374.} \nbparatext{Lunar Ecliptic Limit.}---The lunar \textit{ecliptic limit} is the greatest
distance from the node of the moon's orbit at which the sun
can be \DPtypo{consistently}{consistent} with having an eclipse. This limit depends upon
the inclination of the moon's orbit, which varies a little, and also upon
the radius of the shadow at the time of the eclipse and the moon's
apparent semi-diameter, which quantities are still more variable.
Hence we recognize two limits, the major and minor. If the distance
of the sun from the node at the time of full moon exceeds the
major limit, an eclipse is impossible; if it is less than the minor, an
eclipse is inevitable. The major limit is found to be $12°\, 5'$; the minor,
$9°\, 30'$.  Since the sun passes over an arc of $12°\, 5'$ in less than
thirteen days, it follows that an eclipse of the moon cannot possibly
take place more than thirteen days before or after the time when the
sun crosses the node.

\includegraphicsmid{illo127}{\textsc{Fig.~127.}---Lunar Ecliptic Limit.}

\begin{fineprint}
\nbarticle{375.} In \figref{illo127}{Fig.~127} let $NE$ be the ecliptic, and $NM$ the orbit of the moon,
the point $N$ being the node, and the angle at $N$ the inclination of the moon's
orbit. $E$ is the centre of the earth's shadow. The sun, of course, is directly
opposite, and its distance from the opposite node is equal to $EN$. $M$ is the
%% -----File: 244.png---Folio 233-------
centre of the moon. Call the semi-diameter of the moon $S'$; then $EM$ (the
greatest possible distance between $E$ and $M$ which permits an eclipse) equals
the sum of the semi-diameters of the moon and shadow, or $S' + (P + p - S)$,
and the corresponding ecliptic limit $EN$ is found by solving the spherical
triangle $MNE$, having given $ME$ and the angle at $N$, which is about $5\frac{1}{4}°$.
We must also know one other angle, and with sufficient approximation for
such purposes we may regard the angle at $M$ as a right angle. The solution
will give the value of the ecliptic limit by assigning the proper values to the
quantities involved. The limit is always very nearly \textit{eleven} times $EM$, because
the inclination of the moon's orbit is nearly $\frac{1}{11}$ of a ``radian.''
\end{fineprint}

\nbarticle{376.} \nbparatext{Phenomena of a Total Lunar Eclipse.}---Half an hour or so
before the moon reaches the shadow its eastern limb begins to be
sensibly darkened, and the edge of the shadow itself, when it is first
reached, looks nearly black by contrast with the bright parts of the
moon's surface. To the naked eye the outline of the shadow appears
reasonably sharp; but with even a small telescope it is found to be
indefinite and hazy, and with a large instrument and high magnifying
power it becomes entirely indistinguishable. It is impossible to determine
the exact moment when the edge of the shadow reaches any
particular point on the moon within half a minute or so.

\includegraphicsmid{illo128}{\textsc{Fig.~128.}---Light bent into Earth's Shadow by Refraction.}

After the moon has wholly entered the shadow her disc is usually
still distinctly visible, illuminated with a dull, copper-colored light,
which is sunlight, deflected around the earth into the shadow by the
refraction of our own atmosphere, or rather by that portion of our
atmosphere which lies within ten or fifteen miles of the earth's surface.
Since the ordinary horizontal refraction is $34'$,
it follows that
light which just grazes the earth's surface will be bent inwards by
twice that amount, or $1°\, 8'$. Now, the maximum ``radius of the
shadow'' is only $1°\, 2'$. In an extreme case, therefore, even when
the moon is exactly central
in the largest possible shadow, it receives
some sunlight coming around the edge of the earth, as shown by
Fig.~128. To an observer stationed on the moon, the disc of the
earth would appear to be surrounded by a narrow ring of brilliant
sunshine, colored with sunset hues by the same vapors which tinge
%% -----File: 245.png---Folio 234-------
terrestrial sunsets, but acting with double power because the light
has traversed a double thickness of our air. If the weather happens
to be clear at this portion of the earth (upon its \textit{rim} as seen
from the moon), the quantity of light transmitted through the atmosphere
is very considerable, and the moon is strongly illuminated.
If, on the other hand, the weather happens to be stormy
in this region, the clouds cut off nearly all the light. In the lunar
eclipse of 1886 the moon was absolutely invisible to the naked eye,
a very unusual circumstance on such an occasion. At the eclipse
of Jan.~28, 1888, Pickering found that the \textit{photographic power} of
the centrally eclipsed moon was about $\frac{1}{400\,000}$ that of the moon
when uneclipsed.

\begin{fineprint}
\nbarticle{377.} \nbparatext{Uses made of Lunar Eclipses.}---In astronomical importance a
lunar eclipse cannot be at all compared with a solar eclipse. It has its uses,
however. \textit{a}.~Many dates in chronology are fixed by reference to certain
lunar eclipses. For instance, the date of the Christian era is determined by
a lunar eclipse which happened upon the night that Herod died. \textit{b}.~Before
better methods were devised, lunar eclipses were made use of to some extent
in determining the longitude. Unfortunately, as has been said (\artref{Art.}{119}),
it is impossible to note the critical instants with any degree of accuracy, on
account of the indefiniteness of the moon's shadow. \textit{c}.~The study of the
spectrum of the eclipsed moon gives us some data as to the constitution of
our own atmosphere. We are thus enabled to examine light which has
passed through a greater thickness of air than is obtainable in any other
way. \textit{d}.~The study of the heat radiated by the moon during the different
phases of an eclipse gives us some important information as to the absorbing
power and temperature of its surface. Observations have been made
at Lord Rosse's observatory of all the recent lunar eclipses, with this end in
view. \textit{e}.~Finally, at the time when the moon is eclipsed, it is possible to observe
its passage over small stars which cannot be seen at all when near the
moon except at such a time. Observations of these star occultations made
at different parts of the earth, furnish the best possible data for computing
the dimensions of the moon, its parallax, and for determining its precise
position in its orbit at the time of observation. The eclipses of the last
few years have been very carefully observed in this way by concert between
the different leading observatories.
\end{fineprint}

\sloppy
\nbarticle{378.} \nbparatext{Computation of a Lunar Eclipse.}---Since all the phases of
a lunar eclipse are seen everywhere at the same absolute instant
wherever the moon is above the horizon, it follows that a single computation
giving the Greenwich times of the different phenomena is all
that is needed, and can be made and published once for all.  Each
observer has merely to correct the predicted time by simply adding
%% -----File: 246.png---Folio 235-------
or subtracting his longitude from Greenwich in order to get the
true local time. The computation is very simple, but lies rather
beyond the scope of this work.

\fussy
\section*{ECLIPSES OF THE SUN.}

\nbarticle{379.} \nbparatext{Dimensions of the Moon's Shadow.}---By the same method
as used for the shadow of the earth (merely substituting in the
formulę the radius of the \textit{moon} for that of the earth), we find that
the length of the moon's shadow at any time is $\frac{1}{399.55}$ of its distance
from the sun, and at new moon averages 232,150 miles. It varies
not quite 4000 miles each way, and so ranges from 236,050 to
228,300. The semi-angle of the moon's shadow is practically equal
to the semi-diameter of the sun at the earth, or very nearly $16'$.

\includegraphicsmid{illo129}{\textsc{Fig.~129.}---The Moon's Shadow on the Earth.}

\nbarticle{380.} \nbparatext{The Moon's Shadow on the Earth's Surface.}---Since the
mean length of the shadow is less than the mean distance of the
moon from the earth (which is 238,800 miles), it is obvious that
\textit{on the average} it will not reach the earth. On account of the
eccentricity of the moon's orbit however, our satellite is much of
the time considerably nearer than this mean distance, and may come
within 221,600 miles from the earth's centre, or about 217,650
miles from its surface. The shadow, also, under favorable circumstances,
may have a length of 236,050 miles. Its point may therefore
at times extend nearly 18,400 miles beyond the earth's surface.
The cross-section of the shadow where the earth's surface cuts it
(at $o$ in \figref{illo129}{Fig.~129}) will then be 167~miles. \textit{This is the largest value
possible}.

\begin{fineprint}
Of course, if the shadow strikes obliquely on the surface of the earth, as
it must except when the moon is in the zenith, the shadow spot will be \textit{oval}
instead of circular, and the length of the oval along the earth's surface may
much exceed the true cross-section of the shadow.
\end{fineprint}

\nbarticle{381.} \nbparatext{The ``Negative'' Shadow.}---Since the distance of the moon
may be as great as 252,970 miles from the earth's centre, or nearly
%% -----File: 247.png---Folio 236-------
249,000 miles from its surface, while the shadow may be as short as
228,300 miles, we may have the state of things indicated by placing
the earth at $B$ in the \figref{illo129}{figure}. The vertex of the shadow, $V$, will then
fall 21,700 miles short of the surface, and the cross-section of the
``\textit{shadow produced}'' will have a diameter of 206 miles where the
earth's surface cuts it. When the shadow falls near the edge of the
earth, this cross-section may be as great as 230 miles. The shadow-spot
which is formed by the intersection of the produced shadow-cone
with the earth's surface is sometimes called the \textit{negative shadow},
because in calculating an eclipse its radius comes out from the formulę
as a \textit{minus} quantity in case the shadow does not reach the earth.

\nbarticle{382.} \nbparatext{Total and Annular Eclipses.}---To an observer within the
true shadow cone, that is, between $V$ and the moon in \figref{illo129}{Fig.~129}, the
sun will be \textit{totally} eclipsed; but an observer in the produced cone
beyond $V$ will see the moon projected on the sun, leaving an uneclipsed
ring around it. He will have what is called an \textit{annular}
eclipse. These annular eclipses are considerably more frequent than
total eclipses---nearly in the ratio of three to two.

\includegraphicsmid{illo130}{\textsc{Fig.~130.}---Width of the Penumbra of the Moon's Shadow.}

\nbarticle{383.} \nbparatext{The Penumbra and Partial Eclipses.}---The penumbra can
easily be shown to have a diameter on the line $CD$ (\figref{illo129}{Fig.~129}) of very
nearly twice the moon's diameter. We may take it as having an average
diameter at this point of 4400 miles; but as the earth is often beyond
$V$, its cross-section \textit{at the earth} is sometimes as much as 4800
miles. An observer situated within the penumbra observes a partial
eclipse: if he is near the shadow cone, the sun will be mostly covered
by the moon; but if near the outer limit of the penumbra, the moon
will only slightly encroach on the sun's disc. While, therefore, total
and annular eclipses are visible as such only by an observer within
the narrow path traversed by the shadow-spot, the same eclipse
will be visible as a partial one everywhere within 2000 miles on
%% -----File: 248.png---Folio 237-------
either side of the shadow path; and the 2000 miles is to be reckoned
\textit{perpendicularly} to the axis of the shadow. When, for instance,
the penumbra falls, as shown in \figref{illo130}{Fig.~130}, the distance $BC$ \textit{measured
along the earth's surface} will be over 3000 miles, although $BF$ is only
2000.

\nbarticle{384.} \nbparatext{Velocity of the Shadow and Duration of Eclipses.}---The
moon advances along its orbit very nearly 2100 miles an hour, and
were it not for the earth's rotation this is the rate at which the
shadow would pass the observer. The earth, however, is rotating
towards the east in the same general direction as that in which the
shadow moves, and its surface moves at the rate of about 1040
miles an hour at the equator. An observer, therefore, on the earth's
equator, with the moon near the zenith, would be passed by the
shadow with a speed of about 1060 miles per hour ($2100 - 1040$);
and this is its slowest velocity, which is about equal to that of a
cannon-ball.

\includegraphicsmid{illo131}{\textsc{Fig.~131.}---Track of the Moon's Shadow, Eclipse of July, 1878.}

In higher latitudes, where the velocity of the earth's rotation is less,
the relative speed of the shadow is higher; and where the shadow falls
very obliquely, as it does when an eclipse occurs near sunrise or sunset,
%% -----File: 249.png---Folio 238-------
the advance of the shadow along the earth's surface may become
exceedingly swift,---as great as 4000 or 5000 miles per hour. \figref{illo131}{Fig.~131},
which we owe to the courtesy of the publishers of Langley's
``New Astronomy,'' shows the track of the moon's shadow during the
eclipse of July 29, 1878.

\nbarticle{385.} \nbparatext{Duration of an Eclipse.}---A \textit{total} eclipse of the sun observed
at a station near the equator under the most favorable conditions possible
(the shadow-spot having its maximum diameter of 167 miles),
may continue total for \textit{seven minutes and fifty-eight seconds}. In latitude
$40°$ the duration of totality can barely equal six and one-quarter
minutes. The greatest possible excess of the radius of the moon
over that of the sun is only $1'\, 19''$.

An \textit{annular} eclipse may last for $12^{\text{m}}\, 24^{\text{s}}$ at the equator. The
maximum width of the ring of the sun visible around the moon
is $1'\, 37''$.

In the observation of an eclipse four ``contacts'' are noted: the
first when the edge of the moon first touches the edge of the sun; the
second, when the eclipse becomes total or annular; the third, at
the cessation of the total or annular phase; and the fourth, when
the moon finally leaves the disc of the sun. From the first contact
to the fourth the duration may be a little over two hours.

\includegraphicsmid{illo132}{\textsc{Fig.~132.}---Solar Ecliptic Limits.}

\nbarticle{386.} \nbparatext{The Solar Ecliptic Limits.}---It is necessary, in order to have
an eclipse of the sun, that the moon should encroach on the cone
$ACBD$ (\figref{illo132}{Fig.~132}), which envelops earth and sun. In this case the
``true'' angular distance between the centres of the sun and moon---that
is, their distance as seen from the centre of the earth---would
be the angle $MES$ in the \figref{illo132}{figure}. This is made up of three angles:
$MEF$, which equals the moon's semi-diameter, $S'$; $AES$, the sun's
semi-diameter; $S$; and $FEA$. This latter angle is equal to the difference
%% -----File: 250.png---Folio 239-------
between $CFE$ and $FAE$. $CFE$ is the moon's horizontal parallax
(the semi-diameter of the earth seen from the moon), and $FAE$
or $CAE$ is the sun's parallax. $FEA$, therefore, equals $P - p$; and
the whole angle $MES$ equals $S + S' + P - p$. This angle may range
from $1°\, 34'\, 13''$ to $1°\, 24'\, 19''$, according to the changing distances\footnote
  {We give herewith in a table the different quantities which determine the
dimensions of the shadows of the earth and moon, as well as the ecliptic limits
and the duration of eclipses.\\[1ex]
\begin{tabular*}{\textwidth} {l@{\extracolsep{\fill}}c@{\extracolsep{1pt}} |c|c|c}
\hline\hline
&& Greatest. & \PadTo{Greatest.}{Least.} & \PadTo{Greatest.}{Mean.}\rule[-1.5ex]{0pt}{4.5ex}\\
\cline{2-5}
Apparent semi-diameter of sun
&& $16'\, 18''$ & $15'\, 45''$ & $16'\, 02''$\rule{0pt}{3ex}
\\
Apparent semi-diameter of moon
&& $16'\, 46''$ & $14'\, 44''$ & $15'\, 45''$
\\
Horizontal parallax of the sun
&& $9''.05$ & $8''.55$ & $8''.80$
\\
Horizontal parallax of the moon
&& $61'\, 18''$ & $53'\, 58''$ & $57'\, 38''$
\\
Inclination of moon's orbit
&& $\phantom{0}5°\, 19'\,$ & $\phantom{0}4°\, 57'\,$
& $5°\, 8'\, 40''$
\\
\multicolumn{5}{c}{Sun's radius, 433,200 miles; earth's (mean), 3956; moon's, 1081.5.\rule[-1.5ex]{0pt}{4.5ex}}\\
\hline\hline
\end{tabular*}}
of
the sun and moon from the earth.

The corresponding distances of the sun from the node, calculated in
the same way as the lunar ecliptic limits (taking the maximum inclination
of the moon's orbit as $5°\, 19'$ and the minimum as $4°\, 57'$,
according to Neison), give $18°\, 31'$ and $15°\, 21'$ for the major and
minor ecliptic limits.

In order that an eclipse may be \textit{central} (total or annular) at any
part of the earth, it is necessary that the moon should lie wholly
inside the cone $ACBD$, as at $M'$. In this case the angle $M'ES$ will
be $S - S' + P - p$, and the corresponding major and minor \textit{central}
ecliptic limits come out $11°\, 50'$ and $9°\, 55'$.

\nbarticle{387.} \nbparatext{Phenomena of a Solar Eclipse.}---There is nothing of special
interest until the sun is mostly covered, though before that time the
shadows cast by foliage begin to look peculiar. The light shining
through every small interstice among the leaves, instead of forming a
little \textit{circle} on the earth, makes a little \textit{crescent}---an image of the
partly covered sun.

Some ten minutes before totality the darkness begins to be felt, and
the remaining light, coming as it does from the edge of the sun only,
is much altered in quality, producing an effect very like that of
a calcium light rather than sunshine. Animals are perplexed, and
birds go to roost. The temperature falls a few degrees, and sometimes
dew appears.
%% -----File: 251.png---Folio 240-------

In a few moments, if the observer is so situated that his view commands
a distant western horizon, the moon's shadow is seen coming
much like a heavy thunder-storm. It advances with almost terrifying
swiftness until it envelops him.

For a moment the air appears to quiver, and on every white surface
bands or ``\textit{fringes},'' alternately light and dark, appear. They
are a few inches wide and from a foot to three feet apart, and on the
whole seem to be parallel to the edge of the shadow. Probably they
travel with the \textit{wind}; but observations on this point are as yet hardly
decisive. The phenomenon is not fully explained, but is probably
due to irregular atmospheric refraction of the light coming from the
indefinitely narrow strip of the sun's limb on the point of disappearing.

\nbarticle{388.} \nbparatext{Appearance of the Corona and Prominences.}---As soon as the
shadow arrives, and sometimes a little before it, the corona and prominences
become visible. The stars of the first three magnitudes make
their appearance at the same time.

The suddenness with which the darkness descends upon the observer
is exceedingly striking; the sun is so brilliant that even the
small portion which remains visible up to within a very few seconds
of the time of totality so dazzles the eye that it is not prepared for
the sudden transition. In a few moments, however, the vision becomes
accustomed to the changed circumstances, and it is then found
that the darkness is not really very intense. If the totality is of
short duration,---that is, if the diameter of the moon exceeds that
of the sun by less than a minute of arc,---the lower parts of the corona
and chromosphere, which are very brilliant, give a light at least three
or four times as great as that of the full moon. Since the shadow also
in such a case is of small diameter, a large quantity of light is sent in
from the surrounding air, where thirty or forty miles away the sun
is still shining; and what may seem remarkable, this intrusion of
outside light is greatest when the sky is clouded. In such an eclipse
there is not much difficulty in reading an ordinary watch-face. In
an eclipse of long duration (say five or six minutes) it is much
darker, and lanterns are necessary.

\begin{fineprint}
\nbarticle{389.} \nbparatext{Observations of an Eclipse.}---A total solar eclipse offers an
opportunity of making an immense number of observations of great importance
which are possible at no other time, besides certain others which can
also be made during a partial eclipse. We mention (\textit{a})~\textit{Times of the four
contacts, and direction of the line joining the cusps during the partial phases}.
%% -----File: 252.png---Folio 241-------
These observations determine accurately the relative position of the sun and
moon at the time, and so furnish the means for correcting the tables of their
motion. (\textit{b})~\textit{The search for intra-mercurial planets}. It has been thought
likely that there may be one or more planets between the orbit of Mercury
and the sun, and during a total eclipse they would become visible, if ever.
On the whole, however, the observations, so far made, negative the existence
of any body of considerable size in this region, though in 1878, Professor
Watson and Mr.~Swift, it was thought, had discovered one, if not two, such
planets. (\textit{c})~\textit{Observations on the fringes}, which have been described as showing
themselves at the commencement of totality. Probably the phenomenon
is merely atmospheric and of little importance, but it is not yet sufficiently
understood. (\textit{d})~\textit{Photometric measurements} of the intensity of the light at
different stages of the eclipse and during totality. (\textit{e})~\textit{Telescopic observations
of the details of the prominences and of the corona}. (\textit{f})~\textit{Spectroscopic observations},
both visual and photographic, upon the spectra of the lower atmosphere
of the sun, the prominences, and the corona. (\textit{g})~\textit{Observations with the
polariscope} upon the polarization of the light of the corona, for the purpose
of determining the relation between the reflected and intrinsic light, and
perhaps the size of the reflecting particles which are distributed through the
corona. (\textit{h})~\textit{Photography}, both of the partial phases and of the corona.

\nbarticle{390.} \nbparatext{Calculation of a Solar Eclipse.}---The calculation of a solar
eclipse cannot be dealt with in any such summary way as that of a lunar
eclipse, owing to the moon's parallax, which makes the times of contact and
other phenomena different at every different station. The moon's apparent
path in the sky, \textit{relative to the centre of the sun}, is not even a portion of a
great circle, nor is it described with a uniform velocity. Moreover, since
the phenomena of a solar eclipse admit of very accurate observations, it is
necessary to take account of numerous little details which are of no importance
in a lunar eclipse.

Certain data for each solar eclipse hold good wherever the observer may
be. These are calculated beforehand and published in the nautical almanacs;
and from them, with the knowledge of his geographical position, the
observer can work out the results for his own station. But the calculations
are somewhat complicated and lie beyond our scope. The reader is referred
to any work on practical astronomy; Chauvenet and Loomis treat the matter
very fully.
\end{fineprint}

\nbarticle{391.} \nbparatext{Number of Eclipses in a Year.}---The least possible number is
\textit{two}, both central eclipses of the sun. The largest possible number is
\textit{seven}, \textit{five} of the sun and \textit{two} of the moon. The eclipses each year
happen at two seasons (which may be called the ``eclipse months''),
half a year apart---about the times, of course, at which the sun in its
annual path crosses the two nodes of the moon's orbit. If these nodes
were stationary, the eclipse months would be always the same; but
%% -----File: 253.png---Folio 242-------
because the nodes retrograde around the ecliptic once in about nineteen
years, the eclipse months are continually changing. The time
required by the sun in passing around from a node to the same
node again is 346.62 days, which is sometimes called the ``eclipse
year.''

\includegraphicsouter{illo133}{\textsc{Fig.~133.}---Number of Eclipses Annually.}

\sloppy
\nbarticle{392.} \nbparatext{Number of Lunar Eclipses.}---Representing the ecliptic by a
circle (\figref{illo133}{Fig.~133}) with the two opposite nodes $A$ and $a$, it is easy to see
\textit{first}, that there can he but two \textit{lunar} eclipses in a year (omitting for
a moment one exceptional case). The major lunar ecliptic limit is
$12°\, 15'$; hence there is only a space of twice that amount, or $24°\, 30'$,
between $L$ and $L'$, at each ``node
month,'' within which the occurrence
of a full moon might give
a lunar eclipse. Now, in a synodic
month the sun moves along
the ecliptic $29°\, 6'$, while the node
moves in the opposite direction
$1°\, 31'$, giving the \textit{relative} motion
of the sun referred to the node
equal to $30°\, 37'$; \textit{i.e., the full-moon
points on the circle would fall
at a distance of $30°\, 37'$ from each
other}. Only \textit{one} full moon, therefore,
can possibly occur within
the lunar ecliptic limits each time that the sun passes the node.

\fussy
Since the \textit{minor} ecliptic limit for the moon is only $9°\, 30'$, it may
easily happen that \textit{neither} of the full moons which occur nearest to
the time when the sun is at the node will fall within the limit. There
are accordingly many years which have no lunar eclipses.

\textit{Three} lunar eclipses, however, may possibly happen in one calendar
year in the following way. Suppose the first eclipse occurs about
Jan.~1, the sun passing the node about that time; the second may then
happen about June 25 at the other node, $a$. The first node, $A$, will
run back during the year, so that the sun will encounter it again about
Dec.~13 at $A'$, and thus a third eclipse may occur in December of
the same year. This actually occurred in 1787, the dates of the three
lunar eclipses being Jan.~3, June 30, and Dec.~24.

\sloppy
\nbarticle{393.} \nbparatext{Number of Solar Eclipses.}---Considering now \textit{solar} eclipses,
we find that there must inevitably be \textit{two}. Twice the minor limit
%% -----File: 254.png---Folio 243-------
(\artref{Art.}{386}) of a solar eclipse ($15°\, 21'$) is $30°\, 42'$, which, is more than
the sun's whole motion in a month. One new moon, at least, therefore,
\textit{must} fall within the limiting distance of the node, and two
\textit{may} do so, since in the \figref{illo133}{figure}, $SS'$ is always greater than the distance
between the points occupied by two successive new moons.

\fussy
If the two \textit{new} moons in the two eclipse months happen to fall
very near a node, the two \textit{full} moons, a fortnight earlier and later,
will both be very likely to fall outside the lunar limit. In that case
the year will have only two eclipses, both solar and both central; \textit{i.e.},
either total or annular. This was the case in 1886.

Again, if in any year two \textit{full moons} occur when the sun is very near
the node, then since the \textit{major} solar limit is $18°\, 31'$, it may happen,
and often does, that there will be two partial solar eclipses, one a
fortnight before, the other a fortnight after, each of the lunar eclipses,
and so the year will have three eclipses in each eclipse month---six
eclipses in all, two lunar and four solar. A \textit{fifth} solar eclipse may
also come in near the end of the year, if the node was passed about
Jan.~15, in the same way that sometimes happens with a lunar eclipse:
the year will then have \textit{seven} eclipses. This was the case in 1823.
The most usual number of eclipses is four or five.

\nbarticle{394.} \nbparatext{Relative Frequency of  Solar and Lunar Eclipses.}---Although,
\textit{taking the whole  earth into account}, the  solar  eclipses are
the most numerous, about in the proportion of four to three, it is
not so with the eclipses \textit{visible at any given place}. A solar eclipse
can be seen only from a limited portion of the globe, while a lunar
eclipse is visible over considerably more than half the earth, either at
the beginning or end, if not throughout its whole duration; and this
more than reverses the proportion between lunar and solar eclipses
for any given station.

\nbarticle{395.} \nbparatext{Recurrence of Eclipses, and the Saros.}---It is not known how
early it was discovered that eclipses recur at a regular interval of
eighteen years and eleven and one-third days (\textit{ten} and one-third days,
if there happen to be five leap years in the interval); but the Chaldeans
knew the period very well, and called it the \textit{Saros} (which means
``restitution'' or ``repetition''), and used it in predicting the recurrence
of these phenomena. It is a period of 223 \textit{synodic months},
which is almost exactly equal to nineteen eclipse years. The eclipse
year is $346^{\text{d}}.6201$, and  nineteen of them equal $6585^{\text{d}}.78$, while 223
months equal $6585^{\text{d}}.32$.
%% -----File: 255.png---Folio 244-------

The difference is only $\frac{46}{100}$ of a day (about 11~hours) in which time
the sun moves $28'$. If, therefore, an eclipse should occur to-day at
new moon, with the sun exactly at the node, then after 223 months
(18~years 11~days) a new moon will occur again with the sun only $28'$
west of the node; so that the circumstances of the first eclipse will
be pretty nearly repeated. It would however occur about eight hours
of longitude further west on the earth's surface, since the 223 months
exceed the even 6585 days by $\frac{32}{100}$ of a day, or $7^{\text{h}}\ 42^{\text{m}}$.

\begin{fineprint}
\nbarticle{396.} \nbparatext{Number of Recurrences of a Given Eclipse.}---It is usual to
speak of eclipses recurring at this regular interval as ``repetitions'' of one
and the same eclipse. A lunar eclipse is usually thus ``repeated'' 48 or 49
times. Beginning as a very small partial eclipse, with the sun about $12°$
\textit{east} of the node, it will be a little larger at its next occurrence eighteen
years later; and after 13 or 14 repetitions the sun will have come so near
the node that the eclipse will have become total. It will then be repeated
as a total eclipse 22 or 23 times, after which it will become partial again
with the sun \textit{west} of the node, and after 13 more returns as a partial eclipse
will finally dwindle away and disappear, having thus recurred regularly once
in every 223 months during an interval of $865\frac{1}{2}$ years.

The same thing happens with the solar eclipses, only since the solar
ecliptic limit is larger than the lunar, a solar eclipse has from 65 to 70 returns,
occupying some 1200 years. Of these about 25 are only partial
eclipses, the sun being so near the ecliptic limit that the \textit{axis} of the shadow
does not reach the earth at all. The 45 eclipses in the middle of the period
are central somewhere or other on the earth, about 18 of them being total,
and about 27 annular. These numbers vary somewhat, however, in different
cases.
\end{fineprint}

\nbarticle{397.} It is to be noticed that the Saros exhibits not only a close
commensurability of the \textit{synodic} months with the \textit{eclipse} years, but
also with the \textit{nodical}\footnote
  {The \textit{nodical} month is the time of the moon's revolution from one of its \textit{nodes}
to the same node again, and is equal to $27^{\text{d}}.21222$; the \textit{anomalistic} month is the
time of revolution from \textit{perigee} to perigee again, and equals $27^{\text{d}}.55460$. See
  Arts.~\arnref{454}, \arnref{455}.}
and \textit{anomalistic} months: 242 nodical months
equal 6585.357 days; 239 anomalistic months equal 6585.549 days.
This last coincidence is important. The moon at the end of the
Saros of  223 months not only returns very closely to its original
position \textit{with respect to the sun and the node}, but also with respect to
\textit{the line of apsides} of its orbit. If it was at perigee originally, it will
again be within five hours of perigee at the end of the Saros. If this
were not so, the time of the eclipse might be displaced several hours
%% -----File: 256.png---Folio 245-------
by the perturbations of the  moon's motion, to be considered later,
in \chapref{CHAPTERXII}{Chap.~XII.}

\nbarticle{398.} \nbparatext{Number of Eclipses in a Single Saros.}---The total number is
usually about seventy, varying two or three one way or the other, as
new eclipses come in at the eastern limit and go out at the western.
Of the 70, 29 are usually lunar and 41 solar; and of the solar, 27 are
\textit{central}, 17 being annular and 10 total. (These numbers are necessarily
only approximate.) It appears, therefore, that total solar eclipses,
\textit{somewhere or other on the earth}, are not very rare, there being about
ten in eighteen years. Since, however, the shadow track averages
less than 100 miles in width, each total eclipse is visible, \textit{as total},
over only a very small fraction of the earth's whole surface---about
$\frac{1}{200}$ in the mean. This gives about one total eclipse in 360 years, in
the long run, at any given station.

\begin{fineprint}
The total solar eclipses visible in the United States during the present
century are the following:---

June 10, 1806, in New York and New England, duration $4\frac{1}{2}$ minutes;
Nov.~30, 1834, in Arkansas, Missouri, Alabama, and Georgia, duration 2
minutes; July 18, 1860, in Washington Territory and Labrador, 3 minutes;
Aug.~7, 1869, in Iowa, Illinois, Kentucky, North Carolina, $2\frac{3}{4}$ minutes; July 29,
1878, in Wyoming, Colorado, Texas, $2\frac{1}{2}$ minutes; Jan.~11, 1880, in California,
duration 32 seconds; Jan.~1, 1889, in California and Montana, $2\frac{1}{4}$ minutes;
May 27, 1900, from Louisiana to Virginia, 2 minutes.
\end{fineprint}

\nbarticle{399.} \nbparatext{Occultations of Stars.}---In theory, and in the method of computation,
the occultation of a star is precisely like a solar eclipse, except
that the shadow of the moon projected by a star is a \textit{cylinder} instead
of a \textit{cone}, since, compared with the distance of the sun, that of a
star is infinite: moreover, the star is a mere point, so that there is
no sensible penumbra. In other words, a star has neither parallax
nor semi-diameter, and  these  circumstances somewhat simplify the
formulę.

As the moon moves always towards the east, the disappearance of
the star always takes place at the eastern limb, and the reappearance
at the western. In the first half of the lunation the eastern limb
is dark and invisible, and the star vanishes without warning. The
suddenness with which it vanishes and reappears has already been
referred to (\artref{Art.}{255}) as proof of the non-existence of a lunar atmosphere.
Observations of this sort determine the moon's plane with
great accuracy, and when corresponding observations are made at
%% -----File: 257.png---Folio 246-------
different places, they supply one of the best possible means of determining
their difference of longitude.

\begin{fineprint}
In some cases observers have reported that a star, instead of disappearing
instantaneously when struck by the moon's limb (faintly visible by earth-shine),
has appeared to cling to the limb for a second or two before vanishing,
and in a few instances they have even reported it as having reappeared
and disappeared a second time, as if it had been for a moment visible through
a rift in the moon's crust. Some of these anomalous phenomena have been
explained by the subsequent discovery that the star was double, or had a
faint companion.
\end{fineprint}
\chelabel{CHAPTERX}

%% -----File: 258.png---Folio 247-------

\Chapter{XI}{Central Forces}
\nbchapterhang{\stretchyspace
CENTRAL FORCES: EQUABLE DESCRIPTION OF AREAS.---AREAL,
LINEAR, AND ANGULAR VELOCITIES.---KEPLER'S LAWS AND
INFERENCES FROM THEM.---GRAVITATION DEMONSTRATED
BY THE MOON'S MOTION.---CONIC SECTIONS AS ORBITS.---THE
PROBLEM OF TWO BODIES.---THE ``VELOCITY FROM
INFINITY,'' AND ITS RELATION TO THE SPECIES OF ORBIT
DESCRIBED BY A BODY MOVING UNDER GRAVITATION.---THE
INTENSITY OF GRAVITATION.}

\nbarticle{400.} \textsc{A moving} body left to itself, according to Newton's first law
of motion (Physics, p.~27), moves on forever in a straight line with a
uniform velocity. If we find a body so moving, we may, therefore,
infer that it is either acted on by \textit{no} force whatever, or, if forces are
acting upon it, that they exactly balance each other.

\begin{fineprint}
It has been customary with some writers to speak of a body thus moving
``uniformly in a straight line'' as actuated by a ``projectile \textit{force},'' a very
unfortunate expression, which is a survival of the Aristotelian idea that rest
is more ``natural'' to matter than motion, and that when a body moves,
some force must operate to keep it moving. The mere uniform rectilinear
motion of a material mass in empty space implies no physical cause, and
demands no explanation. \textit{Change} of motion, either in speed or in direction---this
alone implies \textit{force in operation}.
\end{fineprint}

\includegraphicsouter{illo134}{\textsc{Fig.~134.}---Curvature of an Orbit.}

\nbarticle{401.} If a body moves in a straight line, with swiftness either
increasing or decreasing, we infer a
force acting exactly in the line of motion,
and accelerating or retarding it.
If it moves in a \textit{curve}, we know that
some force is acting \textit{athwart} the line
of motion. If the velocity in the curve
increases, we know that the direction
of the force that acts is \textit{forward}, like
$ab$ (\figref{illo134}{Fig.~134}), making an angle of less
than $90°$ with the ``line of motion'' ${a\,t}$
(the tangent to the path of
the body); and \textit{vice versa}, if the motion of the body is retarded.
%% -----File: 259.png---Folio 248-------
If the speed does not alter at all, we know that the force must act
along the line $ac$, \textit{exactly perpendicular} to the line of motion.

\begin{fineprint}
Here, also, we find many writers, the older ones especially, bringing in
the ``projectile force,'' and saying that when a body moves in a curve it does
so under the action of \textit{two} forces, one a force that draws it sideways, the
other the ``projectile force'' directed along its path. We repeat; this ``projectile
force'' has no present existence or meaning in the problem. Such
a force may have put the body in motion long ago, but its function has
ceased, and \textit{now} we have only to do with the action of one single force,---\textit{the
deflecting force}, which alters the direction of the body's motion. Of
course it is not intended to deny that the deflecting force may itself be the
resultant of any number of forces all acting together; but a single force acting
athwart a body's line of motion is sufficient to cause it to describe a
curvilinear orbit, and from such an orbit we can only infer the \textit{necessary}
existence of \textit{one} such force.
\end{fineprint}

\includegraphicsouter{illo135}{\textsc{Fig.~135.}\\Description of Areas in Uniform Motion.}

\nbarticle{402.} \nbparatext{Description of Areas.}---(\textit{a}) If a body is moving uniformly on
a straight line, and if we connect the points $A$, $B$, $C$, etc., \figref{illo135}{Fig.~135}, which
it occupies at the end of successive units of time with any point whatever,
as $O$, we shall have a series
of triangles, $AOB$, etc., \textit{which
will all be equal}; since their bases
$AB$, $BC$, etc., are equal and on
the same straight line, and they
have a common vertex at $O$.
Calling the line from $A$ to $O$ its
radius vector, and $O$ the ``centre,''
we may say, therefore, that
when a body is moving undisturbed
by any force whatever,
\textit{its radius vector, from any centre arbitrarily chosen, describes equal
areas in equal times around that centre}. The area enclosed in the
triangle described by the radius vector in a unit of time is called
the body's ``\textit{areal} (or ``\textit{areolar}'') velocity,'' and in this case is
constant.

\nbarticle{403.} (\textit{b}) Moreover, \textit{any impulse in the line of the radius vector,
either towards or from the centre, leaves unchanged both the plane of
the body's motion and its areal velocity}.

Suppose a body moving uniformly on the line $AC$ (\figref{illo136}{Fig.~136}) with
such a velocity that it describes $AB$, $BC$, etc., in successive units of
time; then, by the preceding section, the areal velocity will be constant,
%% -----File: 260.png---Folio 249-------
and measured by the area of any one of the equal triangles
$AOB$, $BOC$, or $COL$. Suppose, now, that at $C$ the body receives
an ``impulse'' directed along the radius vector towards $O$---a blow,
for instance, which by itself would make the body describe $CK$ in a
unit of time. The body will now take a new path, which will carry it
to the point $D$, determined  by constructing the ``parallelogram  of
motions'' $CKDL$, and thus combining the new motion $CK$ with the
former motion $CL$, according to Newton's second law (Physics, p.~27).
The new areal velocity, measured by the triangle $OCD$, will be the
same as before, as is easily shown.

\includegraphicsmid{illo136}{\textsc{Fig.~136.}---Description of Areas under an Impulse directed towards a Centre.}

Triangle $BOC =$ triangle $COL$, because $BC = CL$, and $O$ is their
common vertex.

Also, triangle $COL =$ triangle $COD$, because they have the common
base $OC$, and their summits $L$ and $D$ are on a line which was drawn
parallel to this base in constructing the parallelogram of motions.
Hence triangle $BOC =$ triangle $COD$, and the \textit{areal velocity remains
unchanged}.

Also, as may be seen by following out the same reasoning with
$CK'$ and $CD'$, the same result would hold true if the impulse had
been directed \textit{away from} $O$ instead of towards it.

\begin{fineprint}
\nbarticle{404.} This result depends entirely on the fact that the impulse $CK$ or
$CK'$ was \textit{exactly along the radius vector} $CO$. If it has not been so, then in
%% -----File: 261.png---Folio 250-------
constructing the parallelogram of motions to find the points $D$ and $D'$, we
should have had to draw $LD$ or $LD'$ \textit{not} parallel to $CO$, and the two triangles
$BOC$ and $COD$ would necessarily have been unequal. $COD$ would be
greater than $BOC$ if $CK$ were directed ahead of the radius vector, and less
if behind it.
\end{fineprint}

As regards the plane of motion, the point $D$ is on the plane $OCL$,
because $LD$ was drawn through $L$ parallel to $OC$. $OCL$ is a part
of the plane which contains the triangles $BOC$ and $AOB$, and hence
$OCD$ also lies in the same plane.

\sloppy
\nbarticle{405.} (\textit{c}) From this obviously follows the important general proposition
that \textit{when a body is moving under the action of a force always
directed towards or from a fixed centre, the radius vector will describe
equal areas in equal times; and the path of the body will all lie in one
plane}.

\fussy
Such a force constantly acting is simply equivalent to an indefinite
number of separate impulses. Now if no single impulse directed
along the radius vector can alter the areal velocity or plane of motion,
neither can a succession of them. Hence the proposition follows.

In case of a \textit{continuously} acting force the orbit, however, will
become a curve instead of being a broken line.

Observe that this proposition remains true whether the force is
attractive or repulsive, and that it is independent of the \textit{law} of the
force; that is, the force may vary directly \textit{with the distance}, or \textit{inversely
as the square of the distance}, or as \textit{the logarithm} of it, or in
any conceivable way; it may even be \textit{discontinuous}, acting only at
intervals and ceasing between times: and still the law holds good.

\nbarticle{406.} Conversely, \textit{if a body moves in this way, describing equal
areas in equal times around a point, it is easily shown that all the
forces acting upon the body must be directed toward that point}.

We, however, leave the demonstration to the student.

Since the earth moves very nearly in this way in its orbit around
the sun, we conclude that the only force of any consequence acting
upon the earth is directed towards the sun. We say, ``of any consequence,''
because there are other small forces which do slightly
modify the earth's motion, and prevent it from \textit{exactly} fulfilling the
law of areas.

{\centering\nbrule[.5\textwidth]\par}

As a direct consequence of the law of equal areas we have certain
laws with respect to the linear and angular velocities of a body moving
under the action of a central force.
%% -----File: 262.png---Folio 251-------

\includegraphicsouter{illo137}{\textsc{Fig.~137.}\\Linear and Angular Velocities.}

\nbarticle{407.} \nbparatext{Law of Linear Velocity.}---Suppose a body moving under
the action of a force always directed towards $S$ (\figref{illo137}{Fig.~137}), and let
$AB$ be a portion of its path which it describes
in a second. Draw the tangent
$Bb$. Regarding the sector $ASB$ as a
triangle (which it will be, sensibly, since
the curvature of the path in one second
will be very small) the area of this triangle
will be $\frac{1}{2}(AB× Sb)$. Now $AB$,
the distance travelled in a second, is
the \textit{linear velocity} of the body (called
\textit{linear} because it is measured with the
same units as any other \textit{line}; \textit{i.e.}, in
\textit{miles} or in \textit{feet} per second), and $Sb$ is the
distance from the centre of force to the ``line of motion,'' as the tangent
$Bb$ is called. For $Sb$, $p$ is usually written; hence in every
part of the same orbit, $V$ (the velocity in miles per second) $= \dfrac{2A}{p}$, and
is inversely proportional to $p$. If $p$ were to become zero, $V$ would
become infinite, unless $A$ were zero also.

\nbarticle{408.} \nbparatext{Law of Angular Velocity.}---Referring again to the same
\figref{illo137}{figure}, the area of $ASB$ is equal to $\frac{1}{2}(AS× BS×\sin{ASB})$, or
$A = \frac{1}{2}r_{1}r_{2}\sin{\omega}$. If we draw $r$ to the middle point of $AB$, then $r_{1}r_{2} = r^{2}$,
nearly, since in a second of time the distance would not
change perceptibly as compared with its whole length. $\omega$ will also
be a small angle, so that its sine will equal the angle itself expressed
in radians;
\begin{flalign*}
&\text{hence }&& \tfrac{1}{2}r^{2}\omega = A, \text{ and }\omega = \frac{2A}{r^{2}}.  &&\phantom{hence }
\end{flalign*}
Now $\omega$ is the \textit{angular velocity} of the body; that is, the number of
``radians'' which it describes in a second of time, as seen from $S$,
while $r$ is the radius vector.

\nbarticle{409.} In every case, therefore, of motion under a central force,
I.~\textit{The Areal velocity $($acres per second$)$ is constant}; II.~\textit{The
Linear velocity $($miles per second$)$ varies inversely as the distance from
the centre of force to the body's line of motion at the moment}, which
line of motion is the tangent to the orbit at the point where the body
happens to be; III.~\textit{The Angular velocity $($radians, or degrees, per
second$)$ varies inversely as the square of the distance of the body from
the centre of force}.
%% -----File: 263.png---Folio 252-------

\nbarticle{410.} The student will remember that it was found by observation
that the sun's angular velocity varies as the square of its apparent
diameter, and from this (\artref{Art.}{186}) the law of equal areas was
inferred as a fact with respect to the earth's motion. Newton was
the first to point out that a body moving under the action of a
central force must \textit{necessarily} observe this law of areas, and, conversely,
that a body thus observing the law of areas must necessarily
be under the control of a central force.

\nbarticle{411.} \nbparatext{Circular Motion.}---In the case of a body moving in a
\textit{circle} under the action of a central force, the force must be constant,
and (Physics, p.~62) is given by the formula
\[
f = \frac{V^2}{r}, \eqno{(a)}
\]
in which $r$ is the radius of the circle and $V$ the velocity, while $f$ is
the central force measured by the ``acceleration'' of the body
towards the centre; that is, by the number of units of velocity
which the force would generate in the body in a second of time; just
as the force of gravity is expressed by writing, $g = 9.81$ metres.

For many purposes it is desirable to have an expression which
shall substitute for $V$ (a quantity not given directly by observation)
the time of revolution, $t$, which is so given. Since $V$ equals the
circumference of the circle divided by $t$, or $\dfrac{2\pi r}{t}$, we have at once, by
substituting this value for $V$ in equation (\textit{a}),
\[
f = 4\pi^2\left(\frac{r}{t^2}\right). \eqno{(b)}
\]

This, of course, is merely the equivalent of equation (\textit{a}), but is
often more convenient.

\section*{KEPLER'S LAWS.}

\nbarticle{412.} In 1607--1620 Kepler discovered as facts, without an explanation,
three laws which govern the motions of the planets,---laws
which still bear his name. He worked them out from a discussion
of the observations which Tycho Brahe had made through many
preceding years. The three laws are as follows:---

\mbox{\phantom{II}}I\@. \textit{The orbit of each planet is an ellipse, with the sun in one of its foci}.

\mbox{\phantom{I}}II\@. \textit{The radius vector of each planet describes equal areas in equal
times}.
%% -----File: 264.png---Folio 253-------

III\@. The ``Harmonic law,'' so-called. \textit{The squares of the periods
of the planets are proportional to the cubes of their mean distances
from the sun;} i.e., $t_1^2:t_2^2 = a_1^3:a_2^3$.

\begin{fineprint}
\nbarticle{413.} To make sure that the student apprehends the meaning and scope
of this third law, we append a few simple examples of its application.

(1) What would be the \textit{period} of a planet having a mean distance from
the sun of 100 astronomical units; \textit{i.e.}, a distance 100 times that of the earth?
\[
  (\textit{\footnotesize Earth's Dist.} )^3 : (\textit{\footnotesize Planet's Dist.} )^3
= (\textit{\footnotesize Earth's Period})^2 : (\textit{\footnotesize Planet's Period})^2;
\]
\vspace{-1.5\baselineskip}
\begin{flalign*}
&\textit{i.e.,}  &&1^3:100^3 = 1^2(\text{year}):X^2(\text{years}),&&&&\\
&\text{whence}   &&X = 100^\frac{3}{2} = 1000\text{ years}.&&&&\phantom{whence}
\end{flalign*}

(2) What would be the distance of a planet having a period of 125 years?
\begin{flalign*}
&&&(1)^2:125^2 = 1^3:X^3,&&&&\\
&\text{whence}   &&X = 125^\frac{2}{3} = 25 \text{ (Astron.\ units)}.&&&&\phantom{whence}
\end{flalign*}

(3) How long would a planet require to fall to the sun?

If the sun were collected in a single point at its centre, a body starting
from a point on the planet's orbit with a slight \textit{side-motion, i.e.}, motion at right
angles to the radius vector, would describe an extremely narrow ellipse
around the sun, with its perihelion just \textit{at} the sun, and the aphelion at the
starting-point. Practically it would ``\textit{fall to the sun},'' and return just as if it
had \textit{rebounded} from a perfectly elastic surface: the time of ``falling'' would be
just equal to that of returning---the two making up the whole period of the
body in the narrow ellipse. Now the semi-major axis of this narrow ellipse
is evidently \textit{one-half} the radius of the planet's orbit. Hence to find the period
in this ellipse which is $2\tau$ ($\tau$ being taken as the time of ``falling''), we have
\begin{flalign*}
&&  & a^3 : (\tfrac{1}{2}a)^3 = t^2:(2\tau)^2, \text{ or }
      1   : \tfrac{1}{8} = t^2:4\tau^2;  &\phantom{whence }&
\\
&\text{whence }&  & \tau = t\sqrt{\frac{1}{32}} = 0.1768t,
&\llap{$t$ being the planet's period.}&
\end{flalign*}

In the case of the earth $\tau = 365\frac{1}{4}× 0.1768 = 64.56$ days.

(4) What would be the period of a satellite revolving close to the earth's
surface?
\[
(\textit{Moon's Dist.})^3:(\textit{Dist.\ of Satellite})^3 = (27.3\text{ days})^2:X^2,
\]
\vspace{-1.5\baselineskip}
\begin{flalign*}
&\text{or } &&60^3:1^3 = (27.3)^2:X^2,&&&\\
&\text{whence } && X = \frac{27.3\text{ days}}{60^\frac{3}{2}} = 1^\text{h}\ 24^\text{m}.&&&
\end{flalign*}

(5) How much would an increase of 10 per cent in the earth's distance
from the sun increase the length of the year? \hfil\penalty0\hfilneg\null\penalty5000\hfill\mbox{\quad  \textsc{Ans}. 56.13 days.}

(6) What is the distance from the sun of an asteroid which has a period
of $3\frac{1}{2}$ years? \hfil\penalty0\hfilneg\null\penalty5000\hfill\mbox{\quad  \textsc{Ans}. 2.305 Astron.\ units.}
\end{fineprint}
%% -----File: 265.png---Folio 254-------

\nbarticle{414.} Many surmises were made as to the physical meaning of
these laws. More than one astronomer \textit{guessed} that a force directed
toward the sun, or emanating from it, might be the explanation.
Newton proved it. He demonstrated the law of equal areas and its
converse as necessary consequences of the laws of motion. He also
proved (Physics, pp.~64--66) that if a body move in an ellipse having
a centre of force at its focus, then the force at different points in the
orbit must vary inversely as the square of the distance from that
centre. And, finally, he showed that, granting the harmonic law,
the force from planet to planet must also vary according to the same
law of inverse squares.

\begin{fineprint}
\nbarticle{415.} The demonstration of this last proposition for circular orbits is so
simple that we give it, merely adding (without proof) that the proposition
is equally true for elliptical orbits, if for $r$ we put $a$, the semi-major axis of
the orbit.

In a circular orbit, from equation (\textit{b}), (\artref{Art.}{411}), we have
\[
f = 4\pi^{2}\left(\frac{r}{t^{2}}\right),
\]
where $r$ and $t$ are the distance and period of a planet. In the same way the
force acting upon a second planet is found from the equation
\begin{flalign*}
&             && f_{1} = 4\pi^{2}\left(\frac{r_{1}}{{t_{1}}^{2}}\right),  &&\\
&\text{whence}&& \frac{f}{f_{1}} = \frac{r}{t^{2}}× \left(\frac{{t_{1}}^{2}}{r_{1}}\right).  &&\\
\intertext{\indent But by Kepler's third law}
&             && t^{2}:{t_{1}}^{2}::r^{3}:{r_{1}}^{3},  &&\\
&\text{whence}&& {t_{1}}^{2} = \frac{t^{2}{r_{1}}^{3}}{r^{3}}.&&\phantom{\text{whence}}
\end{flalign*}

Substitute this value of ${t_{1}}^{2}$ in the preceding equation; we have
\begin{flalign*}
&&  & \frac{f}{f_{1}}
= \frac{r}{t^{2}} × \frac{t^{2}{r_{1}}^{3}}{r^{3}r_{1}}
= \frac{{r_{1}}^{2}}{r^{2}};  &&\\
&\textit{i.e.},& & f:f_{1} = {r_{1}}^{2}:r^{2}, &&\phantom{i.e. }
\end{flalign*}
which is the law of inverse squares.

\nbarticle{416.} Conversely, the harmonic law is just as easily shown to be a
necessary consequence of the law of gravitation in the case of circular orbits.

From \artref{Art.}{411}, Eq.~(\textit{b}), we have
\[
f = 4\pi^{2} \frac{r}{t^{2}} ;
\]
%% -----File: 266.png---Folio 255-------
also, from the law of gravitation,
\[
f = \frac{M}{r^{2}},\quad\rlap{\text{$M$ being the mass of the sun}.}
\]

Hence, equating the two values of $f$,
\[
\frac{M}{r^{2}} = 4\pi^{2}\frac{r}{t^{2}},\qquad\text{ and }\qquad t^{2} = \frac{4\pi^{2}}{M}r^{3}.
\]

Similarly for another planet,
\[
{t_{1}}^{2} =  \frac{4\pi^{2}}{M}{r_1}^{3}.
\]
\begin{flalign*}
\text{\indent Whence,} &&t^{2}:{t_{1}}^{2} = r^{3}:{r_{1}}^{3}.&&\phantom{\text{Whence, }}
\end{flalign*}

The demonstration for elliptical orbits is a little more complicated,
involving the ``law of areas.'' It is given in all works on Theoretical
Astronomy, and may be found in Loomis's ``Elements of Astronomy,'' p.~134.

\nbarticle{417.} \nbparatext{Correction of Kepler's Third Law.}---The ``harmonic law''
as it stands is not \textit{exactly} true, though the difference is too small to appear
in the observations which Kepler made use of in its discovery. It would be
exactly true if the planets were mere particles of matter; but as a planet's
mass is a sensible, though a very small fraction of the sun's mass, it comes
into account. The planet Jupiter, for instance, attracts the sun as well as is
attracted by it. If at the distance $r$ Jupiter is drawn towards the sun by a
force which would give it in a second an acceleration expressed by $\frac{M}{r^{2}}$ (the
sun's mass being $M$), then the sun in the same time is accelerated towards
Jupiter by the quantity $\frac{m}{r^{2}}$ ($m$ being the mass of Jupiter). The rate at which
the two tend to \textit{approach each other} is therefore expressed by $\frac{M + m}{r^{2}}$. Hence,
in discussing the motions of the planet Jupiter around the \textit{centre of the sun},
instead of writing
\begin{flalign*}
&&f &= \frac{M}{r^{2}}\text{ simply},&&\\
&\text{we must put}& f &= \frac{M + m}{r^{2}};&&\phantom{\text{we must put}}\\
\intertext{but (in the case of entirely circular motion)}
&&f &= \frac{4\pi^{2}r}{t^{2}}.&&\\
\intertext{\indent Hence, we find}
&& t^{2}(M + m) &= 4\pi^{2}r^{3};&&\\
\intertext{or, as a proportion,}
&& t^{2}(M + m)&:{t_{1}}^{2}(M + m_{1}) = r^{3}:{r_{1}}^{3},&&\\
\intertext{which is \textit{strictly} true as long as the planet's motions are undisturbed.}
\end{flalign*}
\end{fineprint}
\vspace{-2\baselineskip}
%% -----File: 267.png---Folio 256-------

\nbarticle{418.} \nbparatext{Inferences from Kepler's Laws.}---From Kepler's laws we
are entitled to infer---

\textit{First} (from the second law), that the \textit{force} which retains the planets
in their orbits \textit{is directed towards the sun}.

\textit{Second} (from the first law), that on any given planet the \textit{force
which acts varies inversely as the square of its distance from the
sun}.

\textit{Third} (from the harmonic law), that the force is the same for one
planet as it would be for another in the same place; or, in other
words, the attracting force \textit{depends only on the mass and distance of
the bodies concerned, and is wholly independent of their physical conditions},
such as their temperature, chemical constitution, etc. It
makes no difference in the motion of a planet around the sun whether
it be made of hydrogen or iron, whether it be hot or cold.

\includegraphicsouter{illo138}{\textsc{Fig.~138.}\\
Verification of the Hypothesis of Gravitation
by Means of the Motion of the Moon.}

\nbarticle{419.} \nbparatext{Verification of ``Gravitation'' by Means of the Moon's Motion.}---When
the idea of gravitation
first occurred to Newton (the
apple story is probably apocryphal),
he endeavored to verify it
by comparing the force which keeps
the moon in her orbit with the force
of gravity at the earth's surface,
reduced in the proper proportion.
For lack, however, of an accurate
knowledge of the earth's dimensions,
he failed at first, there being
a discrepancy of about sixteen per
cent. He had assumed a degree
to be exactly sixty miles in length.
Some years afterward, when Picard's
measure of the arc of a meridian
in Northern France had been
made and reported to the Royal Society, making a degree about
sixty-nine miles long, he saw at once that the new value would
reconcile the discrepancy; and he resumed his unfinished work and
completed it.

\nbarticle{420.} At the earth's surface a body falls about 193 inches in a
second. The distance of the moon being very nearly sixty times the
earth's radius, if gravity really varies inversely as the square of the
%% -----File: 268.png---Folio 257-------
distance, a stone at that distance from the earth should fall $\dfrac{1}{60^{2}}$ or
$\dfrac{1}{3600}$ as far; that is, it ought to fall $\dfrac{193\ \text{inches}}{3600} = 0.0535$ inches,---a
little more than one-twentieth of an inch. Now the distance which
the moon actually \textit{does} fall towards the earth in a second, \textit{i.e.}, the
\textit{deflection of its orbit from a straight line in a second of time}, is easily
found; and if the force which keeps the moon in its orbit is really the
same as that which makes bodies fall towards the centre of the earth,
this deflection ought to come out equal to $0^{\text{in}}.0535$. Let $AE$ (\figref{illo138}{Fig.~138})
be the distance the moon travels in a second $= \dfrac{2\pi r}{t}$, where $r$ is the
radius of the moon's orbit, and $t$ the number of seconds in a month.
Then, since $AEF$ is a right-angled triangle, we have,
\[
AB:AE::AE:AF\ (\text{or }2r);
\]
\[
\text{whence }\qquad AB = \frac{AE^{2}}{2r}.
\]
The calculation is easy enough, though the numbers are rather large.
As a result it gives us $AB = 0.0534$ inches, which is practically equal
to the thirty-six hundredth part of $193$ inches.

If the quantities did not agree in amount, the discrepancy would
disprove the theory, and, as we have said, Newton loyally gave it up
until he was able to show that the apparent discordance was the result
of a mistake in the original data, and disappeared when the data were
corrected. The agreement, however, does not \textit{establish} the theory,
but only renders it probable. It does not establish it completely,
because it is conceivable that the agreement might be a case of accidental
coincidence, while the forces might really differ as much in
their nature as an electrical attraction and a magnetic.

\nbarticle{421.} Newton was not satisfied with merely showing that the principal
motions of the planets and the moon could be explained by the
law of gravitation; but he went on to investigate the converse problem,
and to determine what must be the motions \textit{necessary} under that
law. He found that the orbit of a body moving around a central mass
is not of necessity a circle, or even a nearly circular ellipse like the
planetary orbits, but that it may be a \textit{conic section} of any eccentricity
whatever---a circle, ellipse, parabola, or even an hyperbola; but \textit{it
must be a conic}.
%% -----File: 269.png---Folio 258-------

\nbarticle{422.} For the benefit of those of our readers who are not acquainted
with conic sections we give the following brief account of them
(\figref{illo139}{Fig.~139}):---

\textit{a}. If a cone of any angle be cut \textit{perpendicularly to the axis}, the
section will be a \textit{circle}---$MN$ in the
\figref{illo139}{figure}.

\textit{b}. If it be cut by a plane which
makes with the axis an angle \textit{greater}
than the semi-angle of the cone, \textit{so
that the plane of section cuts completely
across the cone} (as $EF$), the
section, is an \textit{ellipse}; the circle being
merely a special case of the ellipse.
Ellipses, of course, differ greatly in
form, from these which are very
narrow to the perfect circle.

\includegraphicsouter{illo139}{\textsc{Fig.~139.}---The Conics.}

\textit{c}. \textit{The parabola} is formed by
cutting the cone with a plane \textit{parallel
to its side}; \textit{i.e.}, making with the axis
an angle \textit{equal to the semi-angle of
the cone}. $RPO$ is such a plane. As
all circles are alike in form, so are
all parabolas, \textit{whatever the angle
of the cone at $V$ and wherever the
point $P$ is taken}. If the cutting
plane is thus situated, then, no
matter what is the angle of the
cone or the place where the cut is
made, the curve will always be the
same \textit{in shape}, though of course \textit{its
size} will depend upon a variety of
circumstances. We do not stop to
prove the statement, at first a little surprising; but it is true.

\textit{d}. If the cutting plane makes an angle with the axis of the cone
\textit{less than the semi-angle at $V$}, so that the cutting plane gets continually
deeper and deeper into the cone, then the curve is an \textit{hyperbola}; so
called, because the plane in this case ``shoots over'' (\mytextgreek{<up'er b'allein})
and intersects the ``cone produced,'' cutting out of this second cone
a curve precisely like the curve cut from the original, as at $H'G'K'$ in
the \figref{illo139}{figure}. The axis of the hyperbola lies outside of the curve itself,
being the line $HH'$ in the \figref{illo139}{figure}, and the ``\textit{centre}'' of the curve is
also outside of the curve at the middle point of this axis.
%% -----File: 270.png---Folio 259-------

\sloppy
\nbarticle{423.} Philosophically speaking there are therefore but \textit{two} species
of conic sections,---the ellipse and the hyperbola, with the parabola
for a partition between them. (The circle, as has been said before,
is merely a special case of the ellipse.) \figref{illo140}{Fig.~140} will give the reader
perhaps a better idea of the nature of the curves as drawn on a plane.
In the ellipse the sum of the distances from the two foci, $FN + F'N$,
equals the major axis of the curve; in the hyperbola it is the difference
of these two lines ($F''N' - FN'$) that equals the major axis;
in the ellipse the \textit{eccentricity} is \textit{less than unity} (zero in the circle); in
the hyperbola it is \textit{greater than unity}; in the parabola \textit{exactly unity}.

\fussy
\includegraphicsmid{illo140}{\textsc{Fig.~140.}---The Relation of the Conics to Each Other.}

\begin{fineprint}
The general equation of a conic in polar co-ordinates, applying alike to
both the species, is
\[
r = \frac{p}{1 + e \cos{V}},
\]
in which $r$ is the distance $Fn$, or $Fn'$, $e$ is the fraction $\dfrac{FC}{PC}$ or $\dfrac{FC'}{PC'}$, the angle
$V$ is the angle $PFn$, $PFn'$, or $PFn''$, and $p$ is the line $FY$, $FY'$ or $FY''$,
called the ``semi-parameter.'' The word ``\textit{para}meter'' means the \textit{cross}-measure
of a curve, just, as ``\textit{dia}meter'' means the \textit{through}-measure of a curve. If $e$
is zero, the curve is a circle, and $r = p$. If $e < 1$, the curve is an ellipse; if
$e > 1$, the curve is an hyperbola; if $e = 1$, it is a parabola.
\end{fineprint}
%% -----File: 271.png---Folio 260-------

\nbarticle{424.} \nbparatext{Problem of Two Bodies.}---This problem, proposed and
solved by Newton, is the following:---

\textit{Given the masses of two spheres and their positions and motions at
any moment; given, also, the law of gravitation: required their motion
ever afterwards, and the data necessary to compute their place at any
future time}.

The mathematical methods by which the problem is solved require
the use of the calculus, and must be sought in works on analytical
mechanics or theoretical astronomy. Some of the results, however,
are simple and easily stated.

\nbarticle{425.} (1) In the first place the motion of the centre of gravity of
the two bodies will not be affected by their mutual attraction, but it
will move on uniformly through space, as if the bodies were united
into one at that point, and their motions combined under the same
laws which hold good in the case of the collision of inelastic bodies.

\includegraphicsmid{illo141}{\textsc{Fig.~141.}---Motion of Bodies relative to their Centre of Gravity.}

\begin{fineprint}
The motion of this centre of gravity is most easily worked out graphically
as follows: First, in \figref{illo141}{Fig.~141}, join the original places of the bodies $A$ and
$B$ by a straight line, and mark on it $G$, the place of the centre of gravity;
then take the positions $A'$ and $B'$ they would occupy at the end of a unit
of time (if they did not attract each other), and mark the new position of
the centre of gravity $G'$ on the line joining them. The line $GG'$ connecting
the two positions of the centre of gravity will show the direction and rapidity
of its motion; with reference to this point the two bodies will have opposite
motions proportional to their distances from it; that is, they will swing
around this point as if on a rod pivoted there, and will either both move
towards it along the rod, or from it, with speeds inversely proportional
to their masses. These relative motions \textit{with respect to the centre of gravity}
are easily found by drawing through $G$ a line parallel to $A'B'$ and measuring
off on it distances $GA''$ and $GB''$ respectively equal to $G'A'$ and $G'B'$.
$AA''$ and $BB''$ will then be the two motions of $A$ and $B$ \textit{relative to their centre
of gravity $G$}.
\end{fineprint}
%% -----File: 272.png---Folio 261-------

\nbarticle{426.} \nbparatext{The Effect of their Mutual Attraction.}---This will cause
them to describe similar conics around this centre of gravity; the
size of their two orbits being inversely proportional to their masses.
The form of the orbits and dimensions will be determined by the
combined mass of the two bodies, and by their velocities with respect
to the common centre of gravity.

\nbarticle{427.} \nbparatext{The Orbit of the Smaller relative to the Centre of the Larger.}---It
is convenient (though it is not necessary) to drop the consideration
of the centre of gravity of the two bodies, and to consider
the motion of the smaller one around the centre of the larger one.
In reference to that point, it will move precisely as if its mass had
been added to that of the larger body, while itself had become a mere
particle. This relative orbit will in all respects be like the actual one
around the centre of gravity, only magnified in the proportion of
$M + m$ to $M$; \textit{i.e.}, if $m$ is $\frac{1}{10}$ of $M$, the \textit{actual} orbit around $G$ will
be magnified by $\frac{11}{10}$ to produce the \textit{relative} orbit around $M$.

\includegraphicsmid{illo142}{\textsc{Fig.~142.}---Elliptical Orbit determined by Projection.}

\nbarticle{428.} \nbparatext{The Orbit determined by Projection.}---Suppose that in the
figure (\figref{illo142}{Fig.~142}) the body $P$ is moving in the direction of the arrow,
and is attracted by $S$, supposed to be at rest. $P$ will thenceforward
move in a conic, either in an ellipse or hyperbola, according to
its velocity, as we shall see in a moment. $S$ being at one focus of
the curve, the other focus will be somewhere on the line $PN$, which
makes the same angle with $PQ$ that $r$ ($SP$) does (since it is a property
of the conics that a tangent-line at any point of the curve makes
equal angles with the lines drawn from the two foci to that point).
%% -----File: 273.png---Folio 262-------
If we can find the place of the second focus $F$, or the length of the
line $PF$ in the \figref{illo142}{figure}, the curve can at once be drawn.

Now, it can be proved, though the demonstration lies beyond our
scope, that $a$, the semi-major axis of the conic, is determined by
the equation
\[
V^2 = \mu\left(\frac{2}{r} - \frac{1}{a}\right), \nbtag{(Equation 1)}
\]
in which $r$ is the distance $SP$, $V$ is the velocity, and $\mu$ is the attracting
mass at $S$ expressed in proper units.

\begin{fineprint}
(See Watson's ``Theoretical Astronomy,'' p.~49; only for $\mu$ he writes $k^2(1 + m)$).
\end{fineprint}

$V$, $r$, and $\mu$ being given, of course $a$ can be found: we get
\[
a = \mu\frac{r}{2\mu - rV^2}. \nbtag{(Equation 2)}
\]

Then by subtracting $r$ from $2a$ we shall get $r'$, or the distance $PF$,
if the curve is an ellipse. If it is a hyperbola, $a$ will come out negative;
and to find $r'$ we must take $r' = 2a + r$ and measure it off to
$F'$, on the other side of the line of motion. In either case, however,
we easily find the other focus, and the line drawn through the foci
will be the line of apsides; a point half-way between the foci will be
the centre of the curve, and any line drawn through this centre will be
a diameter. Having the two foci and the major axis $2a$, \textit{i.e.}, $AA'$
the curve can at once be drawn.

\nbarticle{429.} \nbparatext{Expression for $a$ in Terms of the ``Velocity from Infinity,'' or
``Parabolic Velocity.''}---The expression for $a$ admits of a more convenient
and very interesting form. It is shown in analytical mechanics
that if, under the law of gravitation, a particle falls towards an
attracting body whose mass is $\mu$, from one distance $s$ to another distance
$r$, its velocity is given by the simple equation
\[
w^2 = 2\mu\left(\frac{1}{r} - \frac{1}{s}\right).\footnote
  {If the difference between $s$ and $r$ is called $h$, this equation becomes
\[
w^2 = 2\mu\left(\frac{1}{r} - \frac{1}{r + h}\right) = 2\mu\left(\frac{h}{r^2 + rh}\right).
\]
Now if $h$ is very small as compared with $r$, this gives
\[
w^2 = \left(\frac{2\mu}{r^2}\right)h,
\]
which is the same as the usual expression for the velocity of a falling body at the
earth's surface, viz., $V^2 = 2gh$,  $2g$ being replaced by the fraction $\dfrac{2\mu}{r^2}$.
} \nbtag{(Equation 3)}
\]
%% -----File: 274.png---Folio 263-------

If in this equation $s$ be made infinite, $w$ does not also become
infinite (that is, a body falling from an infinite distance towards the
sun will not acquire an infinite velocity until it actually reaches the
centre of the sun, and $r$ becomes zero); but we get in this case
\[
w^2 = \frac{2\mu}{r}.
\]
This special value of $w$ is usually called ``the velocity from infinity
for the distance $r$,'' or the ``\textit{parabolic velocity}'' (for a reason which
will appear very soon). $U$ is generally used as its symbol; therefore
\[
U^2 = \frac{2\mu}{r}; \text{ whence }\mu = \frac{rU^2}{2}. \nbtag{(Equation 4)}
\]

Substituting this value of $\mu$ in equation 2, we get
\[
a = \frac{r}{2}\left(\frac{U^2}{U^2 - V^2}\right). \nbtag{(Equation 5)}
\]

\nbarticle{430.} \nbparatext{Relation between the Velocity and the Species of Conic Described.}---From
equation~5 it is obvious how the velocity determines
whether the orbit will be an ellipse or a hyperbola. If $V^2$ is \textit{less} than
$U^2$, the denominator of the fraction will be positive, $a$ will also be
positive, and the curve will be an \textit{ellipse; i.e.}, if the velocity of the
body $P$, at the distance $r$ from the central body $S$, be less than
the velocity acquired by the body falling from infinity to that point,
the body will move around $S$ permanently in an ellipse.

If, on the other hand, $V^2$ is \textit{greater} than $U^2$, the denominator will
become negative, $a$ will also come out negative, and the orbit will be
a \textit{hyperbola}. In this case $P$, after once moving past $S$ at the
perihelion point, will go off never to return; and it will recede towards a
a different region of space from that out of which it came, because
the two legs of the hyperbola never become parallel. There will in
this case be no permanent connection between the two bodies. They
simply pass each other with a little graceful recognition of each other's
presence by a curvature in their paths, and then part company for ever.

If $V^2$ exactly equals $U^2$, the denominator of the fraction becomes
zero, $a$ comes out infinite, and the curve is a \textit{parabola}. In this case,
also, the body will never return; but it will recede from the sun ultimately
towards the same point on the celestial sphere as that from
which it appeared to come, since the two legs of the parabola tend to
parallelism. Obviously, if a body were thus moving in a parabola,
the slightest \textit{increase} of its velocity would transform the orbit into an
%% -----File: 275.png---Folio 264-------
\textit{hyperbola}, and the least \textit{diminution} into an \textit{ellipse}; the bearing of
which remark will become evident when we come to deal with comets.

\nbarticle{431.} Again, since\qquad
$a = \dfrac{r}{2}\left(\dfrac{U^{2}}{U^{2} - V^{2}}\right)$,\\
all bodies \textit{having the same velocity $V$, at the same distance $r$ from the
centre of force, will have major axes of the same length for their orbits,
no matter what may be their direction of motion}.

\begin{fineprint}
They will have the same \textit{period} also, the expression for the period being
\[
t = \frac{2\pi a^\frac{3}{2}}{\sqrt{\mu}}. \nbtag{(Watson, p.~46, Equation 28.)}
\]
\end{fineprint}

If, therefore, a body moving around the sun were to explode at any
point, all of its particles which did not receive a velocity greater than
the ``parabolic velocity'' would come around to the same point again,
and these which were projected with equal velocities would come
around and meet at the same moment, however widely different their
paths might be.

\includegraphicsmid{illo143}{\textsc{Fig.~143.}---Confocal Conics described under Different Velocities of Projection.}
%% -----File: 276.png---Folio 265-------

\begin{fineprint}
\nbarticle{432.} \figref{illo143}{Fig.~143} represents the orbits which would be described by five
bodies projected at $O$ with different velocities along the line $OV$, the distance
$OS$ or $r$ being taken as unity, as well as the parabolic velocity $U^{2}$. The
squares of the velocities are assumed as given below, with the resulting
values of $a$ and $r'$.
\[
{V_{1}}^{2} = \tfrac{1}{4} \text{; whence } a_{1} = \tfrac{2}{3} \text{; and } {r_{1}}' = \tfrac{1}{3}.
\]
This places the empty focus at $F_{1}$.

For the next larger ellipse
\begin{flalign*}
&                              &&{V_2}^2 = \tfrac{1}{2};\ a_2 = 1;\ {r_2}' = 1.   &&\phantom{In the same way}\\
&\indent\text{In the same way }&&{V_3}^2 = \tfrac{3}{4};\ a_3 = 2;\ {r_3}' = 3.   &&\\
&                              &&{V_4}^2 = 1;\ a_4 = \infty ;\ {r_4}' = \infty.   \rlap{\quad\text{(Parabola.)}}\\
&                              &&{V_5}^2 = 2;\ a_5 = -\tfrac{1}{2};\ {r_5}' = -2. \rlap{\quad\text{(Hyperbola.)}}
\end{flalign*}

\nbarticle{433.} \figref{illo144}{Fig.~144} shows how three bodies projected at $P$ with \textit{equal velocities,
but in different directions}, indicated by the arrows, describe three different
ellipses; all, however, having the same period, and the same length of semi-major
axis; namely, $a = 2r$; $V^{2}$ being taken equal to $\frac{3}{4} U^{2}$.

\includegraphicsmid{illo144}{\textsc{Fig.~144.}---Ellipses of the Same Periodic Time.}

For a fourth body, $V^{2}$ is taken as $=\frac{1}{2} U^{2}$, and with the direction of motion
perpendicular to $r$. This body will move in a perfect circle, $a$ coming out
%% -----File: 277.png---Folio 266-------
equal to $r$, when $V^{2} = \frac{1}{2} U^{2}$. In order to have \textit{circular} motion, both conditions
must be fulfilled; namely, $V^{2}$ must equal $\frac{1}{2} U^{2}$, and the direction of motion
must be perpendicular to the radius vector.

\includegraphicsouter{illo145}{\textsc{Fig.~145.}---Van der Kolk's Theorem.}

\nbarticle{434.} \nbparatext{Velocity of a Planet at Any Point in its Orbit.}---If $AA'$
(\figref{illo145}{Fig.~145}) be the major axis
of a planet's orbit, and $KK'$
the diameter of a circle described
around $S$ with $AA'$ as
radius, then the velocity of a
planet at any point, $N$, on its
orbit is equal to that which it
would have acquired by falling
to $N$ from the point $n$ on the
circumference of the circle.
The demonstration is not difficult
and may be found in
No.~1426 of the ``Astronomische
Nachrichten.''

\includegraphicsouter{illo146}{\textsc{Fig.~146.}---Projectiles near the Earth.}

\sloppy
\nbarticle{435.} \nbparatext{Projectiles near the
Earth.}---A good illustration
of the principles stated above is obtained by considering the motion of
bodies projected horizontally from the top of a tower near the earth's surface,
supposing the air to be removed so there will be no resistance to the motion.

\fussy
The ``parabolic velocity'' due to
the earth's attraction equals 6.94
miles per second at the earth's surface;
\textit{i.e.}, a body falling from the
stars to the surface of the earth,
\textit{drawn by the earth's attraction only},
would have acquired this velocity
on reaching the earth's surface.

First. If a body be projected with
a very small velocity, it would fall
nearly straight downwards. If the
earth were concentrated at the point
in its centre so that the body should
not strike its surface, it would move
in a very long narrow ellipse having
the centre of the earth at the further
focus, and would return to the original point after an interval of 29.9 minutes.

Second. With a larger velocity the orbit would be a wider ellipse with a
longer period, $C$ being still at the remoter focus.

Third. $V = U \sqrt{\frac{1}{2}}$, or about 4.9 miles per second. In this case the orbit of
%% -----File: 278.png---Folio 267-------
the body would be a \textit{perfect circle}, and the period would be 1~h.\ 24.7~m. It
will be remembered that we found that if the earth's rotation were 17 times
as rapid, thus completing a revolution in $1^{\text{h}}\, 24^{\text{m}}.7$, the centrifugal force
at the equator would become equal to gravity (\artref{Art.}{154}). Also, \artref{Art.}{413} (4),
this same time, $1^{\text{h}}\, 24^{\text{m}}.7$, was found from Kepler's third law as the period of
a satellite revolving close to the earth's surface.

Fourth. $V = U = 6.94$ miles. In this case the projectile would go off in
a \textit{parabola}, never to return.

Fifth. $V > 6.94$. In this case, also, the body would never return, but
would pass off in a \textit{hyperbola}.
\end{fineprint}

\sloppy
\nbarticle{436.} \nbparatext{Intensity of Solar Attraction.}---The attraction between
the sun and the earth from some points of view looks like a very
feeble action. It is only able, as has been before stated (\artref{Art.}{278}),
to bend the earth out of a rectilinear course to the extent of about
one-ninth of an inch in a second, while she is travelling nearly
nineteen miles; and yet if it were attempted to replace by bonds
of steel the invisible gravitation which holds the earth to the sun,
we should find the surprising result that it would be necessary to
cover the whole surface of the earth with wires as large as telegraph
wires, and only about half an inch apart from each other, in order
to get a metallic connection that could stand the strain. This ligament
of wires would be stretched almost to the breaking point. The
attraction of the sun for the earth expressed as tons of \textit{force} (not
tons of \textit{mass}, of course) is \DPtypo{3,600,000}{3,600000} %for consistency with other use of commas
millions of millions of tons (36 with seventeen ciphers); and similar stresses act through the
apparently empty space in all directions between all the different
pairs of bodies in the universe.

\fussy
\begin{fineprint}
\nbarticle{436*.} \nbparatext{Note to \artref{Art.}{429}.}---It is worth noting that $U^{2}$ (the square of
the parabolic velocity at any point) is simply \textit{twice the gravitation potential
due to the sun's attraction at that point}. The ``potential'' may be defined as
the \textit{energy} which would be acquired by a mass of one unit, in falling to the
point in question from a place where the potential (and attraction) is zero,
\textit{i.e.}, from infinity (Physics, pp.~26 and 28). Now $\frac{1}{2} m V^{2}$ is the general expression
for the kinetic energy of a mass, $m$, moving with velocity $V$; if in
this expression we make $m=1$, and $V=U$, we shall have, for the case in
hand, Energy $= \frac{1}{2} U^{2}$, which equals the \textit{Potential at the point}.
\end{fineprint}
\chelabel{CHAPTERXI}

%% -----File: 279.png---Folio 268-------

\Chapter{XII}{The Problem of Three Bodies}
\nbchapterhang{\stretchyspace
THE PROBLEM OF THREE BODIES.---DISTURBING FORCES:
LUNAR PERTURBATIONS AND THE TIDES.}

\nbarticle{437.} \textsc{The} problem of \textit{two} bodies is completely solved; but if,
instead of two spheres attracting each other, we have \textit{three} or more,
given completely in respect to their positions, masses, and velocities,
the general problem of finding their subsequent motions and predicting
their positions at any future date transcends the present power of
our mathematics.

This problem of \textit{three} bodies is in itself just as determinate and
capable of solution as that of two. Given the initial data,---that
is, the \textit{positions, masses, and motions} of the three bodies at a given
instant,---then their motions for all the future, and the positions
they will occupy at any given date, are absolutely predetermined.
The difficulty of the problem lies simply in the inadequacy
of our present mathematical methods, and it is altogether probable
that some time in the future this difficulty will be overcome---very
possibly by the invention of new functions and numerical tables
which shall bear some such relation to our present tables of logarithms,
sines, etc.\ as these do to the old multiplication table of
Pythagoras.

\nbarticle{438.} But while the \textit{general} problem of \textit{three} bodies is thus intractable,
all the special cases of it which arise in the consideration of
the moon's motion and in the motions of the planets have been solved
by special methods of approximation. Newton himself led the way;
and the strongest proof of the truth of his theory of gravitation lies
in the fact that it not only accounts for the \textit{regular} elliptic motions of
the heavenly bodies, but also for the apparent \textit{irregularities} of these
motions.

\nbarticle{439.} \nbparatext{The Disturbing Force.}---In the case where two bodies are
revolving around their common centre of gravity, and the third body
is either \textit{very much smaller} than the central one, or very \textit{remote}, the
%% -----File: 280.png---Folio 269-------
motion of the two will be but slightly modified by the action of the
third; and in such a case the small differences between the actual
motion and the motion as it would be if the third body were not
present, are technically called ``disturbances'' and ``perturbations,''\footnote
  {The student will bear in mind that these terms (``perturbations'' and ``disturbances'')
are mere figures of speech; that philosophically the purely elliptical
motion of two mutually attracting bodies alone in space is no more ``\textit{regular}''
than the (at present) incomputable motion of three or more attracting bodies.
We have in mind a theologian of some note who once maintained that the ``perturbations''
in the solar system are a consequence of Adam's fall. Hence the
  caution.}
and the force which produces them is called the ``disturbing force.''
This disturbing force is not the \textit{attraction} of the disturbing body, but
only a \textit{component} of that attraction, and usually only a small fraction
of it.

\textit{The disturbing force of the attracting body depends upon the difference
of its attraction upon the two bodies it disturbs; difference either
in amount or in direction, or in both.} For instance, if the sun
attracted the earth and moon exactly alike (\textit{i.e., equally and along
parallel lines}), it would not disturb their relative motions in the least,
no matter how powerful its attraction might be. The sun's maximum
\textit{disturbing force} on the moon, as we shall see, is only about one
eighty-ninth of the earth's attraction; and yet the sun's \textit{attraction} for
the moon is actually much greater than that of the earth.

\begin{fineprint}
Since the sun's mass is 330,000 times that of the earth, and its distance
just about 389 times that of the moon from the earth, its attraction on the
moon equals the earth's attraction $× \frac{330,000}{389^{2}} = 2.18$; \textit{i.e.}, the \textit{sun's attraction
on the moon is more than double that of the earth}.
\end{fineprint}

\nbarticle{440.} \nbparatext{Why the Sun does not take the Moon away from the Earth.}---If
at the time of new moon, when the moon is between the earth
and sun, the sun attracts the moon more than twice as much as the
earth does, it is a natural question why the sun does not draw the
moon away entirely, and rob us of our satellite. It would do so if
it were the case of a ``tug of war''; that is, if earth and sun were
\textit{fixed in space}, pulling opposite ways upon the moon between them.
But it is not so; neither sun nor earth has any \textit{foothold}, so to speak;
but all three bodies are free to move, like chips floating on water.
The sun attracts the \textit{earth} almost as much as he does the moon, and
both earth and moon fall towards him freely; though of course this
%% -----File: 281.png---Folio 270-------
falling motion towards the sun is continually combined with whatever
other motion the earth or moon possesses. The only effective disturbance
is produced by the fact that, in the case considered, \textit{the new
moon, being nearer the sun than the earth is} by about $\frac{1}{389}$ part of the
whole distance, falls towards the sun a trifle faster than the earth,
and so on that account the curvature of its orbit toward the earth is,
for the time being, diminished.

At the half-moon the two bodies are equally attracted towards the
sun, but on \textit{converging lines}; and so as they fall towards the sun
they \textit{approach} each other slightly; and for this reason, at quadrature,
the moon's orbit is a little more curved towards the earth than it
would be otherwise.

\nbarticle{441.} \nbparatext{Diagram of the Disturbing Force.}---A very simple diagram
enables us to find graphically the disturbing force produced by a
third body.

\begin{fineprint}
(What follows applies \textit{verbatim et literatim} to either of the two diagrams
of \figref{illo147}{Fig.~147}.)
\end{fineprint}

\includegraphicsmid{illo147}{\textsc{Fig.~147.}---Determination of the Disturbing Force by Graphical Construction.}

Let $E$ be the earth, $M$ the moon, and $S$ the disturbing body (the
sun in this case); and let the sun's attraction on the moon be represented
by the line $MS$. On the same scale the attraction of the sun
on the earth will be represented by the line $EG$, $G$ being a point so
taken that $EG:MS = MS^{2}:ES^{2}$;  that is,---

\begin{fineprint}
\textit{The sun's attraction on the earth} is to \textit{the sun's attraction on the moon} as \textit{the
square of the sun's distance from the moon} is to \textit{the square of the sun's distance
from the earth}, according to the law of gravitation. ($MS$ has to do double
duty in this proportion: in the first ratio it represents a \textit{force}; in the second,
a \textit{distance}.)
\begin{flalign*}
&\indent\text{From this proportion }&
&  EG = MS×\frac{MS^{2}}{ES^{2}}.
&&\phantom{From this proportion }
\end{flalign*}
%% -----File: 282.png---Folio 271-------

In \figref{illo147}{figure (\textit{a})} the moon is nearer to the sun than the earth is, and so $EG$
comes out \textit{less} than $MS$. In \figref{illo147}{figure (\textit{b})} the reverse is the case, and therefore
in this case $EG$ is \textit{larger} than $MS$.
\end{fineprint}

Now if the force represented by the line $MS$ \textit{were parallel and equal
to that represented by} $EG$, there would be no disturbance, as has been
said. If, then, we can resolve the force $MS$ into two components,
one of which is equal and parallel to $EG$, this component will be innocent
and harmless, and the other one will make all the disturbance.

To effect this resolution, draw through $M$ the line $MK$ parallel
and equal to $EG$. Join $KS$, and draw $ML$ parallel and equal to it.
$ML$ \textit{is then the disturbing force on the same scale as} $MS$; \textit{i.e.}, the line
$ML$ shows the true \textit{direction} of the disturbing force, and in \textit{amount}
the disturbing force is equal to the \textit{sun's attraction for the moon multiplied
by the fraction} $\left(\dfrac{ML}{MS}\right)$. The diagonal of the parallelogram
$MLSK$ is $MS$, which represents the resultant of the two forces $MK$
and $ML$, that form its sides.

\begin{fineprint}
For the sake of clearness the lines which represent forces in the \figref{illo147}{figures}
are indicated by herring-bone markings.
\end{fineprint}

\nbarticle{442.} At first it seems a little strange that in \figref{illo147}{figure (\textit{b})} the disturbing
force should be directed \textit{away from the sun}; but a  little
reflection justifies the result. If $E$ and $M$ were connected by a rod,
and the $E$-end of the rod were pulled towards the right more swiftly
than the $M$-end, it is easy to see that the latter would he \textit{relatively}
thrown to the left, as the \figref{illo147}{figure} shows.

\nbarticle{443.} The sun is the only body that sensibly disturbs the moon.
The planets, of course, act upon the moon to disturb it, but their
mass is so small compared with that of the sun, and their distances
so great, that in no case is their \textit{direct} action sensible. It is true,
however, that some of the lunar perturbations are affected by the
existence of one or two of the planets. While they cannot disturb
the moon \textit{directly}, they do so \textit{indirectly}: they disturb the earth in
her orbit sufficiently to make the sun's action different from what it
would be if the planets did not exist. In this way the planets
Venus, Mars, and Jupiter make themselves felt in the lunar theory.
There are also a few small disturbances that depend upon the fact
that the earth is not a perfect sphere.

\nbarticle{444.} Since the distance of the sun is nearly 400 times that of the
moon from the earth, the construction of the disturbing force $ML$,
%% -----File: 283.png---Folio 272-------
Fig.~147, admits of considerable simplification. It is only necessary
to drop the perpendicular $MP$ upon the line that joins the earth and
the sun, and take the point $L$ upon this line, so that $EL$ equals \textit{three
times} $EP$. The line $ML$ so determined will then \textit{very approximately}
(but not exactly) be the true disturbing force.

\begin{fineprint}
To prove this relation, let $MS$, in \figref{illo147}{Fig.~147}, be $D$, $ES = R$, $ME = r$, and
$EP = p$, also $R = D + p$, \textit{very nearly}, $p$ being \textit{negative} when $MS > ES$. $EG$
was taken equal to $MS×\dfrac{MS^{2}}{ES^{2}} = \dfrac{D^{3}}{R^{2}}$.

Now,
\[
EL = GS = (ES - EG)
     = R - \dfrac{D^{3}}{R^{2}} = \dfrac{R^{3} - D^{3}}{R^{2}}
     = \dfrac{(D + p)^{3} - D^{3}}{(D + p)^{2}}.
\]

Developing this expression, we have
\[
EL = \frac{3D^{2}p + 3Dp^{2} + p^{3}}{D^{2} + 2Dp + p^{2}}.
\]
Since $p$ is very small compared with $D$, all the terms except the first
nearly vanish both in numerator and denominator, and we have
\[
EL = \frac{3D^{2}p}{D^{2}} = 3p \text{ (very nearly).}
\]
\end{fineprint}

\nbarticle{445.} \nbparatext{Resolution of the Disturbing Force into Components.}---In
discussing the effect of the disturbing force it is more convenient to
resolve it into three components known as the \textit{radial}, the \textit{tangential},
and the \textit{orthogonal}. The first of these \textit{acts in the direction of the
radius vector}, tending to draw the moon either towards or from the
earth. The second, the tangential, \textit{operates to accelerate or retard
the moon's orbital velocity}.

\includegraphicsmid{illo148}{\textsc{Fig.~148.}---Radial and Tangential Components of the Disturbing Force.}

\begin{fineprint}
Fig.~148 exhibits these two components at different points of the moon's
orbit.
\end{fineprint}
%% -----File: 284.png---Folio 273-------

The orthogonal component has no existence in cases where the
disturbing body lies in the plane of the disturbed orbit; but whenever
it lies outside of that plane, the disturbing force $ML$ will generally
also lie outside of the orbit-plane, and will have a component
tending to \textit{draw the moving body out of the plane of its orbit}. The
motion of the moon's node and the changes of the inclination of its
orbit are due to this component of the sun's disturbing force, which
could not be conveniently represented in the \figref{illo148}{figure}.

\nbarticle{446.} The radial force in the case of the moon's orbit is a maximum
at syzygies and quadratures; in fact, at quadratures the whole
disturbing force is radial, the tangential and orthogonal components
both vanishing. At syzygies (new moon and full moon) the radial
force is \textit{negative}; that is, it draws the moon \textit{from} the earth, diminishing
the earth's attraction by about \textit{one eighty-ninth}\footnote
  {At syzygies $ML = NL_{0} = 2× EN$ (\figref{illo147}{Fig.~147}); but $EN = \dfrac{MS}{389}$. Therefore
$ML = \dfrac{2}{389}$ of the \textit{sun's} attraction on the moon. Now the sun's attraction is 2.18
  times the earth's; hence $NL_{0} = $ the \textit{earth's} attraction multiplied by $\dfrac{2× 2.18}{389} = \dfrac{1}{89.2}$.}
of its whole
amount.

At quadrature or half-moon the radial force is \textit{positive}; and since
$L$ then falls at $E$, it is represented by the line $QE$, and is just half
what it is at syzygies; that is, it equals about \textit{one one hundred and
seventy-eighth} of the earth's attraction.

It becomes zero at four points $54°\, 44'$ on each side of the line of
syzygies.

\begin{fineprint}
This angle is found from the condition that, the disturbing force $M_{1}L_{1}$,
etc., in \figref{illo148}{Fig.~148}, must be perpendicular to the radius $EM_{1}$ at this point,
which gives us $EP_{1}:P_{1}M_{1}::P_{1}M_{1}:P_{1}L_{1}$. But $P_{1}L_{1} = 2EP_{1}$; therefore
$P_{1}{M_{1}}^{2} = 2{EP_{1}}^{2}$, and $\dfrac{P_{1}M_{1}}{EP_{1}} = \tan{M_{1}EL_{1}} = \sqrt{2}$.
\end{fineprint}

\nbarticle{447.} The \textit{tangential component} starts at zero at the time of full
moon, rises to a maximum at the critical angle of $45°$ (having at
that point a value of $\frac{1}{119}$ of the earth's attraction), and disappears
again at quadratures. During the first and third quadrants this
force is \textit{negative}; that is, it \textit{retards} the moon's motion;  in the second
and fourth it is \textit{positive} and accelerates the motion.
%% -----File: 285.png---Folio 274-------

\nbarticle{448.} \nbparatext{Lunar Perturbations.}---So far it has been all plain sailing,
for nothing beyond elementary mathematics is required in determining
the disturbing force at any point in the moon's orbit; \textit{but to
determine what will be the effect of this continually varying force after
the lapse of a given time, upon the moon's place in the sky} is a problem
of a very different order, and far beyond our scope. The reader who
wishes to follow up this subject must take up the more extended
works upon theoretical astronomy and the lunar theory. A few
points, however, may be noted here.

\nbarticle{449.} In the first place, it is found most convenient to consider
the moon as never deviating from an elliptical orbit, but to consider
\textit{the orbit itself as continually changing in place and form}, writhing and
squirming, so to speak, under the disturbing forces; just as if the
orbit were a material hoop with the moon strung upon it like a bead
and unable to get away from it, although she can be set forward and
backward in her motion upon it.

\nbarticle{450.} In the next place, it is found possible to represent nearly all
the perturbations by \textit{periodical formulę}---the same values recurring
over and over again indefinitely at regular intervals. This is because
the sun, moon, and earth keep coming back into the same, or nearly
the same, relative positions, and this leads to recurring values of the
disturbing force itself, and also of its effects.

\nbarticle{451.} Third, the number of these separate perturbations which
have to be taken account of is very large. In the computation of
the moon's \textit{longitude} in the American Ephemeris about \textit{seventy} different
inequalities are reckoned in, and about half as many in the
computation for the \textit{latitude}. Theoretically the number is infinite,
but only a  certain number produce effects sensible to observation.
It is of no use to compute disturbances that do not displace the moon
as much as one-tenth of a second of arc; \textit{i.e.}, about 500 feet in her
orbit.

\nbarticle{452.} Fourth, in spite of all that has been done, the lunar theory
is still incomplete, or in some way slightly erroneous. The best
tables yet made begin to give inaccurate results after fifteen or
twenty years, and require correction. The almanac place of the
moon at present is not unfrequently ``out'' as much as $3''$ or $4''$ of
arc; \textit{i.e.}, about three or four miles. Astronomers are continually at
%% -----File: 286.png---Folio 275-------
work on the subject, but the computations by our present methods
are exceedingly tedious and liable to numerical error.

\nbarticle{453.} The principal effects of the sun's disturbing action on the
moon are the following:--

\textit{First: Effect on Length of Month.}---Since the radial component of
the disturbing force is \textit{negative} more than half the way round ($54°\,
44'$ on each side of the line of syzygies) and is twice as great at
syzygies as the positive component is at quadrature, the net result is
that, taking the whole month through, the earth's attraction for the
moon \textit{is lessened by nearly $\frac{1}{360}$ part}. The effect of this is to increase
the major axis of the moon's orbit, and to make the month a
little longer than it otherwise would be; that is, if the sun were
taken away, the month would immediately shorten by about three
hours.

\includegraphicsouter{illo149}{\textsc{Fig.~149.}\\
Advance of the Apsides of
the Moon's Orbit.}

\nbarticle{454.} \textit{Second: The Revolution of the Line of Apsides.}---This is
due mainly to the \textit{radial} component of the disturbing force.    When
the moon comes to \textit{apogee} at the time of new or full moon, the
diminution of the earth's effective attraction for the moon causes it
to move on farther than it would otherwise do
before turning the corner, so to speak, the consequence
being that the line of apsides \textit{advances}
in the line of the moon's motion. When the
moon passes \textit{perigee} at that time, the effect is
reversed, and the apsides \textit{regress}; but since the
disturbing force is greater at apogee than at
perigee, in the long run the advancing motion
predominates.  When perigee or apogee is
passed at the time of \textit{quadrature}, the line of apsides
is  also  disturbed; \textit{but} the disturbances
thus produced \textit{exactly balance each other} in the long run. The net
result, as has been stated before (\artref{Art.}{238}), is that the line of apsides
completes a direct revolution once in about nine years (8.855 years---\textit{Neison}).
It does not move forward steadily and uniformly, but
its motion is made up of alternate advance and regression. \figref{illo149}{Fig.~149}
illustrates this motion of the moon's apsides.

\begin{fineprint}
For a fuller discussion of the subject, see Herschel's ``Outlines of Astronomy,''
sections 677--689; or Airy's ``Gravitation,'' pp.\ 89--100.
\end{fineprint}

\sloppy
\nbarticle{455.} \textit{Third: The Regression of the Nodes.}---The \textit{orthogonal component
generally} (not always) \textit{tends to draw the moon towards the plane}
%% -----File: 287.png---Folio 276-------
\textit{of the ecliptic}. Whenever this is the case at the time when the moon
is passing a node, the effect (as is easily seen from \figref{illo150}{Fig.~150}) of such
a force $P_1O_1$, acting upon the moon at $P_1$, is to shift the node backward
from $N_1$ to $N_2$, the moon taking the new path $P_1b_1N_2$. As the
moon is approaching the node, the inclination of its orbit is also increased;
but as the moon leaves the node, it is again diminished, the
path $N_2P_2$ being bent at $P_2$ back to $P_2b_2$, parallel to $P_1a_1$: so that
while by both operations the node is made to recede from $N_1$ to $N_3$,
the inclination suffers very little change, if the orthogonal component
remains the same on both sides of the node.

\fussy
Since the orthogonal component vanishes twice a year,---when the
sun is at the nodes of the moon's orbit,---and also twice a month,---when
she is in quadrature,---the rate at which the nodes regress
is extremely variable. In the long run it makes its backward revolution
once in about nineteen years (Arts.~\arnref{249} and \arnref{391}). [18.5997
years.---\textit{Neison}.]

\begin{fineprint}
See Herschel's ``Outlines of Astronomy,'' section 638 seqq.
\end{fineprint}

\includegraphicsmid{illo150}{\textsc{Fig.~150.}---Regression of the Nodes of the Moon's Orbit.}

\sloppy
\nbarticle{456.} \textit{Fourth: The Evection.}---This is an irregularity which at the
maximum puts the moon forward or backward about $1\frac{1}{4}°$ $(1°\, 16'\, 27''.01\textit{---Neison})$,
and has for its period the time which is occupied by the
sun in passing from the line of apsides of the moon's orbit to the
same line again; \textit{i.e.}, about a year and an eighth. This is the largest
of the moon's perturbations, and was earliest discovered, having been
detected by Hipparchus about 150 years \textsc{b.c.}, and afterwards more
fully worked out by Ptolemy, though of course without any understanding
of its cause. It was the only lunar perturbation known to
the ancients. It depends upon the \textit{alternate increase and decrease of
the eccentricity} of the moon's orbit, which is always a maximum when
%% -----File: 288.png---Folio 277-------
the sun is passing the moon's line of apsides, and a minimum when
the sun is at right angles to it.

\fussy
This inequality may affect the time of an eclipse by nearly six
hours, making it anywhere from three hours early to three hours late,
as compared with the time at which it would otherwise occur; it was
this circumstance which called the attention of Hipparchus to it.

\begin{fineprint}
See Herschel's ``Outlines of Astronomy,'' sections 748 seqq.
\end{fineprint}

\nbarticle{457.} \textit{Fifth: The Variation.}---This is an inequality due mainly to
\textit{the tangential component} of the disturbing force. It has a period of
one month, and a maximum amount of $39'\, 30''.70$, attained when
the moon is half-way between the syzygies and quadratures, at the
so-called ``octants.'' At the first and third octants the moon is $39\frac{1}{2}'$
\textit{ahead of} her mean place (about an hour and twenty minutes); at the
second and fourth she is as much \textit{behind}. This inequality was detected
by Tycho Brahe, though there is some reason for believing
that it had been previously discovered by the Arabian astronomer,
Aboul Wefa, five centuries earlier. This inequality does not \textit{affect
the time of an eclipse}, being zero both at the syzygies and quadratures,
and therefore was not detected by the Greek astronomers.

\begin{fineprint}
See Herschel's ``Outlines of Astronomy,'' sections 705 seqq.
\end{fineprint}

\sloppy
\nbarticle{458.} \textit{Sixth: The Annual Equation.}---The one remaining inequality
which affects the moon's place by an amount visible to the
naked eye, is the so-called ``annual equation.'' When the earth is
nearer the sun than its mean distance, the sun's disturbing force is,
of course, greater than the mean, and the month is \textit{lengthened a
little}; during that half of the year, therefore, the moon keeps falling
behindhand; and \textit{vice versa} during the half when the sun's distance
exceeds the mean. The maximum amount of this inequality is
$11'\, 9''.00$, and its period one anomalistic year.

\fussy
\begin{fineprint}
See Herschel's ``Outlines of Astronomy,'' sections 738 seqq.
\end{fineprint}

There remains one lunar irregularity among the multitude of lesser
ones, which is of great interest theoretically, and is still a bone of
contention among mathematical astronomers; namely,---

\nbarticle{459.} \textit{Seventh: The Secular Acceleration of the Moon's Mean Motion.}---It
was found by Halley, early in the last century, by a comparison
of ancient with modern eclipses, that the month is now
%% -----File: 289.png---Folio 278-------
certainly shorter than it was in the days of Ptolemy, and that the
shortening has been progressive, apparently going on continuously,---\textit{in
sęcula sęculorum},---whence the name. In 100 years the
moon, according to the results of Laplace, gets in advance of its
mean place about $10''$, \textit{and the advance increases with the square of
the time}, so that in a thousand years it would gain nearly $1000''$,
and in 2000 years $4000''$, or more than a degree. The moon at
present is supposed to be just about a degree in advance of the position
it would have held if it had kept on since the Christian era
with precisely the rate of motion it then had. If this acceleration
were to continue indefinitely, the ultimate result would be that the
moon would fall upon the earth, as the quickened motion corresponds
to a shortened distance.

\nbarticle{460.} It was nearly 100 years after Halley's discovery before
Laplace found its explanation in the decreasing eccentricity of the
earth's orbit. Under the action of the other planets this orbit is now
growing more nearly circular, without, however, changing the length
of its major axis. Thus \textit{its area becomes larger}, and the earth's \textit{average}
distance from the sun becomes greater (although the \textit{mean} distance,
technically so-called, does not change, the ``mean distance'' being
simply half the major axis). As a result of this rounding up of the
earth's orbit, the \textit{average disturbing force of the sun is therefore diminished},
and this diminution allows the month to come nearer the length
it would have if there were no sun to disturb the motion; that is to
say, the month keeps shortening little by little, and it will continue
to do so until the eccentricity of the earth's orbit begins to increase
again, some 25,000 years hence.

\begin{fineprint}
\nbarticle{461.} But the theoretical amount of this acceleration, about $6''$ in a
century, does not agree with the value obtained by comparing the most
ancient and modern eclipses, which is about $12''$; and this value, again, does
not agree with the one derived by comparing modern observations of the
moon with these made by the Arabians about a thousand years ago, which,
according to recent investigations by Professor Newcomb, indicate an
acceleration of only about $8''$.

So long as the actual acceleration was considered to be $12''$, it was generally
supposed that the discrepancy between the theoretical and observed
result is due to \textit{a retardation of the earth's rotation by the friction of the
tides, and a consequent lengthening of the day}. Evidently if the day and
the seconds become a little longer, there will be fewer of them in each
month or year, and the \textit{apparent} effect of such a change would be to shorten
all really constant astronomical periods by one and the same percentage.
%% -----File: 290.png---Folio 279-------
As matters stand to-day it is hardly possible to assert with confidence
that there is any real discrepancy to be accounted for between the theoretical
and observed values, the latter being considerably uncertain. In Newcomb's
``Popular Astronomy'' (pp.\ 96--102) there will be found an interesting
and trustworthy discussion of the subject.

Questions like this, and those relating to the remaining discrepancies
between the lunar tables and the observed places of our satellite, lie on the
very frontiers of mathematical astronomy, and can be dealt with only by
the ablest and most skilful analysts.
\end{fineprint}

\section*{THE TIDES.}

\nbarticle{462.} Just as the disturbing force due to the sun's attraction
affects the motions of the moon in her orbit, so the disturbing
forces due to the attractions of the moon and sun acting upon the
fluids of the earth's surface produce \textit{the tides}. These consist of the
regular rise and fall of the water of the ocean, the average interval
between successive high waters being $24^\text{h}\ 51^\text{m}$, which is precisely
the same as the average interval between two successive passages of
the moon across the meridian. This coincidence, maintained indefinitely,
of itself makes it certain that there must be some causal connection
between the moon and the tides. As some one has said, the
odd 51 minutes is the moon's ``\textit{ear-mark}.''

\nbarticle{463.} \nbparatext{Definitions.}---When the water is rising, it is ``\textit{flood}'' tide;
when falling, it is ``\textit{ebb}.'' It is ``\textit{high water}'' at the moment when
the tide is highest, and ``\textit{low water}'' when it is lowest. ``\textit{Spring
tides}'' are the highest tides of the month (which occur near the
times of new and full moon), while ``\textit{neap tides}'' are the smallest,
which occur when the moon is in quadrature. The relative heights of
the spring and neap tides are about as 7 to 4. At the time
of spring tides the interval between the corresponding tides of successive
days is less than the average, being only about $24^\text{h}\ 38^\text{m}$,
and then the tides are said to ``\textit{prime}.'' At neap tides the interval
is $25^\text{h}\ 6^\text{m}$, which is greater than the mean, and the tides ``\textit{lag}.''

\textit{The ``establishment'' of a port} is the mean interval between the
time of high water at that port and the next preceding passage of
the moon across the meridian. At New York, for instance, this
``establishment'' is $8^\text{h}\ 13^\text{m}$,
although the actual interval varies about
22 minutes on each side of the mean at different times of the month.

That the moon is largely responsible for the tides is also shown by
the fact that the tides, at the time when the moon is in perigee, are
nearly twenty per cent higher than those which occur when she is in
%% -----File: 291.png---Folio 280-------
apogee. The highest tides of all happen when a \textit{new or full moon
occurs at the time the moon is in perigee, especially if this occurs
about January 1st, when the earth is nearest to the sun}. Since, as
we shall see, the ``tide-raising'' force varies inversely as the \textit{cube} of
the distance, slight variations in the distance of the moon and sun
from the earth make much greater variations in the height of the tide---greater
nearly in the ratio of 3 to 1.

\nbarticle{464.} \nbparatext{The Tide-Raising Force.}---This is the \textit{difference} between the
attractions of the sun and moon (mainly the latter) on the main
body of the earth, and the attractions of the same bodies on particles
at different parts of the earth's surface. The tide-raising force
is but a very small part of the whole attraction.

\includegraphicsmid{illo151}{\textsc{Fig.~151.}---The Moon's Tide-Raising Force on the Earth.}

The amount of this disturbing force for a particle at any point
on the earth's surface can be found approximately by the same geometrical
construction which was used for the lunar theory (\artref{Art.}{441}).
Draw a line from the moon through the centre of the earth.
At the points $A$ and $B$, \figref{illo151}{Fig.~151}, where the moon is directly over
head or under foot, the tide-raising force is directly opposed to gravity,
and equals nearly $\frac{1}{30}$ of the moon's whole attraction, since the line $Aa$
represents the disturbing force on the same scale as the line from $A$
to the moon represents the moon's attraction, and this line, $AM$, is
about sixty times the earth's radius, while $Aa$ is just double it, because
$Ca$ has to be taken equal to $3× CA$ (\artref{Art.}{444}).

Since the moon's mass is only about $\frac{1}{80}$ of the earth's, and its distance
is sixty radii of the earth, this \textit{lifting force} under the moon,
\textit{expressed as a fraction of the earth's gravity}, equals
\[
  \tfrac{1}{30} × \tfrac{1}{80} × \tfrac{1}{3600}
= \tfrac{1}{8640000} ;
\]
\textit{i.e.}, a body weighing \textit{four thousand tons} loses about \textit{one pound} of its
weight when the moon is over head or under foot.

At $D$ and $E$, anywhere on the circle of the earth's surface which
is $90°$ from $A$ and $B$, the moon's disturbing force \textit{increases} the
%% -----File: 292.png---Folio 281-------
weight of a body by just half this amount, the disturbing force being
measured by the lines $DC$ and $EC$. At a point $F$, situated anywhere
on a circle drawn around either $A$ or $B$ with a radius of $54°\, 44'$, the
\textit{weight} of a body is neither increased nor decreased, but it is urged
towards $A$ or $B$ with a horizontal force expressed by the line $Ff$,
which force is equal to about $\frac{1}{12\:000000}$ of its weight.

In the same way the tidal forces at $G$ and $H$ are expressed by the
lines $Gg$ and $Hh$.

\begin{fineprint}
\nbarticle{465.} The same result for the lifting-force directly under the moon may
be obtained more exactly as follows. The distance from the moon to the
centre of the earth is sixty times the earth's radius, and therefore the
distance from the moon to the points $A$ and $B$ respectively will be 59 and
61. The moon's attraction at $A$, $C$, and $B$, expressed as fractions of the
earth's gravity, will be as follows:---
{\renewcommand{\minalignsep}{9pt}
\begin{flalign*}
&&  \text{Attraction of moon on particle at } A
&= g ×  \frac{\frac{1}{80}}{59^2} = 0.0000035910× g.&&
\\
&&  \text{Attraction of moon on particle at } C
&= g ×  \frac{\frac{1}{80}}{60^2} = 0.0000034723× g.&&
\\
&&  \text{Attraction of moon on particle at } B
&= g ×  \frac{\frac{1}{80}}{61^2} = 0.0000033593× g.&&
\\
&\rlap{Hence }&  A - C = 0.0000001187\ g
&= \frac{1}{8,\!424,\!000}\ g.&&
\\
&&  C - B = 0.0000001130\ g
&= \frac{1}{8,\!835,\!000}\ g.&&
\end{flalign*}
}

This is more correct than the preceding, which is based on an approximation
that considers the moon's distance as \textit{very large} compared with the
earth's radius, while it is really only sixty times as great, and sixty is hardly
a ``very large'' number in such a case.

Attempts have been made to \textit{observe} directly the variations in the force of
gravity produced by the moon's action, but they are too small to be detected
with certainty by any experimental method yet contrived. Both Darwin
and Zöllner found that other causes which they could not get rid of produced
disturbances more than sufficient to mask the whole action of the moon.

\nbarticle{466.} It is worth while to note in this connection that the maximum
lifting-force due to the attraction of a distant body varies inversely as the
\textit{cube} of its distance, as is easily shown, thus:---calling $D$ the distance of the
disturbing body from the earth's centre, and $r$ the earth's radius, we have
\begin{flalign*}
&\begin{alignedat}{2}
&\text{Attraction at } A
&&= \frac{M}{(D - r)^2};\qquad \text{attraction at } C = \frac{M}{D^2}.\\
&\text{Tide-raising force at } A
 && = M\left\{ \frac{1}{(D-r)^2}	- \frac{1}{D^2}      \right\}\\
&&& = M\left\{ \frac{2Dr - r^2}{D^2(D^2 - 2Dr + r^2)} \right\}\\
&&&= M\left\{ \frac{2Dr - r^2}{D^4 - 2D^3r + D^2r^2} \right\}\\
&&&  = M\left\{ \frac{2r}{D^3} \right\}, \text{ nearly},
\end{alignedat}  &&
\end{flalign*}
when $r$ is a small fraction of $D$.
\end{fineprint}
%% -----File: 293.png---Folio 282-------

\nbarticle{467.} It is very apt to puzzle the student that the moon's action
should be a \textit{lifting} force at $B$ as well as at $A$ (\figref{illo151}{Fig.~151}). He is
likely to think of the earth as fixed, and the moon also fixed and
attracting the water upon the earth, in which case, of course, the
moon's attraction, while it would decrease gravity at $A$, would increase
it at $B$.

\includegraphicsouter{illo152}{\textsc{Fig.~152.}---The Statical Theory of the Tides.}

The two bodies are not fixed,
however. Let him think of the three
particles at $A$, $C$, and $B$, \figref{illo152}{Fig.~152},
as unconnected with each other, and
falling freely towards the moon;
then it is obvious that they would
separate; $A$ would fall faster than
$C$, and $C$ than $B$. Now imagine
them connected by an elastic cord.
It is obvious that they will still draw apart until the tension of the
cord prevents any further separation. Its tension will then measure
the ``lifting force'' of the moon which tends to draw both the particles
$A$ and $B$ away from $C$.

\nbarticle{468.} \nbparatext{The Sun's Action.}---This is precisely like that of the moon,
except that the sun's distance, instead of being only sixty times the
earth's radius, is nearly 23,500 times that quantity. Since the tide-raising
power varies as the \textit{cube} of the distance inversely, while the
attracting force varies only with the inverse \textit{square}, it turns out that
although the sun's attraction on the earth is nearly 200 times as
great as that of the moon, its \textit{tide-raising power is only about two-fifths
as much}. When the sun is over head or under foot, his disturbing
force diminishes gravity by about $\frac{1}{19\: 600000}$.

\sloppy
\nbarticle{469.} \nbparatext{Statical Theory of the Tides.}---If the earth were wholly
composed of water, and if it kept always the same face towards the
moon (as the moon does towards the earth), so that every particle on
the earth's surface was always subjected to the same disturbing force
from the moon, then, leaving out of account the sun's action, a permanent
tide would be raised upon the earth, distorting it into a
lemon-shaped form with the point towards the moon. It would be
permanently higher water at the points $A$ and $B$ (\figref{illo152}{Fig.~152}) directly
under the moon, and low water all around the earth on the circle $90°$
from these points, as at $D$ and $E$. The difference of the level of
the water at $A$ and $D$ would in this case be about two feet.
%% -----File: 294.png---Folio 283-------

\fussy
The sun's action would produce a similar tide superposed upon
the lunar tide and having about two-fifths of the same elevation, if
the two tide summits should coincide, the resulting elevation of the
high water would be the sum of the two separate tides. If the sun
were $90°$ from the moon, the waves would be in opposition, and the
height of the tide would be decreased, the solar tide partly filling up
the depression at the low water due to the moon's action.

Suppose now the earth to be put in rotation. It is easy to see
that these tidal waves would \textit{tend} to move over the earth's surface,
following the moon and sun at a certain angle dependent on the
inertia of the water, and with a westward velocity precisely equal to
that of the earth's eastward rotation,---about a thousand miles an
hour at the equator. But it is also evident that on account of the
varying depth of the ocean, and the irregular form of the shores, the
tides could not maintain this motion, and that the actual result must
become exceedingly complicated. In fact, the statical theory becomes
utterly unsatisfactory in regard to what actually takes place,
and it is necessary to depend almost entirely upon the results of
observation, using the theory merely as a guide in the discussion of
the observations.

\begin{fineprint}
Yet while this statical theory of the tides worked out by Newton is
certainly inadequate, and in some respects incorrect, it easily furnishes the
explanation of some of the most prominent of the peculiarities of the tides.

\nbarticle{470.} \nbparatext{The Priming and Lagging of the Tides.}---About the time
of new and full moon, as has been stated before (\artref{Art.}{463}), the interval
between the corresponding tides of successive days is about thirteen minutes
less than the average of $24^\text{h}\ 51^\text{m}$, while a week later it is about as much
longer. The reason is found in the combination of the solar and lunar tides.

\begin{center}
\begin{tabular}{c@{\qquad\qquad}c}
  \includegfx{illo153}&
  \includegfx{illo154}\\
  \footnotesize\textsc{Fig.~153.} & \footnotesize\textsc{Fig.~154.}
\end{tabular}
\captionof*{figure}{Priming and Lagging of the Tide.}
\end{center}

On the days of new and full moon the two tides coincide, and the tide
wave has its crest directly under the moon, or rather at the normal distance
behind the moon which corresponds to the ``establishment'' of the port of
observation.
%% -----File: 295.png---Folio 284-------

At quadrature the crest of the solar tide will be just $90°$ from the crest of
the lunar wave, but it will leave the summit of the \textit{combined wave} just where
it would be if there were no solar wave at all: evidently there is no possible
reason why the smaller wave at $S$ and $S'$ should displace the crest of the wave
at $L$ (\figref{illo153}{Fig.~153}) towards the right that would not also require its displacement
towards the left; it will therefore simply \textit{lower} the wave at $L$ \textit{without displacing
it} one way or the other. But when the solar tide wave $SS'$ (\figref{illo154}{Fig.~154})
has its crest at $S_1$ and $S_1'$, $45°$ from $L$ and $L'$, as it will do about three
days after new or full moon, then its combination with the lunar wave will
make the crest of the combined wave take position at a point $X$ between the
two crests, and about half an hour of time ahead (\textit{west}) of the lunar tide;
so that at that time of the month high water will occur about half an hour
\textit{earlier} than if there were no solar tide (since the tide waves travel westward).
And this half-hour has to be gained by diminishing the interval between
the successive tides for the three preceding days. Similar reasoning shows
that when the solar tide crest falls at $S_2$ and $S_2'$, the combined tide wave will
be \textit{east} of the lunar wave, and come later into port.

\includegraphicsouter{illo155}{\textsc{Fig.~155.}---The Diurnal Inequality.}

\nbarticle{471.} \nbparatext{Effect of the Moon's Declination and Diurnal Inequality.}---In
high latitudes on the Pacific Ocean, twice a month, when the moon is
farthest north or south of the celestial equator, the two tides of the day are
very different in magnitude. When the
moon's declination is zero, there is no
such difference: nor is there ever any
difference at ports which are near the
earth's equator.

Fig.~155 makes it clear why it should
be so. When the moon's declination is
zero, things are as in \figref{illo152}{Fig.~152} (\artref{Art.}{469}),
and the two tides of the same
day are sensibly equal at ports in all
latitudes. When the moon is at her
greatest northern declination, say $28°$,
the two tide summits will be at $A$ and $A'$ in \figref{illo155}{Fig.~155}; the tide which
occurs at $B$ when the moon is overhead will be great, while the tide in the
corresponding southern latitude at $B'$ will be small. The tides which
occur twelve hours later will be small at the northern station, then situated
at $C$, and large at the southern station, then at $C'$. For a port on the
equator at $E$ or $Q$ there will be no such difference. In the Atlantic Ocean
the difference is hardly noticeable, because, as we shall see very soon, the
tides in that ocean are mainly (not entirely) due to tide waves propagated
into it from the Pacific and Indian Oceans around the Cape of Good Hope.
\end{fineprint}

\nbarticle{472.} \nbparatext{The Wave Theory of the Tides.}---If the earth were entirely
covered with deep water, except a few little islands projecting here
and there to serve for observing stations, the tide waves would run
%% -----File: 296.png---Folio 285-------
around the globe \textit{regularly}; and if the depth of the ocean were over
thirteen miles, the tide crests, as can be shown, would follow the
moon at an angle of just $90°$. It would be high water just where the
statical theory would give low water. If the depth were (as it really
is) much less than thirteen miles, the tide wave in the ocean could
not keep up with the moon, and the result would be a very complicated
one. The real state of the case is still worse. The continents
of North and South America, with the southern antarctic continent,
make a barrier almost complete from pole to pole, leaving only a
narrow passage at Cape Horn; and the varying depth of the water,
and the irregular contours of the shores are such that it is quite
impossible to determine by theory what the course and character of
the tide wave must be. We must depend upon observation; and
observations are inadequate, because, with the exception of a few
islands, our only possible tide stations are on the shores of continents
where local circumstances largely control the phenomena.

\nbarticle{473.} \nbparatext{Free and Forced Oscillations.}---If the water in the ocean is
suddenly disturbed (as for instance, by an earthquake), and then
left to itself, a ``free'' wave will be formed, which, if the horizontal
dimensions of the wave are large as compared with the depth of the
water, will travel at a rate depending solely on the depth. The velocity
of such a free wave is given by the formula $v = \sqrt{gh}$; that is, it
is equal \textit{to the velocity acquired by a body in falling through half the
depth of the ocean}.

\begin{fineprint}
\begin{tabular} {@{} r@{ } c@{ } c@{ } c@{ } c@{ } c@{ } r@{} c@{ } c@{ } c@{ } c}
Thus a depth of 25 & feet & gives & a & velocity & of & 19 &$+$& miles & per & hour.\\
                100&``&``&``&``&``& 39&&``&``&``\\
             10,000&``&``&``&``&``&388&&``&``&``\\
             40,000&``&``&``&``&``&775&&``&``&``\\
             67,200&\multicolumn{3}{@{ }l@{ }}{($12\frac{3}{4}$
miles)}&``&``&1000&&``&``&``\\
             90,000&``&``&``&``&``&1165&&``&``&``\\
\end{tabular}
\end{fineprint}

Observations upon the waves caused by certain earthquakes in
South America and Japan have thus informed us that between the
coasts of these countries the Pacific averages between two and one-half
and three miles in depth.

\nbarticle{474.} Now, as the moon in its diurnal motion passes across the
American continent each day, and comes over the Pacific Ocean, it
starts such a ``parent'' wave in the Pacific, and the wave once
started moves on nearly (but not exactly) like an earthquake wave.
Not exactly, because the velocity of the earth's rotation being about
%% -----File: 297.png---Folio 286-------
1050 miles an hour at the equator, the moon runs relatively westward
faster than the wave can naturally follow, and so for a while slightly
accelerates it. A second tidal wave is produced daily twelve hours
later when the moon passes \textit{underneath}. The tidal wave is thus, \textit{in
its origin, a forced oscillation, while in its subsequent travel it is pretty
nearly a free one}.

\includegraphicssideways{illo156}{\textsc{Fig.~156.}---Map of Cotidal Lines.}

\nbarticle{475.} \nbparatext{Co-Tidal lines.}---These are lines drawn upon the surface
of the ocean connecting those places which have their high water at
the same moment of Greenwich time. They mark the crest of the
tide wave for each hour of Greenwich time; and if we could draw
them with certainty upon the globe, we should have all necessary
information as to the motion of the wave. Unfortunately we can
obtain no direct knowledge as to the position of these lines in mid-ocean;
we only get a few points here and there on the coasts and
on islands, so that a great deal necessarily remains conjectural. \figref{illo156}{Fig.~156}
is a reduced copy of such a map, borrowed with some modifications
from that given in Guyot's ``Physical Geography.''

\begin{fineprint}
\nbarticle{476.} \nbparatext{Course of Travel of the Tidal Wave.}---On studying the map
we find that the main or ``parent'' wave starts twice a day in the Pacific,
off Callao, on the coast of South America. This is shown on the chart
by a sort of oval ``eye'' in the co-tidal lines, just as a mountain summit is
shown on a topographical chart by an ``eye'' in the contour lines. From
this point the wave travels northwest through the deepest water of the
Pacific, at the rate of about 850 miles per hour, reaching Kamtchatka in
about ten hours. To the west and southwest the water is shallower and
the travel slower,---only 400 to 600 miles per hour,---so that the wave arrives
at New Zealand in about twelve hours. Passing on by Australia, and combining
with the small wave which the moon raises directly in the Indian
Ocean, the resultant tide crest reaches the Cape of Good Hope in about
twenty-nine hours, and enters the Atlantic. Here it combines with the tide
wave, twenty-four hours younger, which has ``backed'' into the Atlantic
around Cape Horn, and it is modified also by the \textit{direct tide} produced by the
moon's action upon the waters of the Atlantic. The resultant tide crest
then travels \textit{northward} through the Atlantic at the rate of nearly 700 miles
per hour. It is about forty hours old when it first reaches the coast of the
United States in Florida, and our coast is so situated that it arrives at all
the principal ports within two or three hours of that time. It is forty-one
or forty-two hours old when it arrives at New York and Boston. To reach
London it has to travel around the northern end of Scotland and through
the North Sea, and is nearly sixty hours old when it arrives at that port and
the ports of the German Ocean,---Hamburg, etc.

In the great oceans there are thus three or four tide crests travelling
%% -----File: 298.png---Folio 287-------
%% -----File: 299.png---Folio 288-------
simultaneously, following each other nearly in the same track, but with
continual minor changes, owing to the variations in the relative positions of
the sun and moon and their changing distances and declinations. If we take
into account the tides in rivers and sounds, the number of simultaneous
tide crests must be at least six or seven; that is, the high water at the
extremity of its travel, up the Amazon River, for instance, must be at least
three or four days old, reckoned from its birth in the Pacific.\footnote
  {We are greatly indebted to Loomis's discussion of the subject in his ``Elements
  of Astronomy.''}
\end{fineprint}

\nbarticle{477.} \nbparatext{Tides in Rivers.}---The tide wave ascends a river at a rate
which depends upon the depth of the water, the amount of friction,
and the swiftness of the stream. It may, and generally does, ascend
until it comes to a \textit{rapid, where the velocity of the water is greater than
that of the wave}. In shallow streams, however, it dies out earlier.

\begin{fineprint}
Contrary to what is usually supposed, it often ascends to an elevation far
above that of the highest crest of the tide wave at the river's mouth. In the
La Plata and Amazon it goes up to an elevation at least one hundred feet
above the sea-level. The velocity of the tide wave in a river seldom exceeds
ten or twenty miles an hour, and is usually less.
\end{fineprint}

\includegraphicsmid{illo157}{\textsc{Fig.~157.}---Increase in Height of Tide on approaching the Shore.}

\nbarticle{478.} \nbparatext{Height of Tides.}---In mid-ocean the difference between high
and low water is usually between two and three feet, as observed
on isolated deep-water islands in the Pacific; but on the continental
shores the height is usually much greater. As soon as the tide wave
touches bottom, so to speak, the velocity is diminished and the height
of the wave is increased, something as in the annexed figure (\figref{illo157}{Fig.~157}).
Theoretically the height varies \textit{inversely as the fourth root of
the depth}. Thus, where the water is 100 feet deep, the tide wave
should be twice as high as at the depth of 1600 feet.

Where the configuration of the shore forces the wave into a corner,
it sometimes becomes very high. At Annapolis, on the Bay of
Fundy, tides of seventy feet are not uncommon, and an altitude of
100 feet is said to be occasionally attained.

\begin{fineprint}
At Bristol, England, in the mouth of the Severn the tide rises fifty feet,
and sometimes ascends the river (as it also does the Seine, in France, and
%% -----File: 300.png---Folio 289-------
the Amazon) as a \textit{breaking} wave, called the ``bore'' or ``eiger'' (French,
\textit{mascaret}), with a nearly vertical front five or six feet in height, crested with
foam, and very dangerous to small vessels. On the east coast of Ireland,
opposite to Bristol, the tide ranges only about two feet.

In mid-ocean the water has no progressive motion, but near the land
it has, running in at the flood to fill up the bays and cover the flats, and
then running out again at the ebb. The velocity of these tidal \textit{currents} must
not be confounded with that of the tide wave itself.
\end{fineprint}

\nbarticle{479.} \nbparatext{Reflection and Interference.}---The tide wave when it reaches
the shore is not entirely destroyed, especially if the coast is bold and
the water deep; but is partly reflected, and the reflected wave goes
back into the ocean to meet and modify the new tide wave which is
coming in. Of course, in such a case we get ``interferences,'' so
that on islands in the Pacific only a few hundred miles apart we find
great differences in the heights of the tides. At one place the direct
waves and the waves reflected from the shores of Asia and South
America may conspire to give a tide of three or four feet, or nearly
double its normal value, while at another they nearly destroy each other.

\begin{fineprint}
There are places, also, which are reached by tides coming by two different
routes. Thus on the east coast of England and Scotland the tide waves
come both around the northern end of Scotland and through the Straits of
Dover. In some places on this coast we have, therefore, a tide of nearly
double height, while at others not very far away there will be hardly any
tide at all; and at intermediate points there are sometimes \textit{four} distinct
high waters in twenty-four hours. As a consequence of this reflection and
interference of the tide waves it follows that if the tide-raising power were
suddenly abolished, the tides would not immediately cease, but would continue
to run for several days, and perhaps weeks, before they gradually died
out.
\end{fineprint}

\nbarticle{480.} \nbparatext{Effect of the Varying Pressure of the Barometer, and of
the Wind.}---When the barometer at a given port is lower than
usual, the level of the water is generally higher than the average,
at the rate of about one foot for every inch of the mercury in the
barometer; and \textit{vice versa} when it is higher than usual.

When the wind blows into the mouth of a harbor, it drives in
the water of the ocean by its surface friction, and may raise the
water several feet. In such cases the time of high water, contrary
to what might at first be supposed, is \textit{delayed}, sometimes as much as
fifteen or twenty minutes.

This result depends upon the fact that the water runs into the
harbor for a longer time than it would do if the wind were not blowing.
%% -----File: 301.png---Folio 290-------
The normal depth of the water on the bar is reached \textit{before}
the predicted time, so that at the predicted time the water is deeper
than it would be if there were no wind, but the \textit{maximum} depth is
not attained until some time later. Of course, the results are the
opposite when the wind blows out of the harbor: the time of high
water comes earlier, and the depth of water on the bar at the predicted
time of high water is less than it otherwise would be.

\begin{fineprint}
\nbarticle{481.} \nbparatext{Tides in Lakes and Inland Seas.}---These are small and difficult
to detect. Theoretically, the range between high and low water in a
land-locked sea should bear about the same ratio to the rise and fall of the
tide in mid-ocean that the length of the sea does to the diameter of the earth.
Variations in the direction of the wind and the barometric pressure cause
continual oscillations in the water-level which, even in a quiet lake, are much
larger than the true tides; so that it is only by taking a long series of observations,
and discussing them with reference to the moon's position in the sky,
that it is possible to separate the real tide from the effects of other causes.
In Lake Michigan, at Chicago, a tide of about one and three-quarters inches
has thus been detected, the ``establishment'' of the port being about thirty
minutes. In Lake Erie, at Buffalo and Toledo, the tide is about three-quarters
of an inch. On the coasts of the Mediterranean the tide averages about
eighteen inches, attaining a height of three or four feet at the head of some
of the bays.
\end{fineprint}

\nbarticle{482.} \nbparatext{The Rigidity of the Earth.}---Sir W.~Thomson has endeavored
to make the tides the criterion of the rigidity of the earth's core.
Evidently if the solid parts of the earth were fluid, there would be no
observable tide anywhere, since the whole surface would rise and fall
together. If the earth were semi-solid, so to speak (that is, viscous,
and capable of yielding more or less to the forces tending to change
its form), the tides would be observable, but to a less degree than if
the earth's core were rigid. And with this further peculiarity---since
a viscous body requires time to change its form, waves of \textit{short period}
would be observable upon the semi-solid earth nearly to their full extent,
while those of \textit{long period} would almost entirely disappear,
owing to the slow yielding of the earth's crust. Now the actual tide
wave, as observed, is really made up of a multitude of component
tide waves of different periods, ranging from half a day upwards.
According to the ``principle of forced vibrations'' every regularly
recurring periodic change in the forces which act on the surface of
the ocean must produce a tide of greater or less magnitude, and of
exactly corresponding period.
%% -----File: 302.png---Folio 291-------

\begin{fineprint}
We have, for instance, the semi-diurnal, solar, and lunar tides; then the
two monthly tides due to the change in the moon's distance and declination,
and the two annual tides due to the changes of the sun's distance and
declination, not to speak of the nineteen-year tide due to the revolution of
the moon's nodes.
\end{fineprint}

A thorough analytical discussion of thirty-three years' tidal observations
at different parts of the world has been made under the
direction of Sir W.~Thomson by Mr.~George Darwin, with the result
that not only do the short waves show themselves, but \textit{the waves of
long period are found to manifest themselves with almost their full
theoretical value}. Thomson's conclusion is that the earth as a whole
``\textit{must be more rigid than steel, but perhaps not quite so rigid as
glass}.'' This result is at variance with the prevalent belief of geologists
that the core of the earth is a molten mass, and has led to much
discussion which we cannot deal with here.

\nbarticle{483.} \nbparatext{Effect of the Tides on the Earth's Rotation.}---If the tidal
motion consisted merely in the upward and downward motion of the
particles of the ocean to the extent of two feet or so twice a day, it
would involve a very trifling expenditure of energy; and this is the
case with the mid-ocean tide. But near the land this almost insensible
mere oscillatory motion is transformed into the bodily travelling
of immense masses of water, which flow in upon the shallows and
then out again to sea with a great amount of fluid friction; and this
involves the expenditure of a very considerable amount of energy
which is dissipated as heat. From what sources does this energy
come? The answer is that it must he derived \textit{mainly from the
earth's energy of rotation}, and the necessary effect is to diminish that
energy by lessening the speed of the rotation. Compared with the
earth's whole stock of rotational energy, however, the loss of it by
tidal friction, even in a century, is very small, and the effect on the
length of the day is extremely slight.

\begin{fineprint}
The reader will recall the remarks upon the subject of the secular acceleration
of the moon's mean motion a few pages back (\artref{Art.}{461}).
\end{fineprint}

While it is certain that the tidal friction \textit{tends} to lengthen the day,
it does not follow that the day really grows longer. There are
counteracting causes:---for example, the earth's radiation of heat
into space, and the consequent shrinkage of her volume.

As matters stand we do not know whether, \textit{as a fact}, the day is
really longer or shorter than it was a thousand years ago. The
%% -----File: 303.png---Folio 292-------
change, if any has really occurred, can  hardly be as great as $\frac{1}{1000}$
of a second.

\nbarticle{484.} \nbparatext{Effect of the Tide on the Moon's Motion.}---Not only does
the tide diminish the \textit{earth's} energy of rotation directly by the tidal
friction, but, theoretically, it also communicates
a minute portion of that energy to the \textit{moon}. It
will be seen that a tidal wave, situated as in \figref{illo158}{Fig.~158},
would slightly accelerate the moon's motion,
the attraction of the moon by the tidal protuberance
$F$ being slightly greater than that of the
tide wave at $F'$---a difference tending to draw
it along in its orbit a little, thus increasing the
major axis of the moon's orbit. The tendency
is therefore to make the moon \textit{recede} from the
earth, and to \textit{lengthen} the month.

\includegraphicsmid{illo158}{\textsc{Fig.~158.}\\
Effect of the Tide on the
Moon's Motion.}

Upon this interaction between the tides and
the motions of the earth and moon Professor
George Darwin has founded his theory of ``\textit{tidal
evolution}''; namely, that the satellites of a planet,
having separated from it millions of years ago,
have been made to recede to their present distances
by just such an action.

\nblabel{art:484*.}% typo in article 917 refers to this note to 484 as 484*
\begin{fineprint}
An excellent popular statement of the theory will be found in the closing
chapters of Ball's ``Story of the Heavens.'' The original papers of Mr.~Darwin
in the ``Philosophical Transactions'' are of course intensely mathematical.
\end{fineprint}
\chelabel{CHAPTERXII}

%% -----File: 304.png---Folio 293-------

\Chapter{XIII}{The Planets: their Motions, Apparent and Real}
\nbchapterhang{\stretchyspace
THE PLANETS: THEIR MOTIONS, APPARENT AND REAL.---THE
PTOLEMAIC, TYCHONIC, AND COPERNICAN SYSTEMS.---THE
ORBITS AND THEIR ELEMENTS.---PLANETARY
PERTURBATIONS.}

\nbarticle{485.} For the most part, the stars keep their relative configurations
unchanged, however much they alter their positions in the sky from
hour to hour. The ``dipper'' remains always a ``dipper'' in whatever
part of the heavens it may be. But while this is true of the
stars in general, certain of the heavenly bodies, and among them
those that are the most conspicuous of all, form an exception. The
sun and moon continually change their places, moving eastward
among the stars; and certain others, which to the eye appear as very
brilliant stars, also move,\footnote
  {When we speak of the motion of the planets, the reader will understand that
  the \textit{diurnal} motion is not taken into account. We speak of their motions \textit{among
  the stars}.}
though not in quite so simple a way.

\nbarticle{486.} These bodies were named by the Greeks the ``\textit{planets}''; that
is, ``wanderers.'' The ancient astronomers counted seven of them.
They reckoned the sun and moon, and in addition Mercury, Venus,
Mars, Jupiter, and Saturn.

Venus and Jupiter are at all times more brilliant than any of the
fixed stars. Mars at times, but not usually, is nearly as bright as
Jupiter; and Saturn is brighter than all but a very few of the stars.
Mercury is also bright, but seldom seen, because always near the sun.

At present the sun and moon are not reckoned as planets; but
the roll includes, in addition to the five other bodies known by the
ancients, the earth itself, which Copernicus showed should be counted
among them, and also two new bodies of great magnitude (though
inconspicuous because of their distance) which have been discovered
in modern times; then there is in addition a host of so-called ``\textit{asteroids}''
which circulate in the otherwise vacant space between the
planets Mars and Jupiter.
%% -----File: 305.png---Folio 294-------

\nbarticle{487.} The list of the planets in the order of distance from the sun
stands thus at present: Mercury, Venus, the Earth, Mars, Jupiter,
Saturn, Uranus, and Neptune; and between Mars and Jupiter, in the
place where a planet would naturally be expected to revolve, there
are at present known nearly 300 little planets, which probably represent
a single one, somehow ``spoiled in the making,'' so to speak, or
burst into fragments.

\sloppy
The planets are all dark bodies, shining only by reflected sunlight,---globes
which, like the earth, revolve around the sun in
orbits nearly circular, moving all in the same direction, and (with
some exceptions among the asteroids) nearly in the common plane of
the ecliptic and sun's equator. All of them but the inner two and the
asteroids are also attended by ``satellites.'' Of these the earth has one
(the moon), Mars two, Jupiter four, Saturn eight, Uranus four, and
Neptune one; \textit{i.e.}, so far as at present known; for although it is hardly
probable, it is not at all impossible that others may yet be found.

\nbarticle{488.} \nbparatext{Relative Distances of Planets from the Sun: Bode's Law.}---There
is a curious approximate relation between the distances
of the planets from the sun, which makes it easy to remember them.
It is usually known as Bode's Law, because Bode first brought it
prominently into notice in 1772, though Titius of Wittenberg seems
to have discovered and enunciated it some years earlier. The law is
this: Write a series of 4's. To the second 4 add 3; to the third add
$3 × 2$, or 6; to the fourth, $3 × 4$, or 12; and so on, doubling the
added number each time, as in the accompanying scheme.

\fussy
\begin{center}
\begin{tabular}{*{8}{c@{\qquad}}c}
  4 & 4 & \phantom{0}4 & \phantom{0}4 & \:\phantom{0}4\:
& \phantom{0}4 & \phantom{00}4 & \phantom{00}4 & \phantom{00}4
\\
& \underline{3\rule[-.5ex]{0pt}{0pt}}
& \underline{\phantom{0}6\rule[-.5ex]{0pt}{0pt}}
& \underline{12\rule[-.5ex]{0pt}{0pt}}
& \underline{\:24\:\rule[-.5ex]{0pt}{0pt}}
& \underline{48\rule[-.5ex]{0pt}{0pt}}
& \underline{\phantom{0}96\rule[-.5ex]{0pt}{0pt}}
& \underline{192\rule[-.5ex]{0pt}{0pt}}
& \underline{384\rule[-.5ex]{0pt}{0pt}}
\\
  4 & 7& 10 & 16 & [28] & 52 & 100 & 196 & 388
\\
  \Mercury & \Venus & \nbEarth &
  \Mars & \;\;\nbAsteroid &
  \Jupiter & \Saturn & \Uranus & \Neptune
\end{tabular}
\end{center}

The resulting numbers, divided by 10, are pretty nearly the true
mean distances of the planets from the sun, in terms of the radius
of the earth's orbit. In the case of Neptune, however, the law
breaks down utterly, and is not even approximately correct.

For the present, at least, the law is to be regarded as a mere
coincidence, there being so far no reasonable explanation of any
such numerical relation.

\begin{fineprint}
The general expression for the nth term of the series is $4 + 3 × 2^{(n-2)}$;
but it does not hold good of the first term, which is simply 4, instead of
being $5\frac{1}{2}$, \textit{i.e.}, $(4 + 3 × 2^{-1})$, as it should be.
% -----File: 306.png---Folio 295-------

\medskip
{\nblabel{art:489.}%\nbarticle{489.}% Article number is within table
\footnotesize
\noindent\centering
\renewcommand{\arraystretch}{1.2}
\begin{tabular} {@{}l @{\,}|@{\,} c @{\,}|@{\,} c @{\,}|@{\,} c @{\,}|@{\,} c @{\,}|@{\,} l@{\:}l @{\,}|@{\,} c@{}}
\multicolumn{8}{c}{\nbparatext{489. Table of Names, Distances, and Periods.}}\\[1ex]
\hline\hline
  \textsc{Name}. & \footnotesize\textsc{Symbol}.
& \footnotesize\textsc{Distance}. & \footnotesize\textsc{Bode}.
& \footnotesize\textsc{Diff}.
& \multicolumn{2}{c|@{\,}}{\footnotesize\textsc{Sid.~Period}.}
& \multicolumn{1}{@{}m{4em}@{}}{\centering \footnotesize\textsc{Syn.\\ Period}.}
%& \multicolumn{1}{c}{\centering \footnotesize\textsc{Syn.\ Period}.}
\\[.5ex]
\hline
  Mercury \dotfill & \Mercury & \phantom{0}0.387 & \phantom{0}0.4
& $-0.013\ $ & \ \ $88^{\text{d}}$ & or \quad $\phantom{\frac12}3^{\text{m}}$
& $116^{\text{d}}$\rule{0pt}{3ex}
\\
  Venus \dotfill & \Venus & \phantom{0}0.723 & \phantom{0}0.7
& $+0.023\ $ & $224.7^{\text{d}}$ & or \quad $7\frac{1}{2}^{\text{m}}$
& $584^{\text{d}}$
\\
  Earth \dotfill & \nbEarth & \phantom{0}1.000 & \phantom{0}1.0
& \phantom{+}$0.000\ $ & $365\frac{1}{4}^{\text{d}}$ & or $1^{\text{y}}$ & \dots
\\
  Mars \dotfill & \Mars & \phantom{0}1.523 & \phantom{0}1.6
& $-0.077\ $ & $687^{\text{d}}$ & or $1^{\text{y}}\, 10\frac{1}{2}^{\text{m}}$
& $780^{\text{d}}$
\\[.5ex]
\hline
  \multicolumn{1}{@{}m{4em}@{}|@{\,}}{Mean Asteroid}& & \phantom{0}2.650 & \phantom{0}2.8 & $-0.150\ $
& \multicolumn{2}{c|@{\,}}{$3^{\text{y}}.1$ to $8^{\text{y}}.0$}
& various\rule{0pt}{3ex}
\\[.5ex]
\hline
  Jupiter \dotfill & \Jupiter & \phantom{0}5.202 & \phantom{0}5.2 & $+0.002\ $
& \multicolumn{2}{c|@{\,}}{$11^{\text{y}}.9$}
& $399^{\text{d}}\rule{0pt}{3ex}$
\\
Saturn \dotfill & \Saturn & \phantom{0}9.539 & 10.0 & $-0.461\ $
& \multicolumn{2}{c|@{\,}}{$29^{\text{y}}.5$} & $378^{\text{d}}$
\\
Uranus \dotfill & \raisebox{\depth}{\fontseries{b}\uranus} \& \Uranus & 19.183 & 19.6 & $-0.417\ $
& \multicolumn{2}{c|@{\,}}{$84^{\text{y}}.0$} & $370^{\text{d}}$
\\
Neptune \dotfill & \Neptune & 30.054 & 38.8
& $-8.746\rlap{\,!}\ $
& \multicolumn{2}{c|@{\,}}{$164^{\text{y}}.8$}
& $\phantom{\frac12}367\frac{1}{2}^{\text{d}}$\\[1ex]
\hline\hline
\end{tabular}\\[1ex]
\renewcommand{\arraystretch}{1}%
}%end \footnotesize

The column headed ``Bode'' gives the distance according to Bode's law; the
column headed ``Diff.,'' the difference between the true distance and that given by
Bode's law.
%% -----File: 307.png---Folio 296-------

\nbarticle{490.} \figref{illo159}{Fig.~159} shows the smaller orbits of the system (including the orbit
of Jupiter) drawn to scale, the radius of the earth's orbit being taken as one
centimetre. On this scale the diameter of Saturn's orbit would be $19^{\text{cm}}.08$,
that of Uranus would be $38^{\text{cm}}.36$, and that of Neptune, $60^{\text{cm}}.11$. The
nearest fixed star on the same scale would be about a mile and a quarter
away. It will be seen that the orbits of Mercury, Mars, and Jupiter are
quite distinctly ``out of centre'' with respect to the sun. This is intentional
and correct. The dotted half of each orbit is that which lies below, \textit{i.e.},
south of, the plane of the ecliptic. The place of perihelion of each planet's
orbit is marked with a $P$. The orbits of five of the asteroids, including
the nearest and the most remote, as well as the most eccentric, are also
given.
\end{fineprint}

\textbf{Periods.}---The \textit{sidereal period} of a planet is the time of its revolution
around the sun from a star to the same star again, as seen from
the sun. The \textit{synodic period} is the time between two successive conjunctions
of the planet with the sun, as \textit{seen from the earth}. The
sidereal and synodic periods are connected by the same relation as
the sidereal and synodic months (\artref{Art.}{232}); namely,---
\[
\frac{1}{S} = \frac{1}{P} - \frac{1}{E},
\]
in which $E$, $P$, and $S$ are respectively the periods of the earth and of
the planet, and the planet's synodic period, and the numerical difference
between $\dfrac{1}{P}$ and $\dfrac{1}{E}$ is to be taken without regard to sign. The
two last columns of the table in \artref{Article}{489} give the approximate
periods, both sidereal and synodic, for the different planets.

\includegraphicsmid{illo159}{\textsc{Fig.~159.}---Plan of the Orbits of the Planets inside of Saturn.}

\nbarticle{491.} \nbparatext{Apparent Motions.}---As viewed from a distant point on the
line drawn through the sun, perpendicular to the plane of the ecliptic,
the planets would be seen to travel in their nearly circular orbits
with a regular motion. As seen from the earth the apparent motion
is much more complicated, being made up of their real motion around
the sun combined with an apparent motion due to the earth's own
movement.

\nbarticle{492.} \nbparatext{Law of Relative Motion.}---The motion of a body relative
to the earth can be very simply stated. \textit{It is always the same, as if
the body had, combined with its own motion, another motion, identical
with that of the earth, but reversed.}
%% -----File: 308.png---Folio 297-------

\includegraphicsmid{illo160}{\textsc{Fig.~160.}---The Relative Motions of Two Bodies.}

\begin{fineprint}
The proof of this is simple. Let $E$, \figref{illo160}{Fig.~160}, be the earth, and $P$ the
planet, its direction and distance being given by the line $EP$. Let $E$ have
a motion which will take it to $E'$ in a unit of time, and $P$ a motion which
will take it to $P'$ in the same time. Then at the end of a unit of time the
distance and direction of $P$ from $E$ will be given by the line $E'P'$. But if
we suppose $E$ to remain at rest, and give to $P$ a motion $Pe$ equal to $EE'$ but
opposite in direction, and combine this motion with $PP'$ by drawing the
parallelogram of motions, we shall get $P''$ for the resulting place of $P$ at
the end of the unit of time; and because the line $EP''$ is parallel and equal
to $E'P'$ (as follows from the construction), the point $P''$, as seen from $E$,
would occupy, in the celestial sphere, precisely the same position as $P'$ seen
from $E'$; since all parallel lines pierce the sphere at one and the same optical
point (\artref{Art.}{7}).
\end{fineprint}

If, therefore, the earth moves in
a circle, every body really at rest
will \textit{appear} to move in a circle of
the same size as the earth's orbit,
but keeping in such a part of its
circle as always to have its motion
precisely opposite to the earth's
own real motion at the moment.
We shall have occasion to use this
principle very frequently.

\nbarticle{493.} \nbparatext{Effect of the Combination
of the Earth's Motion with that of
the Planet.}---As a consequence,
the apparent ``\textit{geocentric}'' motion
of a planet must be made up of two motions,---that of a body moving
once a year around the circumference of a circle equal to the earth's
%% -----File: 309.png---Folio 298-------
orbit, while the centre of this circle itself goes around the sun upon
the real orbit of the planet, and with a periodic time equal to that of
the planet. Jupiter, for instance, appears to move as in \figref{illo161}{Fig.~161},
making eleven loops in each revolution, the smaller circle having a
diameter of about one-fifth that of the larger one, upon which its
centre moves, since the diameter of Jupiter's orbit is about five times
that of the earth's.

\includegraphicsouter{illo161}{\textsc{Fig.~161.}\\
Geocentric Motion of Jupiter from 1708 to
1720. (Cassini.)}

\nbarticle{494.} \nbparatext{Direct and Retrograde Motions of the Planets and Stationary
Points.}---As a consequence of this looped motion we have the peculiar
back-and-forth movement of the planets among the stars which
has been described. Starting from the time when the sun is between
us and the planet,---the time of superior conjunction,\footnote
  {We give \figref{illo162}{Fig.~162} to illustrate the meaning of the different terms, \textit{Opposition},
\textit{Quadrature}, \textit{Inferior and Superior Conjunction}, and \textit{Greatest Elongation}. $E$ is the
position of the earth, the inner circle being the orbit of an \textit{inferior} planet, while
the outer circle is the orbit of a \textit{superior} planet. In general, the angle $PES$ (the
angle at the earth between lines drawn from the earth to the planet and to the
sun) is the planet's \textit{elongation} at the moment. For a superior planet it can have
any value from zero to $180°$; for an inferior it has a maximum value that the
  planet cannot exceed, depending upon the diameter of its orbit.}
as it is called,
because the planet is then \textit{above} the sun, \textit{i.e.}, further from the earth,---the
%% -----File: 310.png---Folio 299-------
planet moves eastward among the stars for a certain time, continually
increasing its longitude (and also its right ascension) until at
last its apparent motion slackens and it becomes \textit{stationary}. The distance
of this stationary point from the sun depends upon the size of
the planet's orbit compared with that of the earth.

Then it reverses its motion and moves westward, or ``\textit{retrogrades},''
for a while, \textit{the middle of the arc of retrogression being passed at the
time when the earth and planet are in line with the sun, and on the
same side of it}. If the planet is one of the outer ones, it will then
be opposite to the sun in the sky like the full moon, and is said to be
``\textit{in opposition}.'' If the planet is one of the inferior planets (Venus
or Mercury), it will then be in ``\textit{inferior conjunction},'' as it is called,
between the earth and sun.

After the planet has completed its arc of retrogression, it again
becomes stationary, turns upon its course, and once more advances
eastward among the stars, until the synodic period is completed by
its re-arrival at superior conjunction.

\includegraphicsmid{illo162}{\textsc{Fig.~162.}---Planetary Configurations and Aspects.}

Both in the number of degrees passed over, and in the time spent
in this motion, the eastward or ``\textit{direct}'' motion always exceeds the
retrograde. In the case of the remoter planets the excess is small---from
$3°$ to $10°$; in the case of the nearest ones, Mars and Venus, it
is from $16°$ to $18°$.

As observed with a \textit{sidereal clock}, all the planets come \textit{later} to the
meridian each night when moving \textit{direct}, since their right ascension is
then increasing; but \textit{vice versa}, of course, when they are \textit{retrograding}.

\nbarticle{495.} \nbparatext{Motion of the Planets with Respect to the Sun's Place in the
Sky. Change of Elongation.}---The visibility of a planet depends
mainly upon its angular distance, or ``\textit{elongation},'' from the sun, because
when near the sun the planet will be above the horizon only by
day, and cannot usually be seen. As regards their motions, considered
from this point of view, there is a marked difference between
the inferior planets and the superior.

\nbarticle{496.} \nbparatext{Behavior of a Superior Planet.}---The superior planets drop
always steadily \textit{westward} with respect to the sun's place in the heavens,
continually increasing their western elongation, or decreasing their
eastern: they therefore \textit{invariably come earlier to the meridian every
successive night}, as observed by a time-piece \textit{keeping solar time}.

\begin{fineprint}
Beginning at superior conjunction, the planet is then moving eastward
among the \textit{stars} with its greatest speed; but even then its eastward motion
%% -----File: 311.png---Folio 300-------
is not so great as the sun's, and so the planet \textit{relatively} falls westward. After
a while it will have fallen behind by $90°$, and will then be in western quadrature,
and on the meridian at sunrise; at the end of half its synodic period
it will have lost $180°$, and will be just opposite the sun at sunset, being then
at its least possible distance from the earth, and at its greatest brilliance.
At this time the difference between the times of its daily culminations is
also the greatest possible, and may be as much (in the case of Mars) as six
minutes, by which amount it arrives at the meridian earlier each successive
night. After opposition the planet is higher in the sky each night at sunset
until it reaches eastern quadrature, when it is $90°$ east of the sun, and
therefore on the meridian at sunset. Thence it drops back, falling more
and more slowly westwards towards the sun, until the synodic period is
completed by a new conjunction.
\end{fineprint}

\nbarticle{497.} \nbparatext{Motion of an Inferior Planet.}---The inferior planets appear,
on the other hand, to \textit{vibrate} across the sun, moving out equal distances
on each side of it, but making the westward swing much
quicker than the eastern.

The reason of this difference is obvious from \figref{illo162}{Fig.~162}. Matters
take place with respect to the earth, sun, and planet as if the earth
were at rest, and the planet revolving around the sun once in a \textit{synodic}
(not sidereal) period. Now, since the distance between the
points of greatest elongation, $V$ and $V'$, is less through inferior conjunction
$I$, than from $V'$ around to $V$ through $C$, the time ought to
be correspondingly shorter, as it is.

\begin{fineprint}
At superior conjunction the planet is moving eastward \textit{faster} than the
sun. Accordingly, it creeps out to the east from the sun's rays, becoming
visible in the twilight as an evening star. As long as its direct motion is
greater than the sun's it keeps receding from it until it reaches its ``\textit{greatest
eastern elongation},'' as it is called, which could in no case be greater than $90°$,
even if the planet's distance from the sun were almost equal to that of the
earth. (In the case of Venus it is actually about $47°$, while for Mercury it
ranges from $18°$ to $28°$.) Then as its eastern motion slackens the sun begins
to overtake it, and when the planet becomes really stationary as regards its
motion among the \textit{stars}, it appears to be slipping westward towards the \textit{sun}
at the rate of about a degree a day. At the stationary point it begins really
to retrograde, and \textit{adds} its motion to the sun's advance, so that from that
point it rushes swiftly towards the inferior conjunction. It passes this and
runs out quickly on the western side, becoming a morning star, and reaching
its western elongation in just the same number of days that it took to
drop from eastern elongation to inferior conjunction. When the elongation
has been gained, the planet turns around to pursue the sun, gradually gains
upon it, and at last overtakes it again at the next superior conjunction, having
completed its synodic period.
\end{fineprint}
%% -----File: 312.png---Folio 301-------

\nbarticle{499.} \nbparatext{Motions in Latitude.}---If the planets' orbits lay precisely in
the same plane with each other and with the earth's orbit, they would
always keep in the ecliptic. But in fact, while they never go \textit{far}
from that circle, they do deviate north or south to the extent of
$5°$ or $6°$, and Mercury sometimes as much as $8°$; their paths among
the stars are consequently loops and kinks like those of \figref{illo163}{Fig.~163},
which represent forms actually observed.

\includegraphicsmid{illo163}{\textsc{Fig.~163.}---Loops of Regression.}

\nbarticle{500.} \nbparatext{The Ptolemaic System.}---The ancient astronomers, for the
most part, never doubted the fixity of the earth, and its position in
the centre of the celestial universe, though there are some reasons to
think that Pythagoras may have done so. Assuming this and the
actual diurnal revolution of the heavens, Ptolemy, who flourished at
Alexandria about 140~\textsc{a.d.}, worked out the system which bears his
name. His \mytextgreek{Meg'alh S'untaxic} (or \textit{Almagest} in Arabic) was for fourteen
centuries the authoritative ``Scripture of Astronomy.'' He showed
that all the apparent motions of the planets could be accounted for by
supposing each planet to move around the circumference of a circle
called the ``\textit{epicycle},'' while the centre of this circle, sometimes
called the ``\textit{fictitious planet},'' itself moved on the circumference of
another and larger circle called the ``\textit{deferent}.'' It was as if the
real planet was carried on the end of a crank-arm which turned
around the fictitious planet as a centre, in such a way as to point
towards or from the earth at times when the planet is in line with the
sun.

\begin{fineprint}
In the case of the superior planets the revolution in the epicycle was
made once a year, so that the ``crank-arm'' was always parallel to the line
joining earth and sun, while the motion around the deferent occupied what
we now call the planet's period. \figref{illo164}{Fig.~164} represents the Ptolemaic System,
except that no attention is paid to dimensions, the ``deferents'' being spaced
%% -----File: 313.png---Folio 302-------
at equal distances. It will be noticed that the epicycle-radii which carry at
their extremities the planets Mars, Jupiter, and Saturn are all always parallel
to the line that joins the earth and sun. In the case of Venus and Mercury
this was not so. Ptolemy supposed that the deferent circles for these planets
lay \textit{between} the earth and the sun, and that the ``fictitious planet'' in both
cases revolved in the \textit{deferent} once a year, always keeping exactly between
the earth and the sun: the motion in the \textit{epicycle} in this case was completed
in the time of the planet's period, as we now know it. He ought to have
seen that, for these two planets, the deferent was really the orbit of the
sun itself, as the ancient Egyptians are said to have understood.

\includegraphicsmid{illo164}{\textsc{Fig.~164.}---The Ptolemaic System.}

\nbarticle{501.} To account for some of the irregularities of the planets' motions
it was necessary to suppose that both the deferent and epicycle, though
circular, are \textit{eccentric}, the earth not being exactly in the centre of the
deferent, nor the ``fictitious planet'' in the exact centre of the epicycle.
In after times, when the knowledge of the planetary motions had become
more accurate, the Arabian astronomers added epicycle upon epicycle until
the system became very complicated. King Alphonso of Spain is said to
have remarked to the astronomers who presented to him the Alphonsine
tables of the planetary motions, which had been computed under his orders,
that ``if he had been present at the creation he would have given some good
advice.''

\nbarticle{502.} Some of the ancient astronomers attempted to account for the planetary
and stellar motions in a mechanical way by means of what were called
the ``\textit{crystalline spheres}.'' The planet Jupiter, for instance, was supposed to
%% -----File: 314.png---Folio 303-------
be set like a jewel on the surface of a small globe of something like glass, and
this itself was set in a hollow made to fit it in the thick shell of a still larger
sphere which surrounded the earth. Thus the planets were supported and
carried by the motions of these invisible crystalline spheres; but this idea,
though prevalent, was by no means universally accepted.
\end{fineprint}

\sloppy
\nbarticle{503.} \nbparatext{Copernican System.}---Copernicus (1473--1543) asserted the
diurnal rotation of the earth on its axis, and showed that it would
fully account for the apparent diurnal revolution of the stars. He
also showed that nearly all the known motions of the planets could
be accounted for by supposing them to revolve around the sun, with
the earth as one of them, in orbits circular, but slightly out of centre.
His system, as he left it, was \textit{nearly} that which is accepted to-day,
and \figref{illo159}{Fig.~159} may be taken as representing it. He was, however,
obliged to retain a few small epicycles to account for certain of the
irregularities.

\fussy
So far, no one dared to doubt the exact circularity of celestial
orbits. It was metaphysically improper that \textit{heavenly} bodies should
move in any but \textit{perfect} curves, and the circle was regarded as the
only perfect one. It was left for Kepler, some sixty-five years later
than Copernicus, to show that the planetary orbits are \textit{elliptical}, and to
bring the system substantially into the form in which we know it now.

\begin{fineprint}
\sloppy
\nbarticle{504.} \nbparatext{Tychonic System.}---Tycho Brahe, who came between Copernicus
and Kepler, found himself unable to accept the Copernican system for two
reasons. One reason was that it was unfavorably regarded by the clergy,
and he was a good churchman. The other was the scientific objection that
if the earth moved around the sun, the \textit{fixed stars all ought to appear to move
in a corresponding manner} (\artref{Art.}{492}), each star describing annually an oval
in the heavens of the same apparent dimensions as the earth's orbit itself,
seen from the star. Technically speaking, they ought to have an ``\textit{annual
parallax}.'' His instruments were by far the most accurate that had so far
been made, and he could detect no such parallax; hence he concluded, not
illogically, but incorrectly, that the earth must be at rest. He rejected the
Copernican system, placed the earth at the centre of the universe, according
to the then received interpretation of Scripture, made the sun revolve around
the earth once a year, and then (this was the peculiarity of his system) made
all the planets except the earth revolve around the sun.

\fussy
This theory just as fully accounts for all the motions of the planets as
the Copernican, but breaks down absolutely when it encounters the aberration
of light, and the annual parallax of the stars, which we can \textit{now} detect
with our modern instruments, although Tycho could not with his. The
Tychonic system never was generally accepted; the Copernican was very
soon firmly established by Kepler and Newton.
\end{fineprint}
%% -----File: 315.png---Folio 304-------

\nbarticle{505.} \nbparatext{Elements of a Planet's Orbit.}---Those elements are the
numerical quantities which must be given in order to describe the
orbit with precision, and to furnish the means of finding the planet's
place in the orbit at any given time, whether past or future. They
are seven in number, as follows:---

1. The semi-major axis, $a$.

2. The eccentricity, $e$.

3. The inclination to the ecliptic, $i$.

4. The longitude of the ascending node, $\nbAscnode$.

5. The longitude of the perihelion, $\pi$.

6. The epoch, $E$.

7. The period $P$, or daily motion $\mu$.

\includegraphicsmid{illo165}{\textsc{Fig.~165.}---The Elements of a Planet's Orbit.}

\nbarticle{506.} Of these, the first five pertain to the orbit itself, regarded as
an ellipse lying in space with one focus at the sun, while two are
necessary to determine the planet's place in the orbit.

The \textit{semi-major axis}, $a$ ($CA$ in \figref{illo165}{Fig.~165}), defines the \textit{Size} of the
orbit, and may be expressed either in ``astronomical units'' (the
earth's mean distance from the sun is the astronomical unit) or in
miles.

The \textit{Eccentricity} defines the orbit's \textit{Form}. It is a mere numerical
quantity, being the fraction $\dfrac{c}{a}$ (usually expressed decimally), obtained
by dividing the distance between the sun and the centre of the orbit
by the semi-major axis. In some computations it is convenient to
use, instead of the decimal fraction itself, the angle $\phi$ which has $e$ for
its sine; \textit{i.e.}, $\phi = \sin^{-1}{e}$.
%% -----File: 316.png---Folio 305-------

The third element, $i$, is the \textit{Inclination} between the plane of the
planet's orbit and that of the earth. In the \figref{illo165}{figure} it is the angle
$KNO$, the plane of the ecliptic being lettered $EKLM$.

The fourth element, \nbAscnode (\textit{the Longitude of the ascending node}),
defines what has been called the ``\textit{aspect}'' of the orbit-plane; \textit{i.e.},
the direction in which it faces. The line of nodes is the line $NN'$ in
the \figref{illo165}{figure}, the intersection of the two planes of the orbit and ecliptic;
and the angle $\aries SN$ is the longitude of the ascending node, the
line $S \aries$ being the line drawn from the sun to the first of Aries. The
planet, moving around its orbit in the plane $ORBT$, and in the direction
of the arrow, passes from the lower or southern side of the plane
of the ecliptic to the northern at the point $n$, which, as seen from $S$,
is in the same direction as $N$.

The fifth, and last, of the elements which belong strictly to the
orbit itself is $\pi$, the so-called \textit{Longitude of the perihelion}, which defines
the \textit{direction} in which the major axis of the ellipse (the line $PA$)
lies on the plane $ORBT$. It is not strictly a \textit{longitude}, but equals
the sum of the two angles $\nbAscnode$ and $\omega$; \textit{i.e.}, $\aries SN$ (in the plane of the
ecliptic) $+\ NSP$ (in the plane of the orbit). It is quite sufficient
to give $\omega$ alone, and in the case of cometary orbits this is usually
done.

\nbarticle{507.} If we regard the orbit as an oval wire hoop suspended in
space, these five elements completely define its position, form, and
size. The \textit{plane} of the orbit is fixed by the elements numbered three
and four, the \textit{position} of the orbit in this plane by number five, the
\textit{form} of it by number two, and finally its \textit{magnitude}, by number one.

The student will recollect that the general equation of curves
of the second degree (the conics) in analytical geometry contains five
constants, and therefore that number of data is enough to define such
a curve completely.

\nbarticle{508.} To determine where the planet will be at any subsequent date
we need two more elements.

Sixth. \textit{The Periodic Time},---we must have the \textit{sidereal period}, $P$,
or else \textit{the mean daily motion}, $\mu$, which is simply $360°$ divided by
the number of days in $P$.

\sloppy
Seventh. And finally; we must have a starting-point, the ``\textit{Epoch},''
so-called; \textit{i.e.}, the longitude of the planet as seen from the sun, at
some given date, usually Jan.~1st, 1850 or 1900, or else some precise
date at which the planet passed the perihelion or node.
%% -----File: 317.png---Folio 306-------

\fussy
\begin{fineprint}
\nbarticle{509.} If it were not for perturbations caused by the mutual interaction
between the planets, these elements would never change, and could be used
directly for computing the planet's place at any date in the past or in the
future; but, excepting $a$ and $P$, they do change on account of such interaction,
and accordingly it is usual to add in tables of the planetary elements,
columns headed $\Delta\nbAscnode$, $\Delta\pi$, $\Delta i$, and $\Delta e$, giving the amount by which the
quantities $\nbAscnode$, $\pi$, $i$, and $e$ respectively change in a century.

\nbarticle{510.} If Kepler's harmonic law were strictly true, we should not need
both $a$ and $P$, because we should have
\[
(\text{Earth's Period})^{2}:P^{2}::1^{3}:a^{3}, \text{or } P = a^{\frac{3}{2}},
\]
$P$ being expressed in years and $a$ in astronomical units. But since the exact
form of the equation is
\[
{P_{1}}^{2}(1 + m_{1}):{P_{2}}^{2}(1 + m_{2})::{a_{1}}^{3}:{a_{2}}^{3}\ (\text{\artref{Art.}{417}}),
\]
it is necessary, in cases where the highest attainable accuracy is required, to
regard $P$ and $a$ as independent, and give them both in the tables.
\end{fineprint}

\nbarticle{511.} \nbparatext{Geocentric Place.}---Our \textit{observations} of a planet's place are
necessarily ``\textit{geocentric},'' or earth-centred; they give us, when properly
corrected for refraction and parallax, the planet's \textit{right ascension}
and \textit{declination} as seen from the centre of the earth, and from them,
if desired, the corresponding geocentric \textit{longitude} and \textit{latitude} are
easily obtained by the method explained in \artref{Article}{180}.

\begin{fineprint}
It often happens that we want the place at some moment of time when the
planet could not be directly observed, as, for instance, in the day time. If we
have a series of observations of the planet made about that time, the place for
the exact moment is readily deduced by a process of \textit{interpolation}, and with
an accuracy actually exceeding that of any single observation of the series.

Graphically it is done by simply plotting the observations actually made.
Suppose, for instance, we want the right ascension of Mars for 8~\textsc{a.m.} on
June~3, and have meridian-circle observations made at 10 o'clock \textsc{p.m.} on
June~1, at $9^{\text{h}}\ 55^{\text{m}}$ on June~2, at $9^{\text{h}}~50^{\text{m}}$ on June~3, and so on. We first lay off
the times of observation as abscissas along a horizontal line taken as the
time-scale, and then lay off the observed right ascensions as ordinates at
points corresponding to the times. Then we draw a smooth curve through
the points so determined, and from this curve we can read off directly, the
right ascension corresponding to any desired moment: similarly for the
declination. Of course, what can be done graphically can be done still
more accurately by computation.
\end{fineprint}

\nbarticle{512.} \nbparatext{Heliocentric Place.}---The \textit{heliocentric} place of a planet is the
place as seen from the sun; and when we know the longitude of the
node of a planet's orbit and its inclination, as well as the planet's distance
%% -----File: 318.png---Folio 307-------
from the sun, this heliocentric place can at once be deduced
from the geocentric by a trigonometrical calculation. The process is
rather tedious, however, and its discussion lies outside the scope of
this work.

\begin{fineprint}
(The reader is referred to Watson's ``Theoretical Astronomy,'' p.~86. An
elementary geometrical treatment of the reduction is also given in Loomis's
``Treatise on Astronomy,'' p.~211.)
\end{fineprint}

\nbarticle{513.} \nbparatext{Determination of a Period of a Planet.}---This can be done
in two ways:

First. \textit{By observation of its node-passage.} When the planet is
passing its node, it is in the plane of the ecliptic, and the earth being
also always in that plane, the planet's latitude, \textit{both geocentric and
heliocentric}, will be zero, no matter what may be the place of the earth
in its orbit. (At any other point of the planet's orbit except the node
its apparent latitude would not be thus independent of the earth's
place, but would vary according to its distance from the earth.) If,
then, we observe the planet at two successive passages of the same
node, the interval between the moments when the latitude becomes
zero will be the planet's period,---\textit{exactly}, if the node is stationary;
very \textit{approximately}, even if the node is not absolutely stationary, as
none of the nodes actually are.

\begin{fineprint}
There are two difficulties with this method.

(\textit{a}) In the case of Uranus the period is eighty-four years, and in that of
Neptune 164 years---too long to wait.

(\textit{b}) Since the orbits all cross the ecliptic at a very small angle, so that the
latitude remains near zero for a number of days, it is extremely difficult to
determine the precise minute and second when it is exactly zero; and slight
errors in the declinations observed will produce great errors in the result.
\end{fineprint}

\nbarticle{514.} Second. \textit{By the mean synodic period of the planet.} The
synodic period is the interval between two successive oppositions or
conjunctions of the planet, the opposition being the moment when
the planet's longitude differs from that of the sun by $180°$.

\begin{fineprint}
This angle between the planet and sun cannot well be measured directly,
but we can make with the meridian circle a series of observations both of
the planet's right ascension and declination for several days before and after
the date of opposition, and reduce the observations to latitude and longitude.
The sun will be observed, of course, at noon, and the planet near midnight;
but from the solar observations we can deduce the longitudes of the sun
corresponding to the exact moments when the planet was observed. From
these, we find the difference of longitude between the planet and the sun at
the time of each planetary observation; and finally from these differences
%% -----File: 319.png---Folio 308-------
of longitude, we find the precise moment when that difference was exactly
$180°$, or the moment of opposition. This can be ascertained within a very
few seconds of time if the observations are good.
\end{fineprint}

Since the orbits are not strictly circular, the interval between two
successive observations will not be the \textit{mean} synodic period, but only
an approximation to it; but when we know it \textit{nearly}, we can compare
oppositions many years apart, and by dividing the interval by the
known number of entire synodic periods (which is easily determined
when we know the approximate length of a single period) we get the
mean synodic period very closely,---especially if the two oppositions
occur at about the same time of the year. Having the synodic
period, the true sidereal period at once follows from the equation
\[
\frac{1}{P} = \frac{1}{E} - \frac{1}{S}.
\]

\nbarticle{515.} \nbparatext{To find the Distance of a Planet in Terms of the Earth's
Distance.}---When we know the planet's sidereal period, this is easily
done by means of two observations of the planet's ``\textit{elongation}''
taken at an interval equal to its periodic time. The ``elongation''
of a planet is the difference between its longitude and that of the
%% -----File: 320.png---Folio 309-------
sun, and a series of meridian-circle observations of sun and planet
will furnish these differences of longitude for any selected moment
included within the term of observation.

\includegraphicsmid{illo166}{\textsc{Fig.~166.}---Determination of the Distance of a Planet from the Sun.}

To find the distance of the planet Mars, for instance, we must
therefore have two observations separated by an interval of 687 days.
Suppose the earth to have been at $A$ (\figref{illo166}{Fig.~166}) at the moment of the
first observation. Then at the time of the second observation she
will be at the point $C$, the angle $ASC$ being that which the earth
will describe in the next $43\frac{1}{2}$ days, which is the difference between
two complete years (or $730\frac{1}{2}$ days) and the 687-day interval between
the two observations.

The angles $SCM$ and $SAM$ are the ``elongations'' of the planet
from the sun, and are given directly by the observations. The two
sides $SA$ and $SC$ are also given, being the earth's distance from
the sun at the dates of observation. Hence we can easily solve
the quadrilateral, and find the length of $SM$, as well as the angle
$ASM$.

\begin{fineprint}
This angle determines the planet's \textit{heliocentric} longitude at $M$, since
we know the direction of $SA$, the longitude of the earth at the time of
observation.

The student can follow out for himself the process by which, from two
elongations of Venus, $SAV$ and $SBV$, observed at an interval of 225 days,
the distance of Venus from the sun (or $SV$) can be obtained.

\includegraphicsouter{illo167}{\textsc{Fig.~167.}}

\nbarticle{516.} In order that this method may apply with strict accuracy it is
necessary that at the moment of observation $M$ should be in the same
plane as $A$, $S$, and $C$; that is, at the node. If it is not so, the process will
give us, not the true distance of the planet
itself from the sun, but that of the ``projection''
of this distance on the plane of
the ecliptic; \textit{i.e.}, the distance from the sun
to the point $m$ (\figref{illo167}{Fig.~167}), where the perpendicular
from the planet would strike
that plane. But when we have determined
$Am$ and the angle $mAM$, the planet's geocentric
latitude, we easily compute $Mm$; and from $Sm$ and $Mm$ we get the
true distance $SM$ and the heliocentric latitude of the planet $MSm$.
\end{fineprint}

\nbarticle{517.} From a series of pairs of observations distributed around
the planet's orbit it would evidently be possible to work out the
orbit completely. It was in this way that Kepler showed that the
orbits of the planets are ellipses, and deduced their distances from
%% -----File: 321.png---Folio 310-------
the sun; and his third, or harmonic law, was then discovered simply
by making a comparison between the distances thus found and the
corresponding periods.

\includegraphicsouter{illo168}{\textsc{Fig.~168.}\\
Distance of an Inferior Planet determined by
Observations of its Greatest Elongation.}

\nbarticle{518.} \nbparatext{Mean Distance of an Inferior Planet by Means of Observations
of its Greatest Elongation.}---By observing from the earth the
greatest elongation $SEV$ (\figref{illo168}{Fig.~168})
of one of the inferior planets,
its distance from the sun can very
easily be deduced if we regard the
orbit as a circle; for the triangle
$SVE$ will be right-angled at $V$, and
$SV = SE × \sin{SEV}$.

\begin{fineprint}
In the case of Venus the orbit is
so nearly circular that the method
answers very well, the greatest elongation
never differing much from $47°$.
Mercury's orbit is so eccentric that the
distance thus obtained from a single
elongation might be very wide of the true mean distance. Since the greatest
elongation, $SEM$, varies all the way from $18°$ to $28°$, it would be necessary
to observe a great many elongations, and take the average result.
\end{fineprint}

\nbarticle{519.} \nbparatext{Deduction of the Orbit of a Planet from Three Observations.}---When
one has command of a great number of observations of a
planet running back many years, and can select such as are convenient
for his purpose, as Kepler could from Tycho's records, it is
comparatively easy to find the elements of a planet's orbit; but when
a new planet is discovered, the case is different. The problem first
arose practically in 1801, when Ceres, the first of the asteroids, was
discovered by Piazzi in Sicily, observed for a few weeks and then
lost in the sun's rays at conjunction, before other astronomers could
be notified of the discovery, in these days of slow communication,
made slower and more uncertain by war.

Gauss, then a young man at Göttingen, attacked the problem, and
invented the method which, with slight modifications, is now universally
used in such cases.

We do not propose to enter into details, but simply say that
\textit{three absolutely accurate observations of a planet's right ascension and
declination are sufficient to determine its orbit}. Three observations,
made only as accurately as is now possible, with intervals of two or
%% -----File: 322.png---Folio 311-------
three weeks between them, will give a very good \textit{approximation} to
the orbit; and it can then be corrected by further observations.

\begin{fineprint}
\nbarticle{520.} Since there are \textit{five} independent coefficients in the equation of a
conic, it is necessary to have \textit{five} conditions in order to determine them.
\textit{Three} such conditions are given by the observations themselves; viz., the
three directions of the planet as seen from the earth at the given moments
of observation; a \textit{fourth} is given by the fact that the sun must be in the
focus of the orbit; and finally, the \textit{fifth} is supplied by the ``law of equal
areas,'' since the areas described by the radius vector between the first and
second observations, and the second and third, must be proportional to the
corresponding intervals of time.

(The student is referred to Gauss's ``Theoria Motus,'' or to Watson's
``Theoretical Astronomy,'' or to Oppolzer's great work on ``The Determination
of Orbits,'' for the full development of the subject.)
\end{fineprint}

\nbarticle{521.} \nbparatext{Planetary Perturbations.}---The attraction of the planets for
each other disturbs their otherwise elliptical motion around the sun.
As in the case of the lunar theory the disturbing forces are, however,
always relatively small, but not for the same reason. The sun's
disturbing force is small because its \textit{distance from the moon is nearly
four hundred times that of the earth}. In the planetary theory the
disturbing bodies are often nearer to the disturbed than is the sun
itself, as, for instance, in the disturbance of Saturn by Jupiter at
certain points of their orbits; but the \textit{mass of the disturbing body in
no case is as great as $\frac{1}{1000}$ part of the sun's mass}, and for this reason
the disturbing force arising from planetary attraction is never more
than a small fraction of the sun's attraction.

\begin{fineprint}
The greatest disturbing force which occurs in the planetary system
(except in the case, of some of the asteroids) is that of Jupiter on Saturn
at the time when the planets are nearest: it then amounts to $\frac{1}{128}$ of the
sun's attraction. When these two planets are most remote from each other,
it amounts to $\frac{1}{357}$. There is no other case where the disturbing force is as
much as $\frac{1}{1000}$ of the sun's attraction (again excepting the asteroids disturbed
by Jupiter).
\end{fineprint}

\nbarticle{522.} In any special case the disturbing force can he worked out
on precisely the same principles that lie at the foundation of the
diagram by which the sun's disturbing force upon the moon was
found (\artref{Art.}{441}, \figref{illo147}{Fig.~147}); but the resulting diagram will look very
differently, because the disturbing body is relatively very near the
disturbed orbit.
%% -----File: 323.png---Folio 312-------

The planetary perturbations which result from the ``integration''
of effects of the disturbing forces, \textit{i.e.}, from their continual action
through long intervals of time, divide themselves into two great
classes,---the \textit{Periodic} and the \textit{Secular}.

\nbarticle{523.} \nbparatext{Periodic Perturbations.}---These are such as depend on the
positions of the planets in their orbits, and usually run through their
course in a few revolutions of the planets concerned. For the most
part they are very small. Those of Mercury never amount to more
than $15''$, as seen from the sun. Those of Venus may reach about
$30''$, those of the earth about $1'$, and these of Mars about $2'$. The
mutual disturbances between Jupiter and Saturn are much larger,
amounting respectively to $28'$ and $48'$; while those of Uranus are
again small, never exceeding $3'$, and those of Neptune are not more
than half as great as that. In the case of the asteroids, which are
powerfully disturbed by Jupiter, the periodical perturbations are
enormous, sometimes as much as $5°$ or $6°$.

\begin{fineprint}
\nbarticle{524.} \nbparatext{Long Inequalities.}---The periodic inequalities of the planets are
so small, because, as a rule, there is a nearly complete compensation effected
at every few revolutions, so that the accelerations balance the retardations.
The line of conjunction falls at random in different parts of the orbits, and
when this is the case, no considerable displacement of either planet can
take place. But when the periodic times of two planets are \textit{nearly commensurable},
their line of conjunction will fall very near the same place in
the two orbits for a considerable number of years, and the small unbalanced
disturbance left over at each conjunction will then accumulate in the same
direction for a long time. Thus, five revolutions of Jupiter roughly equal
two of Saturn; and still more nearly, seventy-seven of Jupiter equal thirty-one
of Saturn, in a period of 913 years. From this comes the so-called
``\textit{long inequality}'' of Jupiter and Saturn, amounting to $28'$ in the place of
Jupiter and $48'$ in that of Saturn, and requiring more than 900 years to
complete its cycle.

In the case of the earth and Venus there is a similar ``long inequality''
with a period of 235 years, amounting, however, to less than $3''$ in the positions
of either of the planets; between Uranus and Neptune there is a similar
inequality with a period of over 4000 years, but this also is very small.
\end{fineprint}

\nbarticle{525.} \nbparatext{Secular Inequalities.}---These are inequalities which depend
not on the position of the planets in their orbits, but \textit{on the relative
position of the orbits} themselves, with reference to each other,---the
way, for instance, in which the \textit{lines of nodes} and \textit{apsides} of two
neighboring orbits lie with reference to each other. Since the planetary
orbits change their positions very slowly, these perturbations,
%% -----File: 324.png---Folio 313-------
although in the strict sense of the word periodic also, are very slow
and majestic in their march, and the periods involved are such as
stagger the imagination. They are reckoned in myriads and hundreds
of thousands of years. From year to year they are insignificant,
but with the lapse of time become important.

\nbarticle{526.} \nbparatext{Secular Constancy of the Periods and Mean Distances.}---It is
a remarkable fact, demonstrated by Lagrange and La~Place
about 100 years ago, that \textit{the mean distances and periods are entirely
free from all such secular disturbance}. They are subject to
slight \textit{periodic} inequalities having periods of a few years, or even
a few hundred years: but in the \textit{long run} the two elements never
change. They suffer no perturbations which depend on the position
of the orbits themselves, but only such as depend on the positions
of the planets in their orbits.

\nbarticle{527.} \nbparatext{Revolution of the Nodes and Apsides.}---The \textit{nodes and perihelia},
on the other hand, move on continuously. The lines of apsides
of all the planets (Venus alone excepted) \textit{advance}, and the nodes of
all without exception (except possibly some of the asteroids), \textit{regress}
on the ecliptic.

\begin{fineprint}
\sloppy
The quickest moving line of apsides---that of Saturn's orbit---\hspace{0pt}completes
its revolution in 67,000 years, while that of Neptune requires 540,000. The
swiftest line of nodes is that of Uranus, which completes its circuit in less
than 37,000 years, while the slowest---that of Mercury---requires 166,000
years.
\fussy
\end{fineprint}

\nbarticle{528.} \nbparatext{The Inclinations of the Orbits.}---These are all slowly changing---some
increasing, and others decreasing; but as La~Place and
Leverrier have shown, all the changes are confined within narrow
limits for all the larger planets: they oscillate, but the oscillations are
never extensive.

\begin{fineprint}
It is not certain that this is so with the asteroids, some of which have
inclinations to the ecliptic of $25°$ and $30°$: it is possible that some of
\textit{these} inclinations may change by a very considerable amount.
\end{fineprint}

\nbarticle{529.} \nbparatext{The Eccentricities.}---These also are slowly changing in the
same way as the inclinations, some increasing and some decreasing;
and their changes also are closely restricted. The periods of the alternate
increase and decrease are always many thousand years in length.

\begin{fineprint}
The asteroids are again to be excepted; the eccentricities of their orbits
may change considerably.
\end{fineprint}
%% -----File: 325.png---Folio 314-------

\nbarticle{530.} \nbparatext{Stability of the Planetary System.}---About the end of the
eighteenth century La~Place and Lagrange succeeded in proving that
\textit{the mutual attraction} of the planets could never destroy the system,
nor even change the elements of the orbit of any one of the larger
planets to an extent which would greatly alter its physical condition.

The nodes and apsides revolve continuously, it is true, but that
change is of no importance. The distances from the sun and the
periods do not change at all in the long run; while the inclinations
and eccentricities, as has just been said, confine their variations
within narrow limits.

\begin{fineprint}
\nbarticle{531.} \nbparatext{The ``Invariable Plane'' of the Solar System.}---There is
no reason, except the fact that \textit{we} live on the earth, for taking the plane of
the \textit{earth's} orbit (the plane of the ecliptic) as the fundamental plane of the
solar system. There is, however, in the system an ``\textit{invariable plane},'' the
position of which remains forever unchanged by any mutual action among
the planets, as was discovered by La~Place in 1784. This plane is defined
by the following conditions,---\textit{that if from all the planets perpendiculars be
drawn to it} (\textit{i.e.}, to speak technically, if the planets be ``projected'' upon it),
\textit{and then if we multiply each planet's mass by the area which the planet's projected
radius vector describes upon this plane in a unit of time, the sum of these
products will be a maximum}. The ecliptic is inclined about $1\frac{1}{2}°$ to this
invariable plane, and has its ascending node nearly in longitude $286°$.

\nbarticle{532.} \nbparatext{La Place's Equations for the Inclinations and Eccentricities.}---La~Place
demonstrated the two following equations, viz.:
\begin{align*}
& (1)\ \ \textstyle\sum(m\sqrt{a} × e^2) = C. &
& (2)\ \ \textstyle\sum(m\sqrt{a} × \tan^2 i) = C'.
\end{align*}
Equation (1) may be thus translated: \textit{Multiply the mass of each planet by the
square root of the semi-major axis of its orbit, and by the square of its eccentricity;
add these products for all the planets, and the sum will be a constant
quantity $C$, which is very small}. It follows that no eccentricity can become
very large, since $e^2$ \textit{in the equation is essentially positive:} there can therefore
be \textit{no counterbalancing of positive and negative eccentricities}; and if the
eccentricity of one planet increases, that of some other planet or planets
must correspondingly decrease.

The second equation is the same, merely substituting $\tan^2i$ for $e^2$, $i$ being
\textit{the inclination of the planet's orbit to the invariable plane}.

The constant in this case also is small, though of course not the same
as in the preceding equation. These two equations, taken in connection
with the invariability of the periods of the major axes of the planetary
orbits, have been called the ``Magna Charta'' of the stability of the solar
system.
\end{fineprint}
%% -----File: 326.png---Folio 315-------



\nbarticle{533.} It does not follow that because the \textit{mutual attractions} of the
planets cannot seriously derange the system it is, therefore, of necessity
securely stable. There are many other conceivable actions
which might end in its ultimate destruction; such, for instance, as
that of a resisting medium, or the entrance into the system of bodies
coming from without.
\chelabel{CHAPTERXIII}
%% -----File: 327.png---Folio 316-------

\Chapter{XIV}{The Planets themselves}
\nbchapterhang{\stretchyspace
THE PLANETS: METHODS OF FINDING THEIR DIAMETERS, MASSES,
ETC.---THE ``TERRESTRIAL PLANETS'' AND ASTEROIDS.---INTRA-MERCURIAL
PLANETS AND THE ZODIACAL LIGHT.}

\textsc{In} discussing the individual peculiarities of the planets, we have
to consider a multitude of different data; for instance, their \textit{diameters},
their \textit{masses}, and \textit{densities}, their \textit{axial rotation}, their \textit{surface-markings},
their \textit{reflecting power} or \textit{``albedo,''} and their \textit{satellite systems}.

\nbarticle{534.} \nbparatext{Diameter.}---The apparent diameter of a planet is ascertained
by measurement with some kind of micrometer (\artref{Art.}{73}). For this
purpose the ``double-image'' micrometer has an advantage over the
wire micrometer because of the effect of irradiation.

\includegraphicsouter{illo169}{\textsc{Fig.~169.}\\
Micrometer Measures of a Planet's Diameter.}

\begin{fineprint}
When we bring two wires to touch the two limits of the planet in the
field of view of the telescope, \figref{illo169}{Fig.~169}, \textit{a}, the bright image of the planet is
always measured too large, because every bright object appears to extend
itself somewhat into the dark surrounding
space, by its physiological
action upon the retina of the
eye. This is known as \textit{irradiation}---well
exemplified at the time of
new moon, when the bright crescent
appears to be much larger
than the ``old moon'' faintly visible
by earth-shine. With small instruments this error is often considerable,
varying with the personal equation of the observer, but it may be
reduced to some extent by using as bright an illumination of the field of
view as the object will bear.

With the double-image micrometer, the observer in measuring has to
bring in contact two discs of equal brightness, as in \figref{illo169}{Fig.~169}, \textit{b}; and in this
case the irradiation almost vanishes at the point of contact.
\end{fineprint}

The diameter thus measured is, of course, only the \textit{apparent} diameter,
to be expressed in seconds of arc, and varies with every change
of distance. To get the real diameter in linear units, we have
\[
\mathrm{Real~diameter} = \frac{\Delta× D''}{206265},
\]
%% -----File: 328.png---Folio 317-------
in which $\Delta$ is the distance of the planet from the earth, and $D''$ the
diameter in seconds of arc. If $\Delta$ is given only in astronomical units,
the diameter comes out, of course, in terms of that unit. To get
the diameter in miles, we must know the value of this unit in miles;
that is, the sun's distance from the earth.

\nbarticle{535.} \nbparatext{Extent of Surface and Volume.}---Having the diameter, the
\textit{surface}, of course, is proportional to its \textit{square}, and is equal to the
earth's surface multiplied by $\left( \dfrac{s}{\rho} \right)^2 $, in which $s$ is the semi-diameter of
the planet and $\rho$ that of the earth.

The \textit{volume} equals $\left( \dfrac{s}{\rho} \right)^3 $ in terms of the earth's volume. (The student
must be on his guard against confounding the \textit{volume} or \textit{bulk} of
a planet with its \textit{mass}.)

The nearer the planet, other things being equal, the more accurately
the above data can be determined. The error of $0''.1$ in measuring
the apparent diameter of Venus, when nearest, counts for less than
thirteen miles in the real diameter of the planet; while in Neptune's
case it would correspond to more than 1300 miles. The student
must not be surprised, therefore, at finding considerable discrepancies
in the data given for the remoter planets by different authorities.

\nbarticle{536.} \nbparatext{Mass of a Planet which has a Satellite.}---In this case its
mass is easily and accurately found by observing the period and
distance of the satellite. We have the fundamental equation
\[
(M + m) = 4 \pi^2 \left( \frac{r^3}{t^2} \right),
\]
in which $M$ is the mass of the planet, $m$ that of its satellite, $r$ the
radius of the orbit of the satellite, and $t$ its period.

\begin{fineprint}
The formula is derived as follows: From the law of gravitation the accelerating
force which acts on the satellite is given by the equation
\[
f = \frac{M + m}{r^2}
\]
(\artref{Art.}{417}), in which $M$ is the mass of the planet and $m$ that of the satellite.
From the law of circular motion (\artref{Art.}{411}, Eq.~$b$) we have
\[
f = 4 \pi^2 \left( \frac{r}{t^2} \right) ;
\]
%% -----File: 329.png---Folio 318-------
whence (equating the two values of $f$) we have
\[
\frac{M + m}{r^2} = 4 \pi^2 \left( \frac{r}{t^2} \right);
\]
and finally
\[
(M + m) = 4 \pi^2 \left( \frac{r^3}{t^2} \right).
\]
This demonstration is strictly good only for circular orbits; but the equation
is equally true, and can be proved for elliptical orbits, if for $r$ we put $a$, the
semi-major axis of the satellite's orbit.
\end{fineprint}

For many purposes a proportion is more convenient than this equation,
since the equation requires that $M$, $m$, $r$, and $t$ be expressed in
properly chosen units in order that it may be numerically true. Converting
the equation into a proportion, we have
\[
(M + m) : (M_1 + m_1) = \frac{r^3}{t^2} :  \frac{r_1^3}{t_1^2};
\]
or, in words, \textit{the united mass of a body and its satellite is to the united
mass of a second body and its satellite as the cube of the distance of
the first satellite divided by the square of its period is to the cube of the
distance of the second satellite divided by the square of its period}. This
enables us at once to compare the masses of any two bodies which
have attendants revolving around them.

The mass of the moon is so considerable as compared with that
of the earth (about $\frac{1}{80}$) that it will not do to neglect it; but in all
other cases the satellite is less than $\frac{1}{1000}$ of the mass of its primary,
and need not be taken into account.

\begin{fineprint}
\nbarticle{537.} \nbparatext{Examples.}---(1) Required the mass of the \textit{sun} compared with
that of the \textit{earth}. The proportion is
\[
(S + \text{earth} ) : (E + \text{moon}) = \frac{ (93000000)^3}{(365 \frac{1}{4})^2} : \frac{(238000)^3}{(27.4)^2}.
\]
The quantities in the last term of the proportion are of course the distance
and period of \textit{the moon}; and it is to be remembered that for the period of
the moon we must use, not the \textit{actual} sidereal period, but the period \textit{as it
would be if the moon's motion were undisturbed},---a period about three hours
longer.

(2) Compare the mass of the earth with that of Jupiter, whose remotest
satellite has a period of $16 \frac{3}{4}$ days, and a distance of 1,160,000 miles. We have
\[
(E + \text{moon} ) : (J + \text{satellite}) = \frac{(238000)^3}{(27.4)^2} : \frac{1160000)^3}{(16 \frac{3}{4})^2},
\]
%% -----File: 330.png---Folio 319-------
which gives the mass of Jupiter about 316 times as great as that of the earth
and moon together.
\end{fineprint}

\nbarticle{538.} It is customary to express the mass of a planet as a certain
fraction of the sun's mass, and the proportion is simply
\[
\text{Sun} : \text{Planet} = \frac{R^3}{T^2} : \frac{r^3}{t^2},
\]
whence \hfill $\text{Planet's mass} = \text{Sun's mass} × \left( \dfrac{r}{R} \right)^3 × \left( \dfrac{T}{t} \right)^2$, \hfill\phantom{whence} \\
where $T$ and $R$ are the \textit{planet's} period and distance from the sun. Since
$R$ and $r$ can both be determined in astronomical units without any
necessity for knowing the length of that unit in miles, the \textit{masses of
the planets in terms of the sun's mass} are independent of any knowledge
of the solar parallax. But to compare them with the \textit{earth}, we
must know this parallax, since the moon's distance from the earth,
which enters into the equations, is found by observation in \textit{miles} or
in \textit{radii of the earth}, and not in astronomical units.

In order to make use of the satellites for this purpose we must
determine by micrometrical observations their distances from the
planets and their periods.

\nbarticle{539.} \nbparatext{Mass of a Planet which has no Satellite.}---When a planet
has not a satellite, the determination of its mass is a very difficult and
troublesome problem, and can be solved only by finding some perturbation
produced by the planet, and then ascertaining, by a sort of
``trial and error'' method, the mass which would produce that perturbation.
Venus disturbs the earth and Mercury, and from these perturbations
her mass is ascertained. Mercury disturbs Venus, and
also one or two comets which come near him, and in this way we get
a rather rough determination of his mass.

\nbarticle{540.} \nbparatext{Density.}---The density of a body as compared with the
earth is determined simply by dividing its mass by its volume; \textit{i.e.},
\[
\text{Density} = \frac{m}{\left( \dfrac{s}{\rho} \right)^3}.
\]

\begin{fineprint}
For example, Jupiter's \textit{diameter} is about eleven times that of the earth
(\textit{i.e.} $\left(\dfrac{s}{\rho} \right) = 11$), so that his \textit{volume} is $11^3$, or 1331 times the earth's. His \textit{mass},
derived from satellite observations, is about 316 times the earth's. The
%% -----File: 331.png---Folio 320-------
\textit{density}, therefore, equals $\frac{316}{1331}$, or about 0.24, of the earth's density, or about
$1 \frac{1}{3}$ times that of water, the earth's density being 5.58 (\artref{Art.}{171}).
\end{fineprint}

\sloppy
\nbarticle{541.} \nbparatext{The Surface Gravity.}---The force of gravity on a planet's
surface as compared with that on the surface of the earth is important
in giving us an idea of its physical condition. If $r$ is the radius of
the planet in terms of the earth's radius, then
\[
\text{Surface gravity, or }G,
= \frac{m}{\left( \frac{s}{\rho} \right)^2},
= \frac{m}{\left( \frac{s}{\rho} \right)^3}
  ×   \left( \frac{s}{\rho} \right).
\]
\textit{i.e.}, it equals the planet's \textit{density}, multiplied by its \textit{diameter} expressed
in terms of the earth's diameter.

\fussy
\begin{fineprint}
For Jupiter, therefore, $G = \dfrac{316}{11^2} = 11 × \text{density} = 11 × 0.24 =2.64$ nearly.
That is, a body at Jupiter would weigh 2.6 as much as one at the earth's
surface.
\end{fineprint}

\sloppy
\nbarticle{542.} \nbparatext{The Planet's Oblateness.}---The ``oblateness'' or ``polar-compression''
is the \textit{difference} between the equatorial and polar
diameters divided by the equatorial diameter. It is, of course,
determined, when it is possible to determine it at all, simply by
micrometric measurements of the difference between the greatest and
least diameters. The quantity is always very small and the observations
delicate.

\fussy
\nbarticle{543.} \nbparatext{The Time of Rotation}, when it can be determined, is found
by observing the passage of some spot visible in the telescope across
the central line of the planet's disc. In reducing the observations to
find the interval between such transits, account has to be taken of the
continual change in the direction of the line which joins the planet
and the earth, and also of the variations in the distance, which will
alter the time taken by light in coming to the earth from the body.

\nbarticle{544.} \nbparatext{The Inclination of the Axis} is deduced from the same observations
which are used in obtaining the rotation period. It is necessary
to determine with the micrometer the paths described by different
spots as they move across the planet's disc. It is possible to
ascertain it with accuracy for only a very few of the planets: Mars,
Jupiter, and Saturn are the only ones that furnish the needed data.

\nbarticle{545.} \nbparatext{The Surface Peculiarities and Topography} of the surface
are studied by the telescope. The observer makes drawings of any
%% -----File: 332.png---Folio 321-------
markings which he may see, and by their comparison is at last able
to discriminate between what is temporary and what is permanent on
the planet. Mars alone, thus far, permits us to make a map of its
surface.

\nbarticle{546.} \nbparatext{Spectroscopic Peculiarities and Albedo.}---The characteristics
of the planet's atmosphere can be to some extent studied by
means of the spectroscope, which in some few cases shows the presence
of water-vapor and other absorbing media, by dark bands in
the planet's spectrum. The \textit{``albedo,''} or reflecting power of a
planet's surface is determined by photometric observations, comparing
it with a real or artificial star, or with some other planet.

\nbarticle{547.} \nbparatext{The Satellite System of a Planet.}---The principal data to be
ascertained are the \textit{distances} and \textit{periods} of the satellites, and the
observations are made by measuring the \textit{apparent} distances and
directions of the satellites from the centre of the planet with the wire
micrometer (\artref{Art.}{73}). Observations made at the times when the
satellite is near its elongation are especially valuable in determining
the distance.

\begin{fineprint}
If the planet and earth were at rest, the satellite's path would appear to
be an ellipse, unaltered in dimensions during the whole series of observations;
but since the earth and planet are both moving, it becomes a complicated
problem to determine the satellite's true orbit from the \textit{ensemble} of
observations.

\nbarticle{548.} With the exception of the moon and the outer satellite of Saturn,
all the satellites of the planetary system move almost exactly in the plane of
the equator of the primary; and all but the moon and the seventh satellite
of Saturn (Hyperion) move in orbits almost perfectly circular. Tisserand
has recently shown that the equatorial protuberance of a planet compels any
satellite which is not very remote from its primary to move nearly in the
equatorial plane, but the almost perfect circularity of the orbits is not yet
explained. When there are a number of satellites in a system, interesting
problems arise in connection with their mutual disturbances; and in a few
cases it becomes possible to determine a satellite's mass as compared with
that of its primary. In several instances satellites show peculiar variations
in their brightness, which are supposed to indicate that they make an axial
rotation in the time of one revolution around the primary, in the same way
as our moon does.
\end{fineprint}

\sloppy
\nbarticle{549.} \nbparatext{Humboldt's Classification of the Planets.}---Humboldt has
divided the planets into two groups: the \textit{terrestrial} planets, so called,
%% -----File: 333.png---Folio 322-------
and the \textit{major} planets. The terrestrial planets are Mercury, Venus,
the earth, and Mars. They are bodies of the same order of magnitude,
ranging from 3000 to 8000 miles in diameter, not very different
in density (though Mercury is much denser than either of
the others), and are probably roughly alike in physical constitution,
and covered with water and air. But we hasten to say that
the differences in the amount of heat and light which they receive
from the sun, and in the force of gravity upon their surfaces, and
probably in the density of their atmospheres, are such as to bar any
positive conclusions as to their being the abode of life resembling the
forms of life with which we are acquainted on the earth.

\includegraphicsmid{illo170}{\textsc{Fig.~170.}---Relative Sizes of the Planets.}

\nbarticle{550.} The four major planets, Jupiter, Saturn, Uranus, and Neptune,
are much larger bodies (ranging in diameter between 30,000
and 90,000 miles), are much less dense, and so far as we can make
out, present to us only a surface of cloud, and may not have
anything solid about them. There are some reasons for suspecting
that they are at a high temperature; in fact, that Jupiter is a sort of
\textit{semi-sun}, but this is by no means yet certain.

\fussy
As for the multitudinous asteroids, the probability is that they
represent a single planet of the terrestrial group which, as has
been intimated, failed for some reason in its evolution, or else has
%% -----File: 334.png---Folio 323-------
been broken to pieces. All of them united would not make a planet
one-half the mass of the earth.

Fig.~170 shows the relative sizes of the different planets.

In what follows, all the numerical data, so far as they depend on
the solar parallax, are determined on the assumption that that parallax
is $8''.80$, and that the sun's mean distance is 92,897000 miles.

\section*{MERCURY.}

\nbarticle{551.} There is no record of the discovery of the planet. It has
been known from remote antiquity; and we have recorded \textit{observations}
running back to \textsc{b.c.}~264.

\begin{fineprint}
For a time the ancient astronomers seem to have failed to recognize it
as the same body on the eastern and western sides of the sun, so that the
Greeks had for a time two names for it,---Apollo when it was morning star,
and Mercury when it was evening star. According to Arago, the Egyptians
called it Set and Horus, and the Hindoos also gave it two names.
\end{fineprint}

It is so near the sun that it is comparatively seldom seen with the
naked eye; but when near its greatest elongation it is easily enough
visible as a brilliant star of the first magnitude low down in the twilight,
perhaps not quite so bright as Sirius, but certainly brighter than
Arcturus. It is usually visible for about a fortnight at each elongation,
and is best seen in the evening at such eastern elongations as
occur in March and April. In Northern Europe it is much more
difficult to observe than in lower latitudes, and Copernicus is said
never to have seen it. Tycho, however, obtained a considerable
number of observations.

\nbarticle{552.} It is exceptional in the solar system in a great variety
of ways. It is the \textit{nearest} planet to the sun, \textit{receives the most light
and heat}, is \textit{the swiftest} in its movement, and (excepting some of the
asteroids) \textit{has the most eccentric orbit}, with the \textit{greatest inclination}
to the ecliptic. It is also the \textit{smallest} in diameter (again excepting
asteroids), has the \textit{least mass}, and the \textit{greatest density} of all the
planets.

\nbarticle{553.} \nbparatext{Distance, Light, and Heat.}---Its \textit{mean distance} from the
sun is 36,000000 miles, but the eccentricity of its orbit is so great
(0.205), that the sun is seven and one-half millions of miles out of
its centre, and the actual distance of the planet from the sun ranges
all the way from 28,500000 to 43,500000, while its velocity in its orbit
%% -----File: 335.png---Folio 324-------
varies from thirty-five miles a second at perihelion to only twenty-three
at aphelion. On the average it receives 6.7 times as much light and
heat as the earth; but the heat received at perihelion is to that at
aphelion in the ratio of 9 to 4. For this reason there must be two
seasons in its year due to the changing distance, even if the equator
of the planet is parallel to the plane of its orbit, which would preclude
seasons like our own. If the planet's equator is inclined at an angle
like the earth's, then the seasons must be very complicated.

\nbarticle{554.} \nbparatext{Period.}---The \textit{sidereal period} is very nearly 88~days, and
the \textit{synodic period}, or the time from conjunction to conjunction again,
is about 116~days. The greatest elongation ranges from $18°$ to $28°$,
and occurs about twenty-two days before and after the inferior
conjunction, or about thirty-six days before and after the superior
conjunction. The planet's arc of retrogression is about $12°$ (considerably
variable), and the stationary point is very near the greatest
elongation.

\nbarticle{555.} \nbparatext{Inclination.}---The inclination of the orbit to the ecliptic is
about $7°$, but the greatest geocentric latitude (that is, the planet's
greatest distance from the ecliptic as seen from the earth) is never
quite so great.

\nbarticle{556.} \nbparatext{Diameter, Surface, and Volume.}---\textit{The apparent diameter}
ranges from $5''$ to about $13''$, according to its distance from us; the
least distance from the earth being about 57,000000 miles ($93 - 36$),
while the greatest is about 129,000000 ($93 + 36$). The \textit{real
diameter} is very near 3000 miles, not differing from that more than
fifty miles either way. It is not easy to measure, and the ``probable
error'' is perhaps rather larger than would have been expected.
With this diameter, its \textit{surface} is $\frac{1}{7}$ of the earth's, and its
\textit{volume} $\frac{1}{18.5}$.

\nbarticle{557.} \nbparatext{Mass, Density, and Surface Gravity.}---Its mass is very difficult
to determine, since it has no satellite, and the values obtained by
La~Place, Encke, Leverrier, and others, range all the way from $\frac{1}{9}$ of
the earth's mass to $\frac{1}{15}$.

\begin{fineprint}
The most recent, and perhaps the most reliable, determination, although
the largest of all, is that of Backlund, by means of the perturbations of
Encke's comet. He makes its mass about $\frac{1}{8}$ that of the earth. This gives
%% -----File: 336.png---Folio 325-------
us a mean density about $2 \frac{1}{4}$ times that of the earth, or nearly $12\frac{1}{2}$ times that
of water, a little less than that of the metal \textit{mercury}. So far as known, the
earth stands next to it in density, but with a wide interval, so that Mercury
is altogether exceptional in this respect, as has been said before. The result,
however, is liable to a considerable error, on account of the uncertainty of
the planet's mass, Leverrier's estimate of the mass and density being only
about one-half as great as Backlund's. The \textit{superficial gravity} is about $\frac{5}{6}$ that
at the earth's surface.
\end{fineprint}

\nbarticle{558.} Its \textit{Albedo}, or reflecting power, as determined by Zöllner is
very low---only 0.13, somewhat inferior to that of the moon.

\begin{fineprint}
In 1878 Mr.\ Nasmyth observed the planet in the same field of view with
Venus; and although Mercury was then not much more than half as far
from the sun as Venus, and therefore four times as brightly illuminated, it
appeared to be less luminous in the telescope. ``Venus was like silver,
Mercury like zinc or lead.''
\end{fineprint}

In the proportion of light given out at its different phases, it
behaves like the moon, flashing out strongly near the full, as if it had
a surface of the same rough structure as that of our satellite.

\sloppy
\nbarticle{559.} \nbparatext{Telescopic Appearance and Phases.}---Seen by the telescope,
the planet looks like a little moon, showing phases precisely similar
to those of our satellite. At inferior conjunction the dark side is
towards us; at superior conjunction the illuminated surface. At
the greatest elongation it appears like a half-moon. Between superior
conjunction and greatest elongation it is gibbous, while between
inferior conjunctions and the elongations it shows the crescent phase.
Fig.~171 illustrates the phases of both Mercury and Venus.

\includegraphicsmid{illo171}{\textsc{Fig.~171.}---Phases of Mercury and Venus.}

\fussy
\begin{fineprint}
For the most part, Mercury can be observed only by daylight; but when
proper precautions are taken to screen the object-glass of the telescope from
%% -----File: 337.png---Folio 326-------
the direct sunlight, the observation is not difficult. The surface presents
very little of interest. There are no markings well enough defined to give
us any trustworthy determination of the planet's rotation, or of its geography.
Occasionally the terminator (\textit{i.e.}, the line which separates the illuminated
and unilluminated portions of the planet) appears to be a little
irregular, instead of a true oval, as it should be; and at times when the phase
is crescent-form, the points, or cusps, are a little blunted; from this some astronomers
have inferred the existence of high mountains upon the planet.
Schröter, a German astronomer, the contemporary of the elder Herschel,
deduced from his observations a rotation period for the planet of 24~h.\ 5~m.;
but later observers, with instruments certainly far more perfect, have
not been able to verify his results, and they are now considered as of little
weight.
\end{fineprint}

\nbarticle{560.} \nbparatext{Atmosphere.}---The evidence upon this subject is not conclusive.
Its atmosphere, if it has one, must, however, be much less
dense than that of Venus. No ring of light is seen surrounding the
disc of the planet when it enters the limb of the sun at the time of
a transit, while in the case of Venus such a ring, due to the atmospheric
refraction, is very conspicuous. On the other hand, Huggins
and Vogel, who have examined the spectrum of the planet, report
that certain lines in the spectrum, due to the presence of water-vapor,
were decidedly stronger than in the spectrum of the air (illuminated
by sunshine), which formed the background for the planet,
making it probable that it has an atmosphere containing water-vapor
like the atmosphere of the earth, but probably less extensive and
dense.

\nbarticle{561.} \nbparatext{Transits.}---Usually at the time of inferior conjunction the
planet passes north or south of the sun, the inclination of its orbit
being $7°$; but if the conjunction occurs when the planet is very near
its node, it will cross the disc of the sun and be visible upon it us a
small black spot---not, however, large enough to be seen without a
telescope, as Venus can under similar circumstances.

\begin{fineprint}
At this time we have the best opportunity for measuring the diameter of
the planet; but unless special precautions are taken, the measured diameter
under these circumstances is likely to be \textit{too small}, on account of the irradiation
of the surrounding background, which encroaches upon the planet's disc.
\end{fineprint}

Since the planet's nodes are in longitudes $227°$ and $47°$, and are
passed by the earth on May~7 and November~9, the transits can
occur only near these days. If the orbit of the planet were strictly
circular, the ``transit limit'' (corresponding to an ecliptic limit)
%% -----File: 338.png---Folio 327-------
would be $2° 10'$; but at the May transits the planet is near its
aphelion and much nearer the earth than ordinarily, so that the limit
is diminished, while the November limit is correspondingly increased.
The May transits are in fact only about half as numerous as the
November transits.

\nbarticle{562.} \nbparatext{Interval between Transits.}---Twenty-two synodic periods of
Mercury are pretty nearly equal to 7~years; 41 still more nearly
equal 13~years; and 145 almost exactly equal 46 years. Hence,
after a November transit, a second one is possible in 7~years, probable
in 13~years, and practically certain in 46. For the May transits
the repetition after 7~years is not possible, and it often fails in
13~years.

\begin{fineprint}
The transits of the present century are the following:---

\textit{May Transits.}---1832, May~5; 1845, May~8; 1878, May~6; 1891, May~9.

\textit{November Transits.}---1802, Nov.~9; 1815, Nov.~11; 1822, Nov.~5; 1835,
Nov.~7; 1848, Nov.~10; 1861, Nov.~12; 1868, Nov.~5; 1881, Nov.~7; 1894,
Nov.~10.

The first transit of Mercury ever observed was by Gassendi, Nov.~7, 1631.

The transits of Mercury are of no particular astronomical importance,
except as giving accurate determinations of the planet's place, by means of
which its orbit can be determined. Newcomb has also recently made an
investigation of all the recorded transits, for the purpose of testing the
uniformity of the earth's rotation. They indicate no perceptible change in
the length of the day.
\end{fineprint}


\section*{VENUS.}

\nbarticle{563.} The next planet in order from the sun is Venus, the brightest
and most conspicuous of all, the earth's twin sister in magnitude,
density, and general constitution, if not also in age, as to which we
have no knowledge. Like Mercury, it had two names among the
Greeks,---\textit{Phosphorus} as morning star, and \textit{Hesperus} as evening star.
It is so brilliant that it is easily seen by the naked eye in the daytime
for several weeks when near its greatest elongation; sometimes it is
bright enough to catch the eye at once, but usually it is seen by daylight
only when one knows precisely where to look for it.

\begin{fineprint}
(It is not, however, the ''\textit{Star of Bethlehem},'' though it has of late been
frequently taken for it.)
\end{fineprint}

\sloppy
\nbarticle{564.} \nbparatext{Distance, Period, and Inclination of Orbit.}---Its mean distance
from the sun is 67,200000 miles. The eccentricity of the
%% -----File: 339.png---Folio 328-------
orbit is the smallest in the planetary system (only 0.007), so that the
greatest and least distances of the planet from the sun differ from
the mean only 470,000 miles each way. Its orbital velocity is
twenty-two miles per second.

\fussy
Its \textit{sidereal period} is 225 days, or seven and one-half months, and
its \textit{synodic period} 584~days---a year and four months. From superior
conjunction to elongation on either side is 220 days, while from
inferior conjunction to elongation is only 71 or 72~days. The arc of
retrogression is $16°$.

The \textit{inclination of its orbit} is only $3 \frac{1}{2}°$.

\nbarticle{565.} \nbparatext{Diameter, Surface, and Volume.}---The apparent diameter
ranges from $67''$ at the time of inferior conjunction to only $11''$ at the
superior. This great difference depends, of course, upon the enormous
change in the distance of the planet from the earth. At
inferior conjunction the planet is only 26,000000 miles from us ($93 - 67$).
No other body ever comes so near the earth except the
moon, and occasionally a comet. Its greatest distance at superior
conjunction is 160,000000 miles ($93 + 67$), so that the ratio between
the greatest distance and the least is more than 6 to 1.

The \textit{real diameter} of the planet is 7700 ($\pm \: 30$) miles. Its \textit{surface,}
as compared with that of the earth, is ninety-five per cent;
its \textit{volume} ninety-two per cent.

\nbarticle{566.} \nbparatext{Mass, Density, and Gravity.}---By means of the perturbations
she produces upon the earth, the \textit{mass} of Venus is found to be
seventy-eight per cent of the earth's; hence her \textit{density} is eighty-six
per cent, and her \textit{superficial} gravity eighty-three per cent of the
earth's.

\nbarticle{567.} \nbparatext{Phases.}---The telescopic appearance of the planet is striking
on account of her great brilliance. When about midway between
greatest elongation and inferior conjunction she has an apparent
diameter of $40''$, so that, with a magnifying power of only forty-five,
she looks exactly like the moon four days old, and of precisely the
same apparent size.

\begin{fineprint}
Very few persons, however, would think so on their first view through
the telescope, for a novice always underrates the apparent size of a telescopic
object: he instinctively adjusts his focus as if looking at a picture
only a few inches away, instead of projecting the object visually into
the sky.
\end{fineprint}
%% -----File: 340.png---Folio 329-------

\includegraphicsmid{illo172}{\textsc{Fig.~172.}---Telescopic Appearances of Venus.}

According to the theory of Ptolemy, Venus could never show us
more than half her illuminated surface, since according to his hypothesis
she was \textit{always between us and the supposed orbit of the sun}
(\artref{Art.}{500}). Accordingly, when in 1610 Galileo discovered that she
exhibited the gibbous phase as well as the crescent, it was a strong
argument for Copernicus. Galileo announced his discovery in a curious
way, by publishing the anagram,---
\begin{verse}
``Haec immatura a me jam frustra leguntur; o.\ y.''
\end{verse}

Some months later he furnished the translation,---
\begin{verse}
``Cynthię figuras ęmulatur mater amorum,''
\end{verse}
which is formed by merely transposing the letters of the anagram.
His object was to prevent any one from claiming to have anticipated
him in this discovery, as had been done with respect to his discovery
of the sun spots.

\begin{fineprint}
Fig.~172 represents the disc of the planet as seen at four points in its
orbit. 1, 3, and 5 are taken at superior conjunction, greatest elongation,
and near inferior conjunction respectively, while 2 and 4 are at intermediate
points.
\end{fineprint}

\nbarticle{568.} \nbparatext{Maximum Brightness.}---The planet attains its maximum
brilliance thirty-six days before and after inferior conjunction, at a
%% -----File: 341.png---Folio 330-------
distance of about $38°$ or $39°$ from the sun, when its phase is like that
of the moon about five days old. It then casts a strong shadow,
and, as has already been said, is easily visible by day with the
naked eye.

\sloppy
\begin{fineprint}
\nbarticle{569.} \nbparatext{Surface Markings.}---These are not at all conspicuous. Near
the limb of the planet, which is always much brighter than the central parts
(as is also the case with Mercury and Mars), they can never be well seen,
although sometimes when Venus was in the crescent phase, intensely bright
spots have been reported near the cusps, as at $a$ and $b$ in No.~4, \figref{illo172}{Fig.~172}.
These may perhaps be \textit{ice-caps} like those which are seen on Mars. Near the
``terminator,'' which is less brilliant and less sharply defined than the limb,
irregular darkish shadings are sometimes seen, such as are indicated by the
dotted lines in the \figref{illo172}{figures}, but without any distinct outline. They may be
continents and oceans dimly visible, or they may be mere atmospheric
objects; observations do not yet decide.
\end{fineprint}

\fussy
\nbarticle{570.} \nbparatext{Rotation of the Planet.}---From the observation of such
markings Schröter deduced the rotation period of $23^\text{h}\: 21^\text{m}$, which has
since been partially confirmed by one or two other observers, and
may be approximately correct. De~Vico concluded that the planet's
equator makes an angle of about $54°$ with the plane of its orbit.
This, however, is extremely doubtful. If the bright spots which
have been referred to in \artref{Art.}{569} are really polar ice-caps, the
equator cannot be very greatly inclined. (See Clerke's ``History of
Astronomy,'' p.~299.)

\begin{fineprint}
No sensible difference has been ascertained between the different diameters
of the planet. If it were really as much flattened at the poles as the
earth is, there should be a difference of $0''.2$ between the polar and equatorial
diameters as measured at the time of the planet's transit.
\end{fineprint}

\sloppy
\nbarticle{571.} \nbparatext{Mountains.}---From certain irregularities occasionally observed
upon the terminator, and especially from the peculiar blunted
form of one of the cusps of the crescent, various observers have concluded
that there are numerous high mountains upon the surface of
the planet. Schröter assigned to some of these near the southern
pole the extravagant altitude of twenty-five or thirty miles, but the
evidence is entirely insufficient to warrant any confidence in the conclusion.

\fussy
\nbarticle{572.} \nbparatext{Albedo.}---According to Zöllner the \textit{Albedo} of the planet is
$0.50$, which is about three times that of the moon, and almost four
%% -----File: 342.png---Folio 331-------
times that of Mercury. It is, however, exceeded by the reflecting
power of the surfaces of Jupiter and Uranus, while that of Saturn
appears to be about the same. This high reflecting power probably
indicates that the surface is mostly covered with cloud, as few rocks
or soils could match it in brightness.

\nbarticle{573.} \nbparatext{Evidences of Atmosphere.}---When the planet is near the
sun, the horns of the crescent extend notably beyond the diameter,
and when very near the sun, a thin line of light has been seen by
several observers, especially Professor Lyman of New Haven, to
complete the whole circumference. This is due to refraction of sunlight
by the planet's atmosphere, a phenomenon still better seen as
the planet is entering upon the sun's disc at a transit, when the
black disc is surrounded by a beautiful ring of light. From the observations
of the transit of 1874, Watson concluded that the planet's
atmosphere must have a depth of about fifty-five miles, that of the
earth being usually reckoned at forty miles. Other observers in
different ways have come to substantially the same results. Its
atmosphere is probably from one and a half to two times as extensive
and dense as our own, and the spectroscope shows evidence of the
presence of water-vapor in it.

\textit{Lights on Dark Portion.}---Many observers have also reported
faint lights as visible at times on the dark portion of the planet's
disc. These cannot be accounted for by reflection, but must originate
on the planet's surface; they recall the Aurora Borealis and
other electrical manifestations on the earth.

\nbarticle{574.} \nbparatext{Satellites.}---No satellite is known, although in the last
century a number of observers at various times thought they had
found one.

\begin{fineprint}
In most cases they observed small stars near the planet, which we can
now identify by computing the place occupied by the planet at the date of
observation. It is not, however, \textit{impossible} that the planet may have some
very minute and near attendants like those of Mars, which may yet be
brought to light by means of the great telescopes of the future, or by photography.
Of course the extreme brilliance of the planet, and the fact that
the necessary observations can be made only in strong twilight, render the
discovery of such objects, if they exist, very difficult.
\end{fineprint}

\nbarticle{575.} \nbparatext{Transits.}---Occasionally Venus passes between the earth
and the sun at inferior conjunction, giving us a so-called ``\textit{transit}.''
%% -----File: 343.png---Folio 332-------
She is then visible (even to the naked eye) as a black spot on the
disc, crossing it from east to west.

As the inclination of the planet's orbit is but $3\frac{1}{2}°$, the ``\textit{transit
limit}'' is small (only $1\frac{3}{4}°$), and the transits are therefore very rare
phenomena. The sun passes the nodes of the orbit on June~5 and
December~7, so that all transits must occur on or near those dates.
When Venus crosses the sun's disc \textit{centrally}, the duration of the
transit is about \textit{eight hours}. Taking the mean diameter of the sun
as $32'$, or $\frac{1}{375}$ of a circumference, and the planet's synodic period
as 584 days, the geocentric duration of a central transit should be
$\frac{277}{723} × \frac{1}{675} × 584^\text{d}$, which equals 0.332 days, or $7^\text{h}\: 58^\text{m}$.

When the transit track lies near the edge of the disc, the duration
is of course correspondingly shortened.

\nbarticle{576.} \nbparatext{Recurrence of Transits.}---Five synodic, or thirteen sidereal,
revolutions of Venus are very nearly equal to eight years, the difference
being only a little more than one day; and still more nearly, in
fact almost exactly, 243 years are equal to 152 synodic, or 395 sidereal,
revolutions. If, then, we have a transit at any time, we \textit{may
have} another at the same node \textit{eight} years earlier or later. Sixteen
years before or after it would be impossible, and no other transit
can occur at the same node until after the lapse of \textit{two hundred and
thirty-five} or \textit{two hundred and forty-three years}.

If the planet crosses the sun nearly centrally, the transit will not
be accompanied by another at an eight-year interval, but the planet
will pass either north or south of the sun's disc, at the conjunctions
next preceding and following. If, however, as is now the case, the
transit path is near the northern or southern edge of the sun, then
there will be a companion transit across the opposite edge of the
disc eight years before or after. Thus, if we have a pair of \textit{June}
transits, separated by an eight-year interval, it will be followed by
another pair at the same node in 243 years; and a pair of \textit{December}
transits will come in about \DPtypo{halfway}{half-way} between the two pairs of June transits.
After a thousand years or so from the present time the transits
will cease to come in pairs, as they have been doing for 2000 years.

\nbarticle{577.} Transits of Venus have occurred or will occur on the following
dates:---
\begin{align*}
\left.\begin{gathered}
\text{December 7, 1631.}\\
\text{December 4, 1639.}\\
\end{gathered}\right\}
&&&
\left.\begin{gathered}
\text{June 5, 1761.}\\
\text{June 3, 1769.}\\
\end{gathered}\right\}
\\
\left.\begin{gathered}
\text{December 9, 1874.}\\
\text{December 6, 1882.}\\
\end{gathered}\right\}
&&&
\left.\begin{gathered}
\text{June 8, 2004.}\\
\text{June 6, 2012.}\\
\end{gathered}\right\}
\end{align*}
%% -----File: 344.png---Folio 333-------

The special interest in these transits consists in the use that has been
made of them for the purpose of finding the sun's
parallax, a subject which will be discussed later on
(\chapref{CHAPTERXVI}{Chap.~XVI.}).

\includegraphicsouter{illo173}{\textsc{Fig.~173.}\\
Transit of Venus Tracks.}

The first observed transit, in 1639, was seen by
only two persons,---Horrox and Crabtree, in England.
The four which have occurred since then
have been extensively observed in all parts of the
world where they were visible, by scientific expeditions
sent out for the purpose by different nations. The
transits of 1769 and 1882 were visible in the United
States. \figref{illo173}{Fig.~173} shows the track of Venus across
the sun's disc at the two transits of 1874 and 1882.

\section*{MARS.}

This planet is also prehistoric as to its discovery. It is so conspicuous
in color and brightness, and in the extent and apparent
capriciousness of its movement among the stars, that it could not
have escaped the notice of the very earliest observers.

\nbarticle{578.} \nbparatext{Orbit.}---Its \textit{mean distance} from the sun is 141,500000
miles, and the \textit{eccentricity} of the orbit is so considerable (0.093) that
this distance varies about 13,000000 miles. The \textit{light and heat}
which it receives from the sun is somewhat less than half of that
received by the earth. The \textit{inclination} of its orbit is small, $1° 51'$.
The planet's \textit{sidereal} period is 687 days, or $1^\text{y}\: {10\frac{1}{2}}^\text{mo}$, which gives
it an average orbital velocity of fifteen miles per second. Its
\textit{synodic} period is 780~days, or $2^\text{y}\: {1\frac{2}{3}}^\text{mo}$. It is the longest in the solar
system, that of Venus (584~days) coming next. Of the 780~days, it
moves eastward during 710, and retrogrades during 70, through an
arc of 18°.

\sloppy
\nbarticle{579.} At opposition its \textit{average} distance from the earth is 48,600000
miles (141,500000 miles minus 92,900000 miles). When the
opposition occurs near the planet's perihelion, this distance is reduced
to 35,500000 miles; if near aphelion, it is increased to over
61,000000. At superior conjunction the average distance from the
earth is 234,400000 miles (141,500000 plus 92,900000).

\fussy
The apparent diameter and brilliancy of the planet, of course, vary
enormously with these great changes of distance.

\begin{fineprint}
If we put $R$ for the planet's distance from the sun, and $\Delta$ for its distance
from the earth, its brightness, neglecting the correction for \textit{phase}, should
%% -----File: 345.png---Folio 334-------
equal $\dfrac{1}{R^2 \Delta^2}$. We find from this that, taking the brightness at conjunction
as unity (at which time the planet is about as bright as the pole-star), it is
more than twenty-three times brighter at the \textit{average} opposition, and fifty-three
times brighter if the opposition occurs at the planet's perihelion. At
an unfavorable opposition Mars, as has been said, may be 61,000000 miles
distant, and its brightness then is only about twelve times as great as at
conjunction,---the difference between favorable and unfavorable oppositions
being more than \textit{four to one}.

These favorable oppositions occur always in the latter part of August (at
which date the sun passes the line of apsides of the planet), and at intervals
of fifteen or seventeen years. The last was in 1877, and the next will be in
1892. A reference to \figref{illo159}{Fig.~159} will show how great is the difference between
the planet's opposition distance from the earth under varying circumstances.
\end{fineprint}

\nbarticle{580.} \nbparatext{Diameter, Surface, and Volume.}---The apparent diameter of
the planet ranges from $3''.6$ at conjunction, to $24''.5$ at a favorable
opposition. Its \textit{real} diameter is very closely 4200 miles,---the error
may be twenty miles one way or the other. This makes its surface
0.28, and its volume 0.147 (equal to $\frac{1}{7}$) of the earth's.

\nbarticle{581.} \nbparatext{Mass, Density, and Gravity.}---Observations upon its satellites
give its mass as $\frac{1}{9.4}$ compared with that of the earth. This
makes its \textit{density} 0.73 and \textit{superficial gravity} 0.38; that is, a body
which weighs 100 pounds on the earth would have a weight of 38
pounds on the surface of Mars.

\nbarticle{582.} \nbparatext{Phases.}---Since the orbit of the planet is outside that of the
earth, it never comes between us and the sun,
\textit{and can never show the crescent phase}; but at
quadrature enough of the unilluminated portion
is turned towards the earth to make the disc
clearly \textit{gibbous} like the moon three or four days
from full. \figref{illo174}{Fig.~174} shows its maximum phase
accurately drawn to scale.

\includegraphicsouter{illo174}{\textsc{Fig.~174.}\\
Greatest Phase of Mars.}

\nbarticle{583.} \nbparatext{The ``Albedo'' of the Planet.}---According
to Zöllner's observations this is 0.26, which
is considerably higher than that of the moon
($\frac{1}{6}$), and just double that of Mercury.

\nbarticle{584.} \nbparatext{Rotation.}---The planet's time of rotation is $24^\text{h}\: 37^\text{m}\: 22^\text{s}.67$.
This very exact determination has been made by Kaiser and Bakhuyzen,
%% -----File: 346.png---Folio 335-------
by comparing drawings of the planet which were made more
than 200 years ago by Huyghens with others made recently.

\sloppy
It is obvious that observations made a few days or weeks apart will
give the time or rotation with only approximate accuracy. Knowing
it thus approximately, we can then determine, without fear of error,
the \textit{whole number of rotations} between two observations separated by
a much longer interval of time. This will give a second and closer
approximation to the true period; and with this we can carry our
reckoning over centuries, and thus finally determine the period within
a very minute fraction of a second. The number given is not uncertain
by more than $\frac{1}{50}$ of a second, if so much.

\nbarticle{585.} \nbparatext{The Inclination of the Planet's Equator to the Plane of its
Orbit.}---This is very nearly $24°\: 50'$ ($26°\: 21'$ to the \textit{ecliptic}), not very
different from the inclination of the earth's equator; so far, therefore,
as depends upon that circumstance, its seasons should be substantially
the same as our own.

\fussy
\nbarticle{586.} \nbparatext{Polar Compression.}---There is a slight but sensible flattening
of the planet at the poles. The earlier observers found for the
polar compression values as large as $\frac{1}{40}$, and even $\frac{1}{10}$. These large
values, however, are inconsistent with the existence of any extensive
surface of liquid upon the planet, and more recent observations
of the writer show the polar compression to be about $\frac{1}{220}$; which is
almost exactly what would be expected from a planet constituted as
we suppose Mars to be.

\includegraphicsouter{illo175}{\textsc{Fig.~175.}---Telescopic Views of Mars.}

\sloppy
\nbarticle{587.} \nbparatext{Telescopic Appearance and Surface-Markings.}---The fact
that we are able to determine the time of rotation so accurately of
course implies the existence of identifiable markings upon the surface.
Viewed through
a powerful telescope,
the planet's disc, as a
whole, is ruddy,---in
fact, almost rose color,---very
bright around
the limb, but not at the
``terminator,'' if there is
any considerable phase.
The central portions of
the disc present greenish and purplish patches of shade, for the
most part not sharply defined, though some of the markings have
%% -----File: 347.png---Folio 336-------
outlines reasonably distinct. On watching the planet for only a few
hours even, the markings pass on across the disc, and are replaced
by others. Some of them are permanent, and recur at regular intervals
with the same form and appearance, while others appear to
be only clouds which for a time veil the surface below, and then
clear away.

\fussy
By comparing drawings made when the same side is turned towards
the earth, it is possible in a short time to ascertain what features
really belong to the planet's geography.

The polar ice-caps, brilliant white patches near the poles, form a
marked feature in the planet's telescopic appearance. These are believed
to be \textit{ice-caps} from the fact that the one which is near the pole
that happens to be turned towards the sun continually diminishes in
size, while the other increases, the process being reversed with the
seasons of the planet.

\nbarticle{588.} The principal peculiarity of the surface of Mars appears to
be the way in which land and water are intermeshed. There seem to
be few great oceans and continents, but there are narrow arms of the
sea, like the Baltic and the Red Sea, penetrating and dividing the
land masses. As to the nomenclature of the principal ``Areographic''
features, Mr.~Proctor in his map has for the most part assigned the
names of astronomers who have taken special interest in the study of
%% -----File: 348.png---Folio 337-------
the planet. Schiaparelli, on the other hand, has taken his names
mostly from classical geography. \figref{illo176}{Fig.~176} is a reduced copy of his
map of the planet, drawn on Mercator's Projection. The student
will remember that in maps of this sort the polar regions are extended
beyond their due proportion, and also that the parts near the
poles can never be seen from the earth nearly as well as those near
the equator, and hence that our knowledge of the details in that part
of the planet is very limited. (See \hyperref[art:608*.]{note to \artref{Art.}{588}}, p.~\pageref{pg:347}.)

\includegraphicsmid{illo176}{\textsc{Fig.~176.}---Schiaparelli's Map of Mars.}

\nbarticle{589.} \nbparatext{Atmosphere.}---At one time it was supposed that the atmosphere
of the planet is very dense, but more recent observations have
shown that this cannot be the case. The probability is that its density
is considerably less than that of our own atmosphere. Dr.~Huggins
has found with the spectroscope unequivocal evidence of the presence
of aqueous vapor. Although the planet receives from the sun less
than half the amount of heat and light per unit of surface that the
earth does, the climate appears, for some reason not yet discovered,
to be much more mild than would be expected. So far as
we can judge, the water on the planet is never frozen except very
near the poles. The earth, as seen from Mars, would present much
more extensive snow-caps.

\nbarticle{590.} \nbparatext{Satellites.}---There are two satellites, which were discovered
in August, 1877, by Professor Hall at Washington with the then
new 26-inch telescope. They are exceedingly minute, and can be
seen only with the most powerful instruments. The outer one,
Deimos, is at a distance of 14,600 miles from the centre of the
planet, and has a period of $30^{\text{h}} \:18^{\text{m}}$, while the inner one, Phobos,
is at a distance of only 5800 miles, and its month is but $7^{\text{h}}\: 39^{\text{m}}$
long, not one-third of the \textit{day} of Mars. Owing to this fact it rises in
the \textit{west} every night for the ``\textit{Marticoli}'' (if there are any people
there) and sets in the \textit{east}, after about $11^{\text{h}}$.

Deimos does not do this; it rises in the east like other stars, but
its orbital eastward motion among the stars is so nearly equal to its
diurnal motion westward, that it is nearly 132 hours between two
successive risings. This is more than four of its months, so that it
undergoes all its changes of phase four times in the interval.

Of course, both the satellites are frequently eclipsed,---the inner
one at every revolution; and it also transits across the sun's disc at
every new moon, as seen from some point or other on the planet's
surface.
%% -----File: 349.png---Folio 338-------

Their orbits appear to be exactly circular, and they move exactly
in the plane of the planet's equator; and they \textit{keep} so, maintained in
their relation to the equator by the action of the ``equatorial bulge''
upon the planet.

\nbarticle{591.} As givers of moonlight they do not amount to much. Their
diameters are too small to be measured with any micrometer; but
from their apparent ``\textit{magnitude}'' (\textit{i.e.}, brightness), as seen from
the earth, and assuming that their surfaces have the same reflective
power as that of the planet, Professor Pickering has estimated the
diameter of Phobos, which is the larger one, as about seven miles,
and that of Deimos, as five or six. The light given by Phobos to
the inhabitants of Mars would be about $\frac{1}{60}$ of our moonlight; that
of Deimos about $\frac{1}{1200}$.

The period of Phobos is by far the shortest period in the solar system.
Next to it is that of Mimas, the inner satellite of Saturn,
which, however, is nearly three times as long,---$22 \frac{1}{2}^\text{h}$. This rapidity
of revolution raises important questions as to the theory of the
development of the solar system, and requires modification of the
views which had been held up to the time of their discovery. If
the nebular hypothesis is true, a shortening of the satellite's period,
or a lengthening of the planet's day, must have occurred since the
satellite came into being, since that hypothesis will not account for the
existence of a satellite having a period shorter than the diurnal rotation
of its primary. \figref{illo177}{Fig.~177} is a diagram of the satellite orbits as
%% -----File: 350.png---Folio 339-------
they appeared from the earth in 1888. It is reduced from the
American Nautical Almanac for that year.

\includegraphicsmid{illo177}{\textsc{Fig.~177.}---Orbits of the Satellites of Mars.}

\section*{THE ASTEROIDS, OR MINOR PLANETS.}

\nbarticle{592.} These are a group of small planets circulating in the space
between Mars and Jupiter. The name ``\textit{asteroid}'' was suggested
by Sir William Herschel early in the century, when the first ones
were discovered. The later term \textit{planetoid} is preferred by many.

It was very early noticed that there is a break in the series of the
distances of the planets from the sun. Kepler, indeed, at one time
thought he had discovered the true law\footnote
  {His supposed law was as follows: Imagine the sun surrounded by a hollow
spherical shell, on which lies the orbit of the earth. Inside of this shell inscribe
a regular \textit{icosahedron} (the twenty-sided regular solid), and within that inscribe a
second sphere. This sphere will carry upon it the orbit of Venus. Inside of the
sphere of Venus inscribe an \textit{octahedron} (the eight-sided solid), and the sphere
which fits within it will carry Mercury's orbit. Next, working outwards from the
earth's orbit, circumscribe around the earth's sphere a \textit{dodecahedron}, circumscribing
around it another sphere, and this will carry upon it the orbit of Mars.
Around the sphere of Mars circumscribe the \textit{tetrahedron}, or the regular pyramid.
The corners of this solid project very far, so that the sphere circumscribed
around the tetrahedron will be at a very great distance from the sphere of Mars.
It carries the orbit of Jupiter. Finally, the cube, or \textit{hexahedron}, circumscribed
around the orbit of Jupiter, gives us in the same way the orbit of Saturn. We
thus obtain a series of distances not enormously incorrect (though by no means
agreeing with fact even as closely us does Bode's law); and, moreover, the theory
had the great advantage to Kepler's mind of accounting for the fact that there
are (so far as was then known) but \textit{seven} planets, there being possible but \textit{five}
  regular solids.}
and the real reason why the
planets' distances are what they are. This theory of his was broached
twenty-two years, however, before he discovered the harmonic law,
and he probably abandoned it when he discovered the elliptical form
of the planets' orbits. At any rate, in later life he suggested that
it was likely that there was a planet between Mars and Jupiter too
small to be seen.

The impression that such a planet existed gained ground when
Bode published the law which bears his name in 1772, and it was
still further deepened when, nine years later, in 1781, Uranus was
discovered, and the distance of the new planet was found to conform
to Bode's law. An association of twenty-four astronomers,
mainly German, was immediately formed to look for the missing
planet, who divided the zodiac between them and began the work.
%% -----File: 351.png---Folio 340-------
Singularly enough, however, the first discovery was made, not by a
member of this association, but by Piazzi, the Sicilian astronomer of
Palermo, who was then engaged upon an extensive star-catalogue.
On January~1, 1801, he observed a seventh-magnitude star which by
the next evening had unquestionably moved, and kept on moving.
He observed it carefully for some six weeks, when he was taken ill;
before he recovered, it had passed on towards superior conjunction,
and was lost in the rays of the sun. He named it \textit{Ceres}, after the
tutelary goddess of Sicily.

\begin{fineprint}
When at length the news reached Germany in the latter part of March
it created a great excitement, and the problem now was to rediscover the
lost planet. The association of planet-hunters began the search in September,
as soon as its elongation from the sun was great enough to give any
prospect of success. During the summer Gauss devised his new method
of computing a planetary orbit, and computed the ephemeris of its path.
Very soon after receiving his results, Baron Von Zach rediscovered Ceres
on December~31, and Dr.~Olbers on the next day, just one year after it was
first found by Piazzi.
\end{fineprint}

\nbarticle{593.} In March, 1802, Dr.~Olbers, who in looking for Ceres had
carefully examined the small stars in the constellation of Virgo, on
going over the ground again, found a second planet, which he named
Pallas, a body of about the same brightness as Ceres. \textit{Two} having
now been found, and Pallas having a very eccentric and much inclined
orbit, he conceived the idea that they were fragments of a broken
planet, and that other planets of the same group could probably be
found by searching near the intersection of their two orbits. Juno,
the third, was discovered by Harding at Lilienthal (Schröter's observatory)
in 1804; and Vesta, the largest and brightest of the whole
group (sometimes visible to the naked eye), was found by Olbers
himself in 1807. The search was kept up for several years after
this, but no more planets were found because they did not look after
small enough stars.

The fifth, Astręa, was discovered in 1845 by Hencke, an amateur
astronomer who for fifteen years had been engaged in studying
the smaller stars in hopes of just the reward he captured. In
1846 no asteroid was found (the discovery of Neptune was glory
enough for that year), but in 1847 three more were brought to light;
and since then not a year has passed without adding from one to
twenty to the number. The list at the end of 1888 counts up 281,
and there is no prospect of diminution in the rate of discovery,
though the new ones are mostly very small,---stars of the twelfth
%% -----File: 352.png---Folio 341-------
and thirteenth magnitudes, which require a large telescope to make
them even visible. All the brighter ones have evidently been already
picked up.

\begin{fineprint}
They have been discovered by comparatively a few observers. Four persons
have found more than 20 each: Palisa, of Vienna, who stands far
in advance of all others, has alone discovered 68, and Dr.~Peters of Clinton,
N.Y., 52; Luther of Düsseldorf, 24; and the late Professor Watson, 22.
The German astronomers have thus far discovered 102; the American, 78;
the French, 66; the English, 19; and the Italians, 16.
\end{fineprint}

\nbarticle{594.} \nbparatext{Method of Search.}---The asteroid-hunter selects certain
portions of the sky, usually near the ecliptic, and prepares charts
covering two or three square degrees, on which he sets down all the
stars his telescope will show. This is a very laborious operation,
which in the future is likely to be very much facilitated by photography.
The chart once made, he goes over the ground from time
to time, comparing every object with the map and looking out for
interlopers. If he finds a star which is not on his chart, the \textit{probability}
is that it is a planet, though it may be a variable star which
was invisible when the chart was made. He very soon settles the
question, however, by measuring with the micrometer the distance of
the new star from some of its neighbors, keeping up the process for
an hour or two. If it is a planet, it will move perceptibly in that
length of time.

\begin{fineprint}
Of course, great care must be taken to be sure that it is a \textit{new} planet, and
not one of the multitude already known. Generally it is possible to decide
very quickly which of the known planets will be in the neighborhood, and
a rough computation will commonly decide at once whether the planet is
new or not. Not always, however, and mistakes in this regard are not very
unusual.
\end{fineprint}

These minor planets are all \textit{named}, the names being derived from
mythology and legend. They are also designated by numbers, and
the symbol for each planet is the number written in a circle. Thus,
for Ceres the symbol is \textcircled{\footnotesize 1}; for Hilda, $(\overline{\underline{\text{\footnotesize 153}\rule[-.3ex]{0pt}{1.7ex}}})$; and so on.

\begin{fineprint}
A full list of them, with the elements of their orbits, is published yearly
in the ``Annuaire du Bureau des Longitudes,'' Paris.
\end{fineprint}

\nbarticle{595.} \nbparatext{Their Orbits.}---The mean distance of the different asteroids
from the sun varies greatly, and of course the periods are correspondingly
different. Medusa, $(\overline{\underline{\text{\footnotesize 149}\rule[-.3ex]{0pt}{1.7ex}}})$, has the smallest mean distance
(2.13, or 198,000000 miles), and the shortest period, $3^{\text{y}}\: 40^{\text{d}}$.
%% -----File: 353.png---Folio 342-------
Thule, $(\overline{\underline{\text{\footnotesize 279}\rule[-.3ex]{0pt}{1.7ex}}})$, is the remotest, with a mean distance of 4.30 or 400,000000
miles, and a period of $8^{\text{y}}\: 313^{\text{d}}$. According to Svedstrup,
the mean distance of the ``mean asteroid'' is 2.65 (246,000000
miles), and its period about $4\frac{1}{3}$ years. Its distance from the earth at
time of opposition would be, of course, 1.65, or 153,000000 miles.

The inclinations of their orbits \textit{average} about $8°$; but Pallas, \textcircled{\footnotesize 2},
has an inclination of $35°$, and Euphrosyne, $(\overline{\underline{\text{\footnotesize 31}\rule[-.3ex]{0pt}{1.7ex}}})$, of $26\frac{1}{2}°$.

Several of the orbits are extremely eccentric. Ęthra, $(\overline{\underline{\text{\footnotesize 132}\rule[-.3ex]{0pt}{1.7ex}}})$, has
an almost cometary eccentricity of 0.38, and nine or ten others
have eccentricities exceeding 0.30. They are distributed quite unequally
in the range of distance, there being, as Kirkwood has
pointed out, very few at such distances that their periods would be
exactly commensurable with that of Jupiter.

\nbarticle{596.} \nbparatext{Diameter and Surface.}---Very little is known as to the real
size of these bodies. The four first discovered, and a few of the
newer ones, show a minute but sensible disc in very powerful telescopes.
Vesta, which is the brightest of the whole group, and is just
visible to the naked eye when near opposition, may be from two hundred
to four hundred miles in diameter. Pickering, by photometric
measurements (assuming the reflecting power of the planet's surface
to be the same as that of Mars), finds a diameter of 319 miles.
The other three of the original four are perhaps two-thirds as large.
As for the rest, it is hardly possible that any one of them can be
as much as 100 miles in diameter, and the smallest, such as are now
being discovered in such numbers, are probably less than ten miles
through,---nothing more than ``mountains broke loose.'' The surface
area of one of the smaller ones would hardly make a large
Western farm.

\nbarticle{597.} \nbparatext{Mass, Density, etc.}---As to the individual masses and densities
we have no certain knowledge. It is probable that the density
does not differ much from the density of the crust of the earth, or the
mean density of Mars. If this is so, the mass of Vesta might possibly
be as great as $\frac{1}{15000}$ or $\frac{1}{20000}$ of the earth. On such a planet
the force of superficial gravity would be about $\frac{1}{25}$ to $\frac{1}{30}$ of gravity on
the earth, and a body projected from the surface with a velocity of
about 2000 feet a second---that of an ordinary rifle-ball---would
fly off into space and never return to the planet, but would circulate
around the sun as a planet on its own account. On the
smallest asteroids, with a diameter of about ten miles, it would be
%% -----File: 354.png---Folio 343-------
quite possible to throw a stone from the hand with velocity enough to
send it off into space.

\nbarticle{598.} \nbparatext{Aggregate Mass.}---Although we can only estimate very
roughly the masses of the individual members of the flock, it is possible
to get some more certain knowledge of their \textit{aggregate} mass.
Leverrier found a motion in the line of apsides of the orbit of Mars
which indicates that the whole amount of matter thus distributed in
the space between Mars and Jupiter is equal to about \textit{one-fourth} of
the mass of the earth. This quantity of matter collected into a body
of the same density as Mars would make a planet of about 5000 miles
in diameter.

\nbenlargepage
\begin{fineprint}
The united masses of these which are already known would make only a
very small fraction of such a body. Up to August, 1880, the united bulk of
the asteroids then discovered was estimated at $\frac{1}{4000}$ part of the earth's bulk,
with a mass probably about $\frac{1}{6000}$ of the earth's. Presumably, therefore, the
number of these bodies remaining undiscovered is exceedingly great---to be
counted by thousands, if not by millions. Most of them, of course, must be
much smaller than those which are already known.
\end{fineprint}

\nbarticle{599.} \nbparatext{Forms, Variations of Brightness, and Atmosphere.}---We
have no definite knowledge on this point, but Dr.~Olbers observed
in the case of Vesta certain fluctuations in her brightness which
seemed to him to indicate that she is not a globe, but an angular
mass,---a splinter of rock. This, however, is not confirmed by the
more recent photometric observations of Müller or Pickering.

\begin{fineprint}
Müller examined seven of the asteroids, and found their changes of brightness
very regular. Four of them, one of which is Vesta, behaved precisely
like Mars, as if their surfaces were comparatively smooth, while three others,
Ceres and Pallas among them, behaved like the moon and the planet Mercury,
as if having a rough surface with very little atmosphere, and nearly
cloudless. Some of the earlier observers reported evidences of an extensive
halo around them. Later observations do not confirm it, and it is
not likely that they carry much air with them.
\end{fineprint}

\sloppy
\nbarticle{600.} \nbparatext{Origin.}---With respect to this we can only speculate. Two
views have been held, as has been already intimated. One is, that the
material, which according to the nebular hypothesis ought to have
been concentrated to form a single planet of the class to which the
earth belongs, has failed to be so collected, and has formed a flock
of small separate masses. It is now very generally believed that the
%% -----File: 355.png---Folio 344-------
matter which at present forms the planets was once distributed in
\textit{rings}, like the rings of Saturn. If so, this ring next outside of Mars
would necessarily suffer violent perturbations from the nearness of
the enormous planet Jupiter, and so would be under very different
conditions from any of the other rings. This, as Peirce has shown,
might account for its breaking up into many fragments.

\fussy
The other view is that a planet about the size of Mars has broken
to pieces. It is true, as has been often urged, that this theory in its
original form, as presented by Olbers, cannot be correct. No \textit{single}
explosion of a planet could give rise to the present assemblage of
orbits, nor is it possible that even the perturbations of Jupiter could
have converted a set of orbits originally all crossing at one point
(the point of explosion) into the present tangle. The smaller orbits
are so small that however turned about they lie wholly inside the
larger, and cannot be made to intersect them. If, however, we
admit a \textit{series} of explosions, this difficulty is removed; and if we
grant an explosion at all, there seems to be nothing improbable in
the hypothesis that the fragments formed by the bursting of the
parent mass would carry away within themselves the same forces and
reactions which caused the original bursting; so that they themselves
would be likely enough to explode at some time in their later
history.

At present opinion is divided between these two theories.

\begin{fineprint}
\nbarticle{601.} The number of these bodies already known is so great, and the
prospect for the future is so indefinite, that astronomers are at their wits'
end how to take care of this numerous family. To compute the orbit and
ephemeris of one of these little rocks is more laborious (on account of the
great perturbations produced by Jupiter) than to do the same for one of the
major planets; and to keep track of such a minute body by observation is
far more difficult. Until recently, the German Jahrbuch has been publishing
the ephemerides of such as came within the range of observation each
year; but this cannot be kept up much longer, and the probability is that
hereafter only the larger ones, or those which present some remarkable
peculiarity in their orbits, will be followed up. One little family of them,
however, is ``endowed.'' Professor Watson, at his death, left a fund to the
American National Academy of Sciences to bear the expense of taking care
of the twenty-two which he discovered.
\end{fineprint}

\section*{INTRA-MERCURIAL PLANETS AND THE ZODIACAL LIGHT.}

It is very probable, indeed almost certain, that there are masses of
matter revolving around the sun within the orbit of Mercury.
%% -----File: 356.png---Folio 345-------

\nbarticle{602.} \nbparatext{Motion of the Perihelion of Mercury's Orbit.}---Leverrier,
in 1859, from a discussion of all the observed transits of Mercury,
found that the perihelion of its orbit has a movement of nearly $38''$
a century. This is more than can be accounted for by the action of
the known planets, and, \textit{so far as known}, could be explained only by
the attraction of a planet, or ring of small planets, revolving inside
this orbit nearly in its plane, with a mass about half as great as that
of Mercury itself.

\begin{fineprint}
We say ``\textit{so far as known},'' because an alternative hypothesis has been
proposed, viz., that the law of gravitation, though \textit{strictly true for bodies at
rest} is not absolutely so \textit{for bodies in motion}; that when bodies are moving
towards each other the attraction is less by a minute fraction than if they
were at rest. The hypothesis is known as the \textit{electro-dynamic theory of gravitation},
but has at present very little to support it. If, however, it were true,
then the peculiar motion of the apsides of Mercury's orbit would be a
necessary consequence.
\end{fineprint}

Subsequent investigations by a number of mathematicians have
fully confirmed Leverrier's results; Mercury's orbit is beyond question
affected as it would be if there were an intra-Mercurial planet,
or a number of them.

\begin{fineprint}
\nbarticle{603.} \nbparatext{Dr.\ Lescarbault's Observation: Vulcan.}---A certain country
physician, living some eighty miles from Paris, Dr.~Lescarbault, on the publication
of Leverrier's result, announced that he had actually seen this planet
crossing the sun nine months before, on the 26th of March of that year,
1859. He was visited by Leverrier, who became satisfied of the genuineness
of his observations, and the doctor was duly congratulated and honored as
the discoverer of ``Vulcan,'' which name was assigned to the supposed new
planet. An interesting account of the matter may be found in Chambers'
``Descriptive Astronomy''; and in many of the works published from twenty
to twenty-five years ago, as well as in some more recent ones, ``Vulcan'' is
assigned a place in the solar system, with a distance of about 13,000000
miles and a period of $19+$ days. Lescarbault described  it as having an
apparent diameter of about $7''$, which would make it over 2500 miles in
diameter.

\nbarticle{604.} Nevertheless, it is nearly certain that Vulcan does not exist.
There are various opinions which we need not here discuss as to the explanation
of this pseudo-discovery. But the planet, if real, ought since
1859 to have been visible on the sun's face at certain definite times which
Leverrier calculated and published; and it has never been seen, though
very carefully looked for. Small, round, dark objects have from time to
%% -----File: 357.png---Folio 346-------
time been indeed reported on the sun's disc, which in the opinion of the
observers at the time were not sun spots; but most of these observations
were made by amateurs with comparatively little experience, with small
telescopes, and with no measuring apparatus by which they could certainly
determine whether or not the spot seen moved like a planet. In most of
these cases photographs or simultaneous observations made elsewhere by
astronomers of established reputation, and having adequate apparatus, have
proved that the problematical ``dots'' were really nothing but ordinary
small sun spots, and the probability is that the same explanation applies to
the rest.
\end{fineprint}

\nbarticle{605.} \nbparatext{Eclipse Observations.}---A planet large enough to be seen
distinctly on the sun by a $2\frac{1}{2}$-inch telescope, such as Lescarbault
used, would be a conspicuous object at the time of a solar eclipse,
and most careful search has been made for the planet on such occasions;
but so far, although \textit{stars} of the third and fourth magnitudes,
and even of the fifth, have been clearly seen by the observers within
a few degrees of the eclipsed sun, no planet has been found.

\begin{fineprint}
One apparent exception occurred in 1878. During the eclipse of that
year, Professor Watson observed two \DPtypo{starlike}{star-like} objects (of the fourth magnitude),
which he thought at the time could not be identified with any known
stars consistently with his observations. Mr.~Swift, also, at the same eclipse,
reported the observations of two bright points very near the sun; but these
from his statement could not (both) have been identical with Watson's stars.
Later investigations of Dr.~Peters have shown that the assumption of a
very small and very likely error in Professor Watson's circle-readings
(which were got in a very ingenious, but rather rough way, without the
use of graduations) would enable his stars to be identified with $\theta$ and $\zeta$
Cancri, and it is almost certain that these were the stars he saw.
Mr.~Swift's observations remain unexplained. With this exception, the eclipse
observations all give negative results, and astronomers generally are now
disposed to consider the ``Vulcan question'' as settled definitely and adversely.
\end{fineprint}

\nbarticle{606.} At the same time it is extremely probable that there are a
number, and perhaps a very great number, of \textit{intra-Mercurial asteroids}.
A body two hundred miles in diameter near the sun would
have an angular diameter of only about $\frac{1}{2}''$, as seen from the earth,
and would not be easily visible on the sun's disc, except with very
large telescopes. It would not be at all likely to be picked up accidentally.
Objects with a diameter of not more than forty or fifty
miles would be almost sure to escape observation, either at a transit
or during a solar eclipse.
%% -----File: 358.png---Folio 347-------

\begin{fineprint}
\nbarticle{607.} \nbparatext{Zodiacal Light.}---This is a faint, soft beam of light that extends
both ways from the sun along the ecliptic. In the evening it is best seen
in February, March, and April, because the portion of the ecliptic which
lies east of the sun's place is then most nearly perpendicular to the western
horizon. During the autumnal months the zodiacal light is best seen in
the morning sky for a similar reason. In our latitudes it can seldom be
traced more than $90°$ or $100°$ from the sun; but at high elevations within
the tropics it is said to extend entirely across the sky, forming a complete
ring, and there is said to be in it at the point exactly opposite to the sun a
patch a few degrees in diameter of slightly brighter luminosity, called the
``Gegenschein'' or ``counter-glow.''

The portions of this object near the sun are reasonably bright, and even
conspicuous at the proper seasons of the year; but the more distant portions
in the neighborhood of the ``counter-glow'' are so extremely faint that it is
only possible to observe them at a distance from cities and large towns, in
places where the air is free from smoke, and where the darkness of the sky
is not affected by the general illumination due to gas and electric lights.

\nbarticle{608.} The cause of the phenomenon is not certainly known, but at present
the theory most generally accepted attributes it to \textit{sunlight reflected by
myriads of small meteoric bodies} which are revolving around the sun nearly in
the plane of the ecliptic, forming a thin, flat sheet like one of Saturn's rings,
and extending far beyond the orbit of the earth. It may be that the denser
portion of this meteoric ring within the orbit of Mercury is the cause of the
motion of the perihelion of that planet which Leverrier detected; it is for
this reason that we deal with the subject here rather than in connection with
meteors. While this theory, however, is at present more generally accepted
than any other, it cannot he said to be established. Some are disposed to
consider the zodiacal light as a mere extension of the sun's corona, whatever
that may be.

\nbarticle{608*.} \nblabel{pg:347}(Note to \artref{Art.}{588}.) \nbparatext{The Canals of Mars.}---According to
Schiaparelli (and his observations are at least partially confirmed by others)
a most characteristic feature of the planet's surface is a series of long,
straight, narrow ``canals'' connecting the larger bodies of water. These
canals were first seen and recognized in 1877. In 1881 they were seen again,
and at that time nearly all of them \textit{double}. If there is not some fallacy in
the observation, the problem as to the nature of these canals, and the cause
of their ``gemination,'' is a very important and perplexing one. It is hoped
that at the next favorable opposition in 1892 it may find its solution.
\end{fineprint}
\chelabel{CHAPTERXIV}
%% -----File: 359.png---Folio 348-------

\Chapter{XV}{The Major Planets}
\nbchapterhang{\stretchyspace
THE PLANETS CONTINUED.---THE MAJOR PLANETS: JUPITER,
SATURN, URANUS, AND NEPTUNE.}

\section*{JUPITER.}

\nbarticle{609.} While this planet is not so brilliant as Venus at her best,
it stands next to her in this respect, being on the average about five
times brighter than Sirius, the brightest of the fixed stars. Jupiter,
moreover, being a ``superior'' planet, is not confined, like Venus,
to the neighborhood of the sun, but at the time of opposition is the
chief ornament of the midnight sky.

\nbarticle{610.} \nbparatext{Orbit.}---The orbit presents no marked peculiarities. The
\textit{mean distance} of the planet from the sun is 483,000000 miles. The
\textit{eccentricity} of the orbit being nearly $\frac{1}{20}$ (0.04825); the greatest and
least distances vary by about 21,000000 miles each way, making
the planet's greatest and least distances from the sun 504,000000
and 462,000000 miles respectively. The average distance of the
planet from the earth at opposition is 390,000000, while at conjunction
it is 576,000000 miles. The minimum opposition distance is
only 369,000000, which is obtained when the opposition occurs about
October~6, Jupiter being in perihelion when its heliocentric longitude
is about $12°$. At an aphelion opposition (in April) the distance
is 42,000000 miles greater; that is, 411,000000.

The relative brightness of Jupiter at an average conjunction and
at the nearest and most remote oppositions is respectively as the
numbers 10, 27, and 18. The average brightness at opposition is,
therefore, more than double that at conjunction; and at an October
opposition the planet is fifty per cent brighter than at an April one.
The differences are considerable, but far less important than in the
case of Mars, Venus, and Mercury.

The \textit{inclination} of the orbit to the ecliptic is small,---only $1°\: 19'$.

\nbarticle{611.} \nbparatext{Period.}---The \textit{sidereal period} is 11.86 years, and the \textit{synodic}
is 399 days (a number easily remembered), a little more than a year
and a month. The planet's orbital velocity is about eight miles a
second.
%% -----File: 360.png---Folio 349-------

\sloppy
\nbarticle{612.} \nbparatext{Dimensions.}---\textit{The planet's apparent diameter} varies from
$50''$ at an October opposition (or $45 \frac{1}{2}''$ at an April one) to $32''$ at
conjunction. The form, however, of the planet's disc is not truly
circular, the polar diameter being about $\frac{1}{17}$ part less than the equatorial,
so that the eye notices the oval form at once. The equatorial
diameter in \textit{miles} is 88,200, the polar being 83,000. Its mean\footnote
  {The mean diameter of an oblate spheroid is $\dfrac{2a + b}{3}$ not $\dfrac{a+b}{2}$. Of the three
  axes of symmetry which cross at right angles at the planet's centre, \textit{one} is the
  axis of rotation, and both the others are equatorial.}
diameter, therefore, is 86,500,---almost eleven times that of the
earth.

\fussy
This makes its surface 119 times, and its volume 1300 times, that
of the earth. It is by far the largest of the planets in the system;
in fact, whether we regard its bulk or its mass, larger than all the rest
put together.

\nbarticle{613.} \nbparatext{Mass, Density, etc.}---Its \textit{mass} is very accurately known,
both by the motions of its satellites, and the perturbations of the
asteroids. It is $\frac{1}{1048}$ of the sun's mass, or very nearly 316 times
that of the earth. Comparing this with its volume, we find its \textit{density}
0.24, less than $\frac{1}{4}$ the density of the earth, and almost precisely the
same as that of the sun. Its mean \textit{superficial gravity} comes out 2.64
times that of the earth; that is, a body on Jupiter would weigh $2 \frac{5}{8}$
times as much as upon the surface of the earth; but on account of
the rapid rotation of the planet and its ellipticity there is a very considerable
difference between the force of gravity at the equator and
at the pole, amounting to $ \frac{1}{5}$ of the equatorial gravity. (On the earth
the difference is only $ \frac{1}{190}$.)

\nbarticle{614.} \nbparatext{Phases and Albedo.}---Its orbit is so much larger than that
of the earth that the planet shows no sensible phases, even at quadrature,
though at that time the edge farthest from the sun shows a
slight darkening.

The reflecting power, or \textit{Albedo}, of the planet's surface is very
high,---0.62 according to Zöllner, that of white paper being only
0.78. The centre of the disc of this planet (and the same is also
true of Saturn) is considerably brighter than the limb---just the
reverse, as will be remembered, from the condition of things upon
the moon, and upon Mars, Venus, and Mercury. This peculiarity
of a darkened limb, in which Jupiter resembles the sun, has suggested
%% -----File: 361.png---Folio 350-------
the idea that it is to some extent \textit{self-luminous}. This, however,
is not a necessary consequence, as a nearly transparent atmosphere
overlying a uniformly reflecting surface would produce the
same effect.

The light which the planet omits, if it emits any, must be very
feeble as compared with sunlight, since the satellites, when they are
eclipsed by entering the shadow, become totally invisible.

\nbarticle{615.} \nbparatext{Axial Rotation.}---The planet rotates on its axis in \textit{about}
$9^{\text{h}}\: 55^{\text{m}}$. The time can be given only approximately, not because
it is difficult to find and observe distinct markings on the planet's
disc, but simply because different results are obtained from different
spots, according to their nature and their distance from the
planet's equator. Speaking generally, spots near the equator indicate
a shorter day than those in higher latitudes, and certain small,
sharply defined, bright, white spots, such as are often seen, give a
quicker rotation than the dark markings in the same latitude.

\begin{fineprint}
For instance, a white spot observed near the equator in 1886 for several
months gives $9^{\text{h}}\: 50^{\text{m}}\: 4^{\text{s}}$, while another one near latitude $60°$ gave $9^{\text{h}} \:
55^{\text{m}}\: 12^{\text{s}}$. The great red spot has given values ranging from $9^{\text{h}}\: 55^{\text{m}} \:
34^{\text{s}}.9$ (in 1879) to $9^{\text{h}}\: 55^{\text{m}}\: 40^{\text{s}}.7$ (in 1886),---the time of rotation as
determined in each case, being certainly accurate within half a second.
The progressive increase has been regular and unmistakable, and is not
due to any possible uncertainty in the observations.
\end{fineprint}

\nbarticle{616.} \nbparatext{The Axis of Rotation and the Seasons.}---The plane of the
equator is inclined only $3°$ to that of the orbit, so that as far as the
sun is concerned there can be no seasons. The heat and light
received from the sun by Jupiter are, however, only about $\frac{1}{27}$ as
intense as the solar radiation at the earth, its distance being 5.2
times as great.

\nbarticle{617.} \nbparatext{Telescopic Appearance.}---Even in a small telescope the
planet is a beautiful object. When near opposition a magnifying
power of only 40 makes its apparent size equal to that of the full
moon (though, as remarked in connection with Venus, no novice
would receive that impression), and with a telescope of 8 or 10 inches
aperture, and with a magnifying power of 300 or 400, the disc is
covered with an infinite variety of beautiful and interesting details
which rapidly shift under the observer's eye in consequence of the
planet's swift rotation. The picture is rich in color, also, browns and
reds predominating, in contrast with olive-greens and occasional
%% -----File: 362.png---Folio 351-------
purples; but to bring out the colors well and clearly requires large
instruments. For the most part the markings are arranged in streaks
more or less parallel to the planet's equator, as shown by \figref{illo178}{Fig.~178}.
With a small telescope the markings usually reduce to two dark and
comparatively well-defined belts, one on each side of the equator,
occupying about the same regions of latitude that the trade-wind
zones do upon the earth; and very likely in Jupiter's case similar
aerial currents have something to do with the appearance, though
upon Jupiter, as has been already said, the solar heat is a comparatively
unimportant factor. The markings upon the planet are almost,
if not entirely, \textit{atmospheric}, as is proved by the manner in which
they change their shapes and relative positions. They are \textit{cloud
forms}. It is hardly probable that we ever see anything upon the
solid surface of the planet underneath, nor is it even certain that
the planet has anything solid about it. In \figref{illo178}{Fig.~178}, the upper
left-hand figure is from a drawing by Trouvelot made in February,
1872; the second is by Vogel in 1880. The small one below represents
the planet as seen in a small telescope.

\includegraphicsmid{illo178}{\textsc{Fig.~178.}---Telescopic Views of Jupiter.}

\nbarticle{618.} \nbparatext{The Great Red Spot.}---While most of the markings on the
planet are evanescent, it is not so with all. There are some which
%% -----File: 363.png---Folio 352-------
are at least ``sub-permanent,'' and continue for years, not without
change indeed, but with only slight changes. The ``great red spot''
is the most remarkable instance so far. It seems to have been first
observed by Prof.\ C.~W.\ Pritchett of Glasgow, Missouri, in July,
1878, as a pale, pinkish, oval spot some $13''$ in length by $3''$ in
width (30,000 miles by 7000). Within a few months it had been
noticed by a considerable number of other observers, though at first
it did not attract any special attention, since no one thought of it as
likely to be permanent. The next year, however, it was by far the
most conspicuous object on the planet. It was of a clear, strong
brick-red color, with a length fully one-third the diameter of the
planet and a width about one fourth of its length.

\includegraphicsmid{illo179}{\textsc{Fig.~179.}---Jupiter's ``Red Spot.'' From Drawings by Mr.~Denning. 1880--85.}

\begin{fineprint}
For two or three years it remained without much change: in 1882--83
it gradually faded out: in 1885 it had become a pinkish oval \textit{ring}, the
central part being apparently occupied with a white cloud. In 1886 it was
again a little stronger in color, and the same in 1887,---an object not difficult
to see with a large telescope, but the merest ghost of what it was in
1880. The present year (1888) its appearance is about the same as in 1887,---perhaps
a little paler. During the ten years its form and size have varied
very little. It lies at the southern edge of the southern equatorial belt, in latitude
%% -----File: 364.png---Folio 353-------
about $35°$, and for some reason the belt seems to be ``notched out'' for
it, so that there has been always a narrow white streak separating the belt
from the spot. Even when the spot was palest and hard to see its \textit{place} was
always evident at once from the indentation in the outline of the belt.

Such phenomena suggest abundant matter for speculation which would
be out of place here. It must suffice to say that no satisfactory explanation
of the phenomena has yet been presented. The unquestionable fact before
mentioned (\artref{Art.}{615}), that the time of rotation of the spot has changed by
more than $5^\text{s}$ in the ten years, greatly complicates the subject. \figref{illo179}{Fig.~179},
from the drawings of Mr.~Denning, represents the appearance of the spot at
four different dates; viz., 1,~1880, Nov.~19; 2,~1882, Oct.~30; 3,~1884,
Feb.~6; 4,~1885, Feb.~25.
\end{fineprint}

\nbarticle{619.} \nbparatext{Temperature and Physical Constitution.}---The rapidity of
the changes upon the visible surface implies the expenditure of a considerable
amount of heat, and since the heat received from the sun is
too small to account for the phenomena which we see, Zöllner, thirty
years ago, suggested that it must come from within the planet, and
that in all probability Jupiter is at a temperature not much short of
incandescence,---hardly yet solidified to tiny considerable extent.
Mr.~Proctor has given special currency to these views among English
readers. The idea that Jupiter might be such a ``semi-sun'' is not
at all new. Buffon, Kant, Nasmyth, and Bond all entertained and
discussed it; but it is only since the investigations of Zöllner that it
has become an accepted item of scientific belief. (See Clerke's
``History of Astronomy,'' p.~335 seqq.)

\nbarticle{620.} \nbparatext{Atmosphere.}---As to the composition of the planet's atmosphere,
the spectroscope gives us rather surprisingly little information.
We get from the planet a good solar spectrum with the solar lines
well marked, but there are no well-defined absorption bands due to
the action of the planet's atmosphere. There are, however, some
\textit{shadings} in the lower red portion of the spectrum that are probably
thus caused. The light, for the most part, seems to come from the
upper surface of the planet's envelope of clouds without having
penetrated to any depth.

\nbarticle{621.} \nbparatext{Satellite System.}---Jupiter has four satellites,---the first
heavenly bodies ever \textit{discovered}---the first revelation of Galileo's
telescope. His earliest observation of them was on Jan.~7, 1610,
and in a very few weeks he had ascertained their true character, and
determined their periods with an accuracy which is surprising when
we consider his means of observation. The number of the heavenly
%% -----File: 365.png---Folio 354-------
bodies was now no longer \textit{seven}, and the discovery excited among
churchmen and schoolmen a great deal of angry incredulity and
vituperation. Galileo called them ``the Medicean stars.''

They are now usually known as the first, second, etc., in the order of
distance from the primary, but they also have names which are sometimes
used; viz., Io, Europa, Ganymede, and Callisto. Their relative
distances range between 262,000 and 1,169,000 miles, being very
approximately 6, 9, 15, and 26 radii of the planet. The distance of
the first from the surface of the planet is almost exactly the same as
that of our own moon from the surface of the earth. Their sidereal
periods range between $1^{\text{d}}\: 18 \frac{1}{2}^{\text{h}}$ and $16^{\text{d}}\:16 \frac{1}{2}^{\text{h}}$ (accurate values in
distances and periods are given in the table in the \hyperref[app:II]{Appendix}). The
orbits are almost exactly circular, and lie in the plane of the planet's
equator.

\begin{fineprint}
The satellites slightly disturb each other's motions, and from these
disturbances their masses can be ascertained in terms of the planet's mass.
The third, which is much the largest, has a mass of about $\frac{1}{11000}$ of the
planet's, a little more than double the mass of our own moon. The mass
of the first satellite appears to be a little less than $\frac{1}{5}$ as much. The second
is somewhat larger than the first, and the fourth is about half as large as the
third; \textit{i.e.}, it has about the mass of our own moon. The densities of the
first and fourth appear to be not very different from that of the planet itself,
while the densities of the second and third are considerably greater.

\nbarticle{622.} \nbparatext{Relation between Mean Motions and Longitudes of the
Satellites.}---In consequence of their mutual interaction a curious relation % [*F1: Is this text intermediate between \small and the main text size?][F2: Surely the printer wasn't that mean.]
(discovered by~La Place) exists between the mean motions of the first three
satellites. The mean motion is of course $360°$ divided by $T$ ($T$ being the
satellite's period). It appears that the mean motion of the first plus twice
the mean motion of the third equals three times that of the second, or
\[
\frac{1}{T_1} + \frac{2}{T_3} = \frac{3}{T_2}.
\]
A similar relation holds for their longitudes:
\[
L_1 + 2 L_3 = 3 L_2 + 180°;
\]
so that they cannot all three come into opposition or conjunction with the
sun at once. These relations are permanently maintained by their mutual
attractions: \textit{exactly} in the long run, though there are slight perturbations
produced by the fourth satellite which disturb the arrangement slightly for
short periods. The fourth satellite does not come into any such arrangement.
\end{fineprint}
%% -----File: 366.png---Folio 355-------

\nbarticle{623.} \nbparatext{Diameters, etc.}---The diameter of the first satellite
is a little more than 2400 miles; the second is almost exactly the
size of our own moon, \textit{i.e.}, between 2100 and 2200 miles;
and the third and fourth have diameters, respectively, of 3600 and
3000 miles, the third, Ganymede, being much larger than either of
his sisters. When Jupiter is in opposition, the fourth satellite is
sometimes nearly $10 \frac{1}{2}'$ away from the planet, or
$\frac{1}{3}$ of the moon's diameter; and in very clear air can be
seen by a sharp eye without telescopic aid. The third, though much
larger, never goes more than $6'$ from the planet, and it is perhaps
doubtful whether it is ever seen with the naked eye, unless when the
fourth happens to be close beside it. A good opera-glass
will easily show them all as minute points of light.

\nbarticle{624.} \nbparatext{Brightness.}---Since the sunlight of Jupiter is only
$\frac{1}{27}$ as intense as ours, the moonlight made by the
satellites is decidedly inferior to our own, although their
reflective power appears to be higher than that of the lunar
surface. They differ among themselves considerably in this respect.
The fourth satellite is of an especially dark complexion, so that it
often looks perfectly black when it passes between us and the
planet, and is projected on the disc. The others, under similar
circumstances, show light or dark according as they have a dark or
light portion of the planet for a background. Even the fourth, when
crossing the disc, is always seen
bright while very near the planet's limb.

\nbarticle{625.} \nbparatext{Markings upon the Satellites.}---The satellites show
sensible discs when viewed with a large telescope, and all of them
but the second sometimes show dark markings upon the surface. These
markings, however, are only visible under the most favorable
circumstances, and it has not been possible to determine whether
they are atmospheric or really geographical, nor
has it yet been possible to deduce from them the satellites' periods of rotation.

\nbarticle{626.} \nbparatext{Variability.}---Galileo noticed variations in the
brightness of the satellites at different times, and subsequent
observers have confirmed his result. In the case of the fourth
satellite there seems to be a regular variation depending upon the
place of the satellite in its orbit, and suggesting that in its
axial rotation it behaves like our own moon, keeping always the same
side next its primary. In addition it shows other \textit{irregular}
changes in its luminosity: so also do the other satellites according
to nearly all authorities, though it is singular that one or two of
the best observers do not find any such irregularity indicated by
their instrumental\footnote{ Clerke's ``History of Astronomy,''
p.~339.} photometric
observations.

\nbarticle{627.} \nbparatext{Eclipses and Transits.}---The satellites' orbits are so
nearly in the plane of the planet's orbit that, excepting the
fourth, they all pass through the shadow of the planet, and suffer
eclipse at every
%% -----File: 367.png---Folio 356-------
revolution. At conjunction, also, they cast their shadows upon the
planet, and these shadows can easily be seen in the telescope as
black dots on the planet's disc, the satellites themselves, which
cross the disc about the same time, being much more difficult to
observe. The fourth satellite escapes eclipse when Jupiter is far
from the node of its orbit. Thus, during 1888 and in the first half
of 1889, there are no eclipses of Callisto at all.

Exactly at opposition or conjunction the planet's shadow lies
straight behind it out of our sight, so that we cannot at that time
observe the eclipses of the satellites, but only their transits
across the disc. Before and after these times, however, the shadow
lies one side of the planet.

\includegraphicsmid{illo180}{\textsc{Fig.~180.}---Eclipses of Jupiter's Satellites, at Western Elongation.}

When the planet is at quadrature and the condition of things is as
represented in \figref{illo180}{Fig.~180} (which is drawn to scale), the shadow
projects so far to one side of the planet that the whole eclipse of
all the satellites, except the first, takes place clear of the
planet's disc,---both the disappearance and reappearance of the
satellite being visible.

\nbarticle{628.} \nbparatext{``Equation of Light.''}---The most important use that has
been made of these eclipses has been to ascertain the time required
by light in traversing the distance between us and the sun, the so-called
%% -----File: 368.png---Folio 357-------
``\textit{equation of light}.'' It was in 1675 that Roemer, the Danish
astronomer (the inventor of the transit instrument, meridian circle,
and prime vertical instrument,---a man nearly a century in advance of
his day), found that the eclipses of the satellites showed a peculiar
variation in their times of occurrence, which he explained as due to
the time taken by light to pass through space. His bold and original
suggestion was rejected by most astronomers for more than fifty
years,---until long after his death,---when Bradley's discovery of
aberration (\artref{Art.}{225}) proved the correctness of his views.

\includegraphicsouter{illo181}{\textsc{Fig.~181.}\\
Determination of the Equation of Light.}

\nbarticle{629.} If the planet and earth remained at an invariable distance
the eclipses of the satellites would recur with unvarying regularity
(their disturbances being very slight), and the mean interval could
be determined, and the times tabulated. But if we thus predict the
times of eclipses for a synodic period of the planet, then, beginning
at the time of opposition,
it will be found that as the
planet recedes from the earth,
the eclipses fall constantly more
and more behindhand, and by
precisely the same amount for
all four of the satellites. The
difference between the tabulated
and observed time continues to
increase until the planet is near
conjunction, when the eclipses
are more than sixteen minutes
late.

\begin{fineprint}
From the insufficient observations
at his command, Roemer
made the difference twenty-two
minutes.
\end{fineprint}

After the conjunction, the eclipses quicken their pace and exactly
make up all the loss; so that when opposition is reached once more,
they are again on time.

It is easy to see from \figref{illo181}{Fig.~181} that at opposition the planet is
nearer the earth than at conjunction by just twice the radius of the
earth's orbit; \textit{i.e.}, $ JB - JA = 2 SA$. The whole apparent retardation
of the eclipses between opposition and conjunction, should therefore
be exactly twice the time required for light to come from the sun to
the earth. This time is very nearly 500 seconds, or $8^{\text{m}}\: 20^{\text{s}}$.
%% -----File: 369.png---Folio 358-------

\begin{fineprint}
Early in the century Delambre, from all the satellite eclipses of which he
could then secure observations, found it to be $493^{\text{s}}$. A few years ago a
redetermination by Glasenapp of Pulkowa made it $501^{\text{s}}$, from fifteen years'
observation of the eclipses of the first satellite. Probably this value is much
nearer the truth than Delambre's.
\end{fineprint}

\sloppy
\nbarticle{630.}\hspace{0em} \nbparatext{Photometric Observations of the Eclipses.}---The eclipses are
\textit{gradual} phenomena, the obscuration of the satellite proceeding continuously
from the time it first strikes the shadow of the planet until
it entirely vanishes. The moment at which the satellite seems to
disappear depends, therefore, on the state of the air and of the
observer's eye, and upon the power of his telescope. The same is
true of the reappearance; so that the observations are doubtful to
the extent of from half a minute for the first satellite (which moves
quickly), to a full minute for the fourth. Professor Pickering has
proposed to substitute for this comparatively indefinite moment of
disappearance or reappearance, \textit{the instant when the satellite has lost
or regained just half its normal light}, and he determines this instant
by a series of photometric comparisons with one of the neighboring
uneclipsed satellites, or with the planet itself.
\fussy

\begin{fineprint}
These comparisons are made with a special photometer devised for the
purpose, and planned with reference to rapid reading: by merely turning a
small button, the observer is immediately able to make the image of the
uneclipsed satellite appear to be of the same brightness as the satellite which
is disappearing, and the observations can be repeated very rapidly with the
help of special contrivances for recording the times and readings. It is
found that this instant of ``half-brightness'' can be deduced from the set of
photometric readings with an error not much exceeding a second or two.
Observations of this kind have now been going at Cambridge (U.~S.)\footnote
  {Professor Pickering has more recently (August, 1888) applied \textit{photography}
  to these observations with most gratifying success. A series of pictures is taken,
  each with an exposure of $10^{\text{s}}$, the time being recorded on a chronograph, and
  they determine with great precision the moment when the satellite's brightness
  had any special value, say fifty per cent of its maximum.}
for
several years. A similar plan has also been devised by Cornu, and is being
carried out at the Paris Observatory under his direction.

A series of such observations covering the planet's whole period of twelve
years, ought to give us a much more accurate determination of the light-equation
than we now have.
\end{fineprint}

\nbarticle{631.} Until 1849 our only knowledge of the velocity of light was
obtained by observations of Jupiter's satellites. By assuming as
%% -----File: 370.png---Folio 359-------
known the earth's distance from the sun, the velocity of light
follows when we know the time occupied by light in coming from
the sun. At present, however, the case is reversed: we can determine
the velocity of light by two independent experimental methods,
and with a surprising degree of accuracy; and then, knowing the
velocity and the light-equation, we can deduce the distance of the sun.

\section*{SATURN.}

\nbarticle{632.} \nbparatext{The Orbit and Period.}---Saturn is the remotest of the
ancient planets, its mean distance from the sun being 9.54 astronomical
units, or 886,000000 miles. This distance varies by nearly
50,000000 miles on account of the eccentricity of its orbit (0.056),
which is a little greater than that of Jupiter.

Its nearest approach to the earth at a December opposition (the
longitude of its perihelion being $90°\: 4'$) is 744 millions of miles, and
its greatest distance at a May conjunction is 1028 millions. It
is so far from the sun that these changes of distance do not so
greatly affect its apparent brightness, as in the case of the nearer
planets, the whole range of variation from this cause being less than
two to one; that is, at the nearest of all oppositions, the planet is
not twice as bright as at the remotest of all conjunctions. The
changing phases of the rings make quite as great a difference as the
variations of distance.

The \textit{orbit is inclined} to the ecliptic about $2\frac{1}{2}°$.

The \textit{sidereal period} of the planet is \textit{twenty-nine and one-half years},
the \textit{synodic period} being 378 days.

The planet itself is unique among the heavenly bodies. The great
belted globe carries with it a retinue of eight satellites, and is surrounded
by a system of rings unlike anything else in the universe
so far as known, the whole constituting the most beautiful and most
interesting of all telescopic objects.

\sloppy
\nbarticle{633.} \nbparatext{Diameter, Volume, and Surface.}---The apparent mean diameter
of the planet varies from $20''$ to $14''$ according to the distance.
We say \textit{mean} diameter because this planet is more flattened at the
pole than any other, its ellipticity being nearly ten per cent,
though different observers vary somewhat in their results. The
equatorial diameter of the planet is about 75,000 miles, and its polar
about 68,000, the mean being very nearly 73,000, or a little more
than nine times that of the earth. Its \textit{surface} is therefore about
eighty-two times, and its \textit{volume} 760 times that of the earth.
%% -----File: 371.png---Folio 360-------

\fussy
\nbarticle{634.} \nbparatext{Mass, Density, and Gravity.}---Its \textit{mass} is only ninety-five
times the earth's mass, from which follows the remarkable fact that
the \textit{density} of Saturn is \textit{only one-eighth that of the earth}, or \textit{only
about five-sevenths that of water}. It is by far the least dense of all
the planets. The \textit{superficial gravity} is 1.2.

\sloppy
\nbarticle{635.} \nbparatext{Axial Rotation.}---It revolves upon its axis in about $10^\text{h}\: 14^\text{m}$
according to a determination of Professor Hall, made in 1876 by
means of a white spot which suddenly appeared upon its surface,
and continued visible for some weeks. Although the surface of the
planet is beautifully marked with belts which often show delicate
rose-colored tints, it is seldom that any well-defined markings present
themselves by which the rotation can be determined.

\fussy
The \textit{inclination of the axis} is about $28°$.

\nbarticle{636.} \nbparatext{Surface, Albedo, and Spectrum.}---As in the case of Jupiter,
the edges of the disc are not quite so brilliant as the central portions,
so that the belts appear to fade out near the limb. These
belts are less distinct and less variable than those of Jupiter; and
are arranged as shown in \figref{illo182}{Fig.~182}, with a very brilliant zone at
the equator, though the engraving much exaggerates the contrast.
The planet's pole is marked by a darkish cap of greenish hue.

According to Zöllner, the \textit{Albedo}, or reflecting power of the surface
is 0.52, almost precisely the same as that of Venus, but a little inferior
to that of Jupiter. The \textit{spectrum} of the planet is the solar spectrum
without any evidence of the presence of water-vapor, so far as
can be made out, but with certain unexplained dark bands in the red
and orange similar to these observed in the spectrum of Jupiter. The
darkest of these bands, however, are not seen in the spectrum of the
ring; this might have been expected, since the ring probably has
but little atmosphere.

\nbarticle{637.} \nbparatext{The Rings.}---The most remarkable peculiarity of Saturn is
his \textit{ring-system}. The planet is surrounded by three, thin, flat, concentric
rings like circular discs of paper pierced through the centre.
Two of them are bright, while the third, the one nearest to the planet,
is dusky and comparatively difficult to see. They are generally referred
to by Struve's notation as $A$, $B$, and $C$, $A$ being the exterior
one.

\includegraphicsmid{illo182}{\textsc{Fig.~182.}---Saturn and his Rings.}

For nearly fifty years this appendage of Saturn was a complete
enigma to astronomers, Galileo, in 1610, saw with his little telescope
%% -----File: 372.png---Folio 361-------
that the planet appeared to have something attached to it on
each side, and he announced the discovery that ``the outermost
planet is triple,''---``ultimam planetam tergeminam observavi.''
Not long afterwards the rings were edgewise to the earth so that they
became invisible to him; and in his perplexity he inquired ``whether
Saturn had devoured his children, according to the legend.'' Huyghens,
%% -----File: 373.png---Folio 362-------
in 1655, was the first to solve the problem and explain the true
structure of the rings. Cassini,\footnote
  {In consequence of a misunderstanding of some expressions used by Ball, an
  English astronomer who observed Saturn in 1665--66, the discovery of the division
  between the rings was for a time attributed to him, and statements to that effect
  will be found in a number of important books. The original drawings belonging
  to his paper in the Philosophical Transactions have, however, recently been
  found, and show that he did not see the division at all, nor, indeed, even understand
  that the appendage was a ring.}
twenty years later, discovered that
the ring was double,---composed of two concentric portions with a
narrow black rift of division between them.

The third, or dusky ring, $C$, is an American discovery, and was
first brought to light by W.~C. Bond at Cambridge, U.~S., in November,
1850. About two weeks later, but before the news had been
published in England, it was also discovered independently by
Dawes.

\begin{fineprint}
For a while there was some question whether it was not really a new
formation; but an examination of old drawings shows that Herschel and
several other astronomers had previously seen it where it crosses the planet,
although without recognizing its character.

\nbarticle{638.} \nbparatext{Dimensions of the Rings.}---The outer ring, $A$, has an exterior
diameter of 168,000 miles, and is a little more than 10,000 miles wide. The
division between it and ring, $B$, is about 1600 miles in width, and apparently
perfectly uniform all around. Ring $B$ is about 16,500 miles wide, and is
much brighter than $A$, especially at its outer edge. At the inner edge it
becomes less brilliant, and is joined without any sharp line of demarcation
by ring $C$, which is sometimes known as the ``\textit{gauze}'' or ``\textit{crape}'' ring,
because it is only feebly luminous and is semi-transparent, allowing the
edge of the planet to be seen through it. The innermost ring is nearly, perhaps
not quite, as wide as the outer one, $A$. There is thus left a clear space
of from 9000 to 10,000 miles in width between the planet's equator and the
inner edge of the gauze ring, the whole ring system having an external
diameter of 168,000 miles, and a width of between 36,000 and 37,000.
\end{fineprint}

The \textit{thickness} of the rings is very small indeed, probably not exceeding
100 miles. If we were to construct a model of them on the
scale of 10,000 miles to the inch, so that the outer one would be nearly
seventeen inches in diameter, the thickness of an ordinary sheet of
writing paper would be about in due proportion. This extreme thinness
is proved by the appearances presented when the plane of the
ring is directed towards the earth, as it is once in every fifteen years.
%% -----File: 374.png---Folio 363-------
At that time the ring becomes invisible except to the most powerful
telescopes.

\nbarticle{639.} \nbparatext{Phases of the Rings.}---The rings are parallel to the equator
of the planet, which is inclined about $27°$ to its orbit, and about $28°$
to the plane of the ecliptic, the two nodes of the ring being in longitude
$166°$ and $346°$, in the constellations of Aquarius and Leo. Now
in the planet's revolution around the sun, the plane of the planet's
equator and of the rings always keeps parallel to itself (as shown
in \figref{illo183}{Fig.~183}), just as does the plane of the earth's equator. Twice,
therefore, in the planet's revolution, when the plane of the ring
passes through the earth, we see it edgewise; and twice at its maximum
width, when it is at the points half-way between the nodes. The
angle of inclination being $28°$, the apparent width of the ring at the
maximum is just about half its length. The last disappearance of
the rings was in February, 1878; the next will be in the autumn of
1891. Near the time of disappearance the ring appears simply as
a thin needle of light projecting on each side of the planet to a
distance nearly equal to its diameter. Upon this the satellites are
threaded like beads when they pass between us and the planet.

\includegraphicsmid{illo183}{\textsc{Fig.~183.}---The Phases of Saturn's Rings.}

\nbarticle{640.} \nbparatext{Irregularities of Surface and Structure.}---When the rings
are edgewise we find that there are notable irregularities upon them.
They are not truly plane, nor quite of even thickness throughout.

\begin{fineprint}
The same thing is indicated by certain peculiarities sometimes reported
in the form of the shadow cast by the planet on the rings; but caution must
be used in accepting and interpreting such observations, because illusions
%% -----File: 375.png---Folio 364-------
are very apt to occur from the least indistinctness of vision or feebleness of
light. The writer has usually found that the better the seeing, the fewer
abnormal appearances were noted, and the experience of the Washington
observers is the same.
\end{fineprint}

It can hardly be doubted that the details of the rings are continually
changing to some extent. Thus the outer ring, $A$, is occasionally
divided into two by a very narrow black line known as ``Encke's
division,'' although more usually there is merely a darkish streak
upon it, not amounting to a real ``crack'' in the surface.

\nbarticle{641.} \nbparatext{Structure of the Rings.}---It is now universally admitted
that the rings are not continuous sheets of either solid or liquid
matter, but are composed of a swarm of separate particles, each a
little independent moon pursuing its own path around the planet.
The idea was suggested long ago, by J.~Cassini in 1715, and by
Wright in 1750, but was lost sight of until Bond revived it in connection
with his discovery of the dusky ring. Professor Benjamin
Peirce soon afterwards demonstrated that the rings could not be continuous
solids; and Clerk Maxwell finally showed that they can be
neither solid nor liquid sheets, but that all the known conditions would
be answered by supposing them to consist of a flock of separate and
independent bodies, moving in orbits nearly circular and in one
plane,---in fact, a swarm of meteors.

\begin{fineprint}
\nbarticle{642.} \nbparatext{Stability of the Ring.}---If the ring were solid it would certainly
not be stable, and the least disturbance would bring it down upon
the planet; nor is it certain that even the swarm-like structure makes it
forever secure. It is impossible to say positively that the rings may not
after a time be broken up. A few years ago there was much interest in a
speculation which Struve published in 1851. All the measures which he
could obtain up to that date appeared to show that a change was actually
in progress, and that the inner edge of the ring was extending itself towards
the planet. His latest series of measurements (in 1885) does not, however,
confirm this theory. They show no considerable change since 1850, and the
measurements of other observers agree with his in this respect.

The researches of Professor Kirkwood of Indiana make it probable that
the divisions in the ring are due to the perturbations produced by the satellites.
They occur at distances from the planet where the period of a small
body would be precisely commensurable with the periods of a number of
the satellites. It will be remembered that similar gaps are found in the
distribution of the asteroids, at points when the period of an asteroid would
be commensurable with that of Jupiter.
\end{fineprint}
%% -----File: 376.png---Folio 365-------

\nbarticle{643.} \nbparatext{Satellites.}---Saturn has eight\footnote
  {Until Herschel's time it was customary to distinguish the satellites as first,
  second, etc., in order of distance from the planet; but as Herschel's new satellites
  were within the orbits of those which were known before, their discovery confused
  matters, and the confusion became worse confounded when the eighth
  appeared. They are now usually designated by names assigned by Sir John
  Herschel as follows, beginning with the most remote, namely: Iap\u{e}tus (Hyperion),
  Titan; Rhea, Dione, Tethys; Enceladus, Mimas. It will be noticed that
  these names, leaving out Hyperion, which was undiscovered when they were
  assigned, form a line and a half of a regular Latin pentameter.}
of these attendants. The
largest of them was discovered by Huyghens in 1655. It appears as
a star of the ninth magnitude, and is easily observable with a three-inch
telescope. Four others were discovered by Cassini before
1700, two by Sir William Herschel near the end of the last century,
and one, Hyperion, the latest addition to the planet's family, by
Bond of Cambridge, in September, 1848, and independently by Lassell
at Liverpool two days later.

The range of the system is enormous. Iapetus has a distance of
2,225000 miles, with a period of 79 days, nearly as long as that of
Mercury. There is a remarkable variation in the brightness of this
satellite. On the western side of the planet it is fully twice as bright
as upon the eastern, which practically demonstrates that, like our own
moon, it keeps the same face towards the planet at all times, one-half
of its surface being much more brilliant than the other.

Mimas, the nearest and smallest of the satellites, coasts around the
edge of the ring at a distance from it of only 34,000 miles, or
118,000 from the planet's centre, having a period of only $22\frac{1}{2}$ hours.
This satellite is so small and so near the planet that it can be seen
only by very large telescopes and under favorable conditions.

Titan, as its name suggests, is by far the largest of the family.
Its distance is about 770,000 miles, and its period a little less than
16 days. It is probably 3000 or 4000 miles in diameter, and according
to Stone, its mass is $\frac{1}{4600}$ of Saturn's.

\begin{fineprint}
\nbarticle{644.} \nbparatext{Peculiar Behavior of Hyperion.}---Hyperion has a distance
of 934,000 miles, and a period of $21\frac{1}{4}$ days. Under the action of Titan its
orbit is rendered considerably eccentric, and \textit{its line of apsides always keeps
itself in the line of conjunction with Titan}, retrograding in a way which
at first seemed to defy theoretical explanation, but turns out to be only
a ``new case in celestial mechanics,'' and a necessary result of the disturbance
by Titan.

The orbit of Iapetus is inclined about $10°$ to the plane of the rings, but all
%% -----File: 377.png---Folio 366-------
the other satellites move exactly in their plane, and all the five inner ones
move in orbits sensibly circular. The orbits of Iapetus, Hyperion, and Titan
have a slight eccentricity. It is not at all impossible or even improbable
that other minute satellites may yet be discovered in the great gap between
Titan and Iapetus.
\end{fineprint}

\section*{URANUS.}

\nbarticle{645.} As the satellites of Jupiter were the first heavenly bodies
to be ``discovered,'' so Uranus was the first ``discovered'' planet,
all the other planets then known having been known from prehistoric
antiquity. On March 13, 1781, the elder Herschel, in sweeping
over the heavens systematically with a seven-inch reflector made
by himself, came upon an object which, by its disc, he saw at once
was not an ordinary star. In a day or two he had ascertained that
it moved, and announced the discovery as that of a \textit{comet}. After
a short time, however, it became obvious from the computations
of Lexell, that its orbit was nearly circular, that its distance was
enormous, and that its path did not at all resemble that ordinarily
taken by a comet; and within a year its planetary character was
recognized and it was formally admitted as a new member of the
solar system. The name of \textit{Uranus}, suggested by Bode, finally prevailed
over other appellations (Herschel himself called it the Georgium
Sidus, in honor of the king), with the symbol \raisebox{-1pt}{\Uranus} or {\fontseries{b}\uranus}. The former is
still generally used by English astronomers.

\begin{fineprint}
The discovery of a new planet, a thing then utterly unprecedented, caused
great excitement. The king knighted Herschel, gave him a pension, and
furnished him with the funds for constructing his great forty-foot reflector
of four feet aperture, with which he afterwards discovered the two inner
satellites of Saturn. It was found on reckoning back from the date of
Herschel's discovery that the planet had been several times before observed
as a star by astronomers who narrowly missed the honor which fell to the
more fortunate and diligent Herschel. Twelve such observations had been
made by Lemonnier alone.
\end{fineprint}

\nbarticle{646.} \nbparatext{Orbit.}---The \textit{mean-distance} of Uranus from the sun is very
nearly 1800 millions of miles, and the \textit{eccentricity} a trifle less than
that of Jupiter's orbit, amounting to about 83,000000. The \textit{inclination}
of the orbit to the plane of the ecliptic is very slight, only $46'$.
The planet's \textit{periodic time} is 84~years, and the \textit{synodic period} (from
opposition to opposition) $369^{\text{d}}\: 8^{\text{h}}$. The \textit{orbital-velocity} is $4 \frac{1}{3}$ miles
per second.
%% -----File: 378.png---Folio 367-------

\sloppy
\nbarticle{647.} \nbparatext{Appearance and Magnitude.}---Uranus is distinctly visible to
the naked eye on a dark night as a small star of the so-called sixth
magnitude. It is so remote, its orbit having a diameter more than
19~times that of the earth's, that there is very little change in its
appearance, and it makes no practical difference whether it is at
opposition or quadrature.

\fussy
In the telescope it shows a sea-green disc of about $4''$ in apparent
diameter, corresponding to a \textit{real diameter} of 32,000 miles. Its \textit{surface}
is about 16~times, and its \textit{volume} about 66~times greater than
that of the earth, so that the earth compares in size with Uranus
about as the moon does with the earth. The \textit{mass} of Uranus is 14.6
times that of the earth, and its \textit{density} and \textit{surface-gravity }are respectively
0.22 and 0.90.

\nbarticle{648.} \nbparatext{Albedo and Light.}---The reflecting power of the planet's
surface is very high, its \textit{albedo}, according to Zöllner, being 0.64, even
exceeding that of Jupiter. It is to be remembered, however, that
sunlight at Uranus is only $\frac{1}{368}$ as intense as at the earth, and only
about $\frac{1}{14}$ as intense as at Jupiter; so that the disc of the planet does
not appear in the telescope even nearly as bright as a piece of Jupiter's
disc of the same apparent size. The greenish blue tint of the planet
is accounted for by the fact that its spectrum shows certain conspicuous
dark bands in its lower portion, bands perhaps identical with
these which are visible in the spectrum of Saturn, but much more
intense. The $F$ line is also specially prominent in the spectrum of
Uranus. These facts probably indicate a dense atmosphere.

\nbarticle{649.} \nbparatext{Polar Compression, Belts, and Rotation.}---The disc of the
planet shows a decided ellipticity---about $\frac{1}{14}$ according to the Princeton
observations of 1883, which agree nearly with those of Schiaparelli.
There are also sometimes visible upon the planet's disc
certain extremely faint bands or belts, much like the belts of Jupiter
viewed with a very small telescope. What is exceedingly singular,
however, is that the trend of these belts seems to indicate a \textit{plane
of rotation  not coinciding with the plane of the satellites' orbits}.
Nearly all the observers who have seen them at all find that they are
inclined to the satellites' orbit-plane at an angle of from $15°$ to $40°$.
Now unless there is some error in Tisserand's investigations upon the
motions of satellites, it is certain that the plane of these orbits must
of necessity nearly coincide with the planet's equator. Probably the
error lies in judging the direction of the belts, which at the best are
at the very limit of visibility.
%% -----File: 379.png---Folio 368-------

One or two observers have assigned to the planet rotation periods
ranging from $9^{\text{h}}$ to $12^{\text{h}}$; but it cannot be said that any determination
of this element yet made is to be trusted.

\nbarticle{650.} \nbparatext{Satellites.}---The planet has four satellites; viz., Ariel,
Umbriel, Titania, and Oberon; Ariel being the nearest to the planet.
The two brightest of them, Oberon and Titania, were discovered by
Sir William Herschel a few years after the discovery of the planet.
He observed them sufficiently to obtain a reasonably correct determination
of their distances and periods.

\begin{fineprint}
It is not certain that he saw either of the other two, though he \textit{thought} he
had found six satellites in all, and a few years ago a popular writer on
astronomy actually credited the planet with \textit{eight} satellites,---the four
whose names have been given, and four others which Herschel supposed he
had seen.

Ariel and Umbriel were first \textit{certainly} discovered by Lassell in 1851, and
have since been satisfactorily observed by numerous large telescopes. They
are telescopically the smallest bodies in the solar system, and the most
difficult to see. In real size, they are, of course, much larger than the satellites
of Mars or many of the asteroids, very likely measuring from 200 to
500~miles in diameter; but they are ten times as far away as the asteroids,
and illuminated by a sunlight not $\frac{1}{50}$ as brilliant as theirs.

\nbarticle{651.} \nbparatext{Satellite Orbits.}---The orbits of the satellites are sensibly circular,
and all lie in one plane, which, as has been said, \textit{ought} to be, and probably
is, coincident with the plane of the planet's equator. They are very
\textit{close-packed} also, Oberon having a distance of only 375,000 miles, with a
period of $13^{\text{d}}\: 11^{\text{h}}$, while Ariel has a period of $2^{\text{d}}\: 12^{\text{h}}$, at a distance of 223,000
miles. Titania, the largest and brightest of them, has a distance of 280,000
miles, somewhat greater than that of the moon from the earth, with a period
of $8^{\text{d}}\: 17^{\text{h}}$. Under favorable circumstances this satellite can be just seen with
a telescope of eight or nine inches aperture.
\end{fineprint}

\sloppy
\nbarticle{652.} \nbparatext{Plane of Revolution.}---The most remarkable thing about
this satellite system remains to be mentioned. The \textit{plane of their
orbits is inclined} $82°.2$ to the plane of the ecliptic, and in that plane
they revolve \textit{backwards}; or we may say, what comes to the same
thing, that their orbits are inclined to the ecliptic at an angle of
$97°.8$, in which case their revolution is to be considered as \textit{direct}.

\fussy
\begin{fineprint}
When the line of nodes of their orbit plane passes through the earth,
as it did in 1840 and 1882, the orbits are seen edgewise and appear as
straight lines. On the other hand, in 1861, they were seen almost \textit{in plan}
%% -----File: 380.png---Folio 369-------
as nearly perfect circles, and will be seen so again in 1903. The year 1882--83
was a specially favorable time for determining the inclination of the orbits
and the position of the nodes, as well as for measuring the polar compression
of the planet.
\end{fineprint}

\section*{NEPTUNE.}

\nbarticle{653.} The discovery of this planet is justly reckoned as the
greatest triumph of mathematical astronomy. Uranus failed to move
precisely in the path which the computers predicted for it, and was
misguided by some unknown influence to an extent which a keen
eye might almost see without telescopic aid. The difference between
its observed place and that prescribed for it had become in 1845
nearly as much as the ``intolerable'' quantity of $2'$ of arc.

\begin{fineprint}
Near the bright star Vega there are two little stars which form with it a
small equilateral triangle, the sides of the triangle being about $1\frac{3}{4}°$ long.
The northern one of the two little stars is the beautiful double-double star
$\epsilon$~Lyrę, and can be seen as double by a keen eye without a telescope, the
two companions being about $3 \frac{1}{2}'$ apart. Now the distance between the
computed place of Uranus and its actual position was, when at its maximum,
just a little more than half of the distance between these components
of $\epsilon$~Lyrę, that only a keen eye can separate. One would almost say that
such a minute discrepancy between observation and theory was hardly worth
minding, and that to consider it \textit{``intolerable''} was what a Scotchman would
call ``sheer pernickittyness.''
\end{fineprint}

But just these minute discrepancies constituted the data which
were found sufficient for calculating the position of a hitherto
unknown planet, and bringing it to light. Leverrier wrote to Galle,
in substance: \textit{``Direct your telescope to a point on the ecliptic in the
constellation of Aquarius, in longitude $326°$, and you will find within
a degree of that place a new planet, looking like a star of about the
ninth magnitude, and having a perceptible disc.''} The planet was
found at Berlin on the night of Sept.~23, 1846, in exact accordance
with this prediction, within half an hour after the astronomers began
looking for it, and only about $52'$ distant from the precise point that
Leverrier had indicated.

\nbarticle{654.} So far as the mathematical operations are concerned, the
honor is to be equally divided between two then young men,---Leverrier
of Paris, and Adams of Cambridge, England. Each took
up the problem, and by perfectly independent and considerably different
methods arrived at substantially the same solution, and each
%% -----File: 381.png---Folio 370-------
promptly communicated the result (Adams some months earlier than
Leverrier) to a practical astronomer provided with the necessary
apparatus for actually detecting the planet.

\begin{fineprint}
Adams, who was then a graduate of three years' standing, a fellow and a
tutor in his college, communicated his results to Challis, his professor of
astronomy at Cambridge, in the autumn of 1845. Challis at once consulted
Airy, the Astronomer Royal, but between them the matter rather lay
in abeyance for some months, until a notice appeared of a preliminary paper
by Leverrier, which indicated that he also had reached substantially the
same conclusions as Adams. Then, at the urgent suggestion of Airy,
Challis decided to begin the search at once, and to capture the planet by
siege, so to speak. If he had had such star-maps as we now possess of the
regions where the planet lay concealed, it would have been comparatively an
easy operation; but as he had not, he decided to go over a space $10°$ wide
by $30°$ long, and to go over it three times. The positions of all fixed
stars would of course be the same at each of the three observations, but a
planet would change its place in the meantime, and so would be surely
detected.

He began his work on July 29, including in his sweep all stars down to
the tenth magnitude. When, on Oct.~1, he learned of the actual discovery
of the planet, he had recorded the positions of something over 3000 stars,
and was preparing to map them. He had already secured, as it turned out,
three observations of the planet on July~30, Aug.~12, and Sept.~29, and
of course it was only the question of a few weeks more or less when the
discussion of the observations would have brought the planet to light.

But while, this rather deliberate process was going on in England, Leverrier
had revised his work, making a second approximation, and had communicated
his results to Galle, at Berlin, substantially as above indicated. The
Berlin astronomers had the great advantage of a new star-chart by Bremiker,
covering that very region of the sky, and therefore did not need to enter
upon any such tedious campaign as that begun by Challis. In less than
half an hour they found a new star, not indicated on the map, and showing
a sensible disc, just as Leverrier had predicted; and within twenty-four hours
its motion proved it to be the planet.
\end{fineprint}

\nbarticle{655.} \nbparatext{Computed Elements Erroneous.}---Both Adams and Leverrier,
besides computing the planet's position in the sky had deduced
elements of its orbit, and a value for its mass, which turned out to be
considerably erroneous. The reason was that they had assumed \textit{that
the mean distance of the new planet from the sun would follow Bode's
law}, a supposition which, as it turned out, is not even roughly true,
although it was entirely warranted by the existing facts, since all the
then known planets, not excepting Uranus, obey it with reasonable
%% -----File: 382.png---Folio 371-------
exactness. This assumption of an erroneous mean distance of thirty-eight
astronomical units, instead of the true distance of thirty, carried
with it errors in all the other elements of the orbit; and the computed
elements are so wide of the truth that great authorities have maintained
that the actual Neptune was not at all the Neptune of Leverrier
and Adams, but an entirely different planet; and even that the
discovery was a ``happy accident.'' It was not an accident at all,
however. While the data and methods employed were not competent
to determine the planet's \textit{orbit} accurately, they \textit{were} sufficient to
determine the \textit{direction} of the unknown body, which was the one
thing needed to insure its discovery. The computers informed the
searchers precisely where to point their telescopes, and could do so
again were a new case of the same kind to appear.

\begin{fineprint}
\nbarticle{656.} \nbparatext{Old Observations of Neptune.}---After a few weeks' observation
of the new planet it became possible to compute an approximate orbit;
and reckoning back by means of this approximate orbit, the approximate
place on any given date for many years preceding could be found. On
examining the observations of stars made by different astronomers in these
regions of the sky, there were found several instances in which they had
observed the planet; a star of the ninth magnitude in the proper place for
Neptune being recorded in their star-catalogues, while the place is now
vacant. These old observations, thus recovered, were of great use in determining
the planet's orbit with accuracy.
\end{fineprint}

\nbarticle{657.} \nbparatext{The Orbit of Neptune.}---The planet's \textit{mean distance} from
the sun is a little more than 2800,000000 of miles, instead of being
over 3600,000000, as it should be according to Bode's law. The
orbit, instead of being considerably eccentric, as it appeared to be
from the computation of Adams and Leverrier, is more nearly circular
than any other in the system except that of Venus, its \textit{eccentricity}
being only $\frac{9}{1000}$. Even this small fraction, however, makes a variation
of over 50,000000 of miles in the planet's distance from the sun
at different parts of its orbit. The \textit{inclination} of the orbit is about
$1 \frac{3}{4}°$. The \textit{period} of the planet is about 164 years, instead of 217, as
it should have boon according to Leverrier's computed orbit. The
\textit{orbital velocity} is about $3 \frac{1}{3}$ miles a second.

\nbarticle{658.} \nbparatext{The Solar System as seen from Neptune.}---At Neptune's
distance the sun itself has an apparent diameter of only a little more
than $1'$ of arc,---only about the diameter of Venus when nearest us,
and too small to be seen as a disc by the eye, if there are eyes on
Neptune. The light and heat received from it are only $\frac{1}{900}$ part of
%% -----File: 383.png---Folio 372-------
what we get at the earth. Still, we must not imagine that, as compared
with starlight or even moonlight, the Neptunian sunlight is feeble.

\begin{fineprint}
Assuming Zöllner's estimate that sunlight at the earth is 618,000 times as
intense as the light of the full moon, we find that the sun, even at Neptune,
gives a light equal to 687 full moons. This is about seventy-eight times the
light of a standard candle at one metre distance, or about the light of a
thousand candle power electric arc at a distance of $10 \frac{1}{2}$ feet---abundant for
all visual purposes. In fact, as seen from Neptune, the sun would look very
much like a large electric arc-lamp at a distance of a few feet. We call
special attention to this, because erroneous statements are not unfrequently
met with that ``at Neptune the sun would be only a first magnitude star.''
It would really give about 44,000000 times the light of such a star.

\nbarticle{659.} From Neptune the four terrestrial planets would be hopelessly
invisible, unless with powerful telescopes and by carefully screening off
the sunlight. Mars would never reach an elongation of $3°$ from the sun;
the maximum elongation of the earth would be about $2°$, and that of Venus
about $1 \frac{1}{2}°$. Jupiter, attaining an elongation of about $10°$, would probably
be easily seen somewhat as we see Mercury. Saturn and Uranus would be
conspicuous, though the latter is the only planet of the whole system that
can be better seen from Neptune than it can be from the earth.
\end{fineprint}

\nbarticle{660.} \nbparatext{The Planet itself.}---Neptune appears in the telescope as a
small star of between the eighth and ninth magnitudes, absolutely
invisible to the naked eye, though easily seen with a good opera-glass.
It shows a greenish disc, having an apparent diameter of about $2''.6$,
which varies very little, since the entire range of variation in the
planet's distance from us is only about $\frac{1}{15}$ of the whole. The real
\textit{diameter} of the planet is about 35,500 miles (but the probable error of
this must be fully 500 miles); the \textit{volume} is a little more than ninety
times that of the earth. Its \textit{mass}, as determined by means of its satellite,
is about 17~times that of the earth, and its \textit{density} 0.20.

The planet's \textit{albedo}, according to Zöllner, is about forty-six per
cent, a trifle lower than that of Saturn and Venus, and considerably
below that of Jupiter and Uranus. There are no visible markings
upon its surface, and nothing is known as to its rotation. The
spectrum of the planet appears to be precisely like that of Uranus.
The light is so feeble that the ordinary lines of the solar spectrum
are difficult to make out, but there are a number of heavy, dark
bands, which indicate the existence of a dense atmosphere, through
which the light, reflected by the cloud-covered surface of the planet,
is transmitted,---an atmosphere which appears to be identically the
%% -----File: 384.png---Folio 373-------
same on both Uranus and Neptune, while some of its constituents are
probably present in Jupiter and Saturn, as shown by the principal
dark band in the red. It is not possible as yet to identify the
substance which produces these bands.

It will be seen that Uranus and Neptune form a ``pair of twins''
very much as the earth and Venus do; being nearly alike in magnitude,
density, and other characteristics.

\nbarticle{661.} \nbparatext{Satellite.}---Neptune has one satellite, discovered by Lassell
within a month after the discovery of the planet itself. Its distance
is 223,000 miles, and its period is $5^{\text{d}}\: 21^{\text{h}}\: 2^{\text{m}}.7$. Its orbit is inclined
$34°\: 53'$, and it moves \textit{backward} in it; \textit{i.e.}, clockwise, from east
towards the west, like the satellites of Uranus. It is a very small
object, appearing of about the same brightness as Oberon, the outer
satellite of Uranus. From its brightness, as compared with that
of Neptune itself, we estimate that its diameter is about the same as
that of our own moon, though perhaps a little larger.

\begin{fineprint}
Since Neptune is so far from the sun, and the planet has no other satellite
of any size (none certainly comparable with this one), its motion must be
practically undisturbed and very nearly uniform. It has been therefore proposed
to make \textit{it a test of the uniformity of other motions in the solar system},
such as the length of the day and the length of the month. It revolves
rapidly enough to admit of very precise observations by large telescopes.
It is, of course, possible that the planet has other undiscovered satellites,
but if so, they must be very minute.

\nbarticle{662.} \nbparatext{Trans-Neptunian Planet.}---Perhaps the breaking down of Bode's
law at Neptune may be regarded as an indication that the system terminates
with him, and that there is no remoter planet. If such a planet exists, however,
it is sure to be found sooner or later, either by means of its disturbing
action upon Uranus and Neptune, or else by the methods of the asteroid
hunters, although, of course, its slow motion will render its detection in this
way difficult.

\sloppy
In 1877, Professor Todd, from a graphical investigation of the outstanding
differences between the computed and observed places of Uranus (after
allowing for Neptune's action), concluded that an undiscovered planet very
probably exists, and that its longitude was then about $170°$. He made a
careful but unsuccessful search for it with the Washington telescope, going
over a region about $40°$ long by $2°$ wide with a power of 400, hoping to
recognize the planet by its disc.

\fussy
\nbarticle{663.} In the \hyperref[app:I]{Appendix}, we give tables containing the most accurate data
of the planetary system at present available, but with renewed cautions to
the student that these data are of very different degrees of accuracy.
%% -----File: 385.png---Folio 374-------

The \textit{distances} (in astronomical units), and the \textit{periods} of the planets
(except perhaps some of the asteroids) are known with extreme precision;
probably the very last figure of the table may be trusted. The other
elements of their \textit{orbits} are also known very closely, if not quite so precisely
as the distances and periods. The \textit{masses, in terms of the sun's mass}, stand
next to the orbit elements in order of precision, with an error probably not
exceeding one per cent (except, however, in the case of Mercury, the mass
of which remains still very uncertain).

The ratio of the \textit{earth's} mass to the sun's is however less accurately known,
being at present subject to an uncertainty of at least two per cent. This is
because its determination involves a knowledge of the solar parallax (\artref{Art.}{278*}),
the \textit{cube} of which appears in the formula for the ratio of the masses.

Of course all the masses of the planets \textit{expressed in terms of the earth's mass}
are subject to the same uncertainty in addition to all other possible causes
of error.

When we come to the \textit{diameters, volumes}, and \textit{densities} of the planets,
the data become less and less certain, as has been pointed out before. For
the nearer and larger planets, say Venus, Mars, and Jupiter, they are reasonably
satisfactory, for the remoter ones less so, and the figures for the density
of the distant planets,---Mercury, Uranus, and Neptune, for instance,---are
very likely subject to errors of ten or twenty per cent, if not more.

\nbarticle{664.} We borrow from Herschel's ``Outlines of Astronomy'' the following
illustration of the relative magnitudes and distances of the members of our
system. ``Choose any well-levelled field. On it place a globe two feet in
diameter. This will represent the sun; Mercury will be represented by a
\textit{grain of mustard-seed} on the circumference of a circle 164 feet in diameter
for its orbit; Venus, a \textit{pea} on a circle of 284~feet in diameter; the Earth also,
a \textit{pea} on a circle of 430~feet; Mars, a rather large \textit{pin's-head} on a circle of
654~feet; the asteroids, \textit{grains of sand} in orbits of 1000 to 1200 feet; Jupiter,
a \textit{moderate-sized orange} in a circle nearly half a mile across; Saturn, a \textit{small
orange} on a circle of four-fifths of a mile; Uranus, a \textit{full-sized cherry} or
\textit{small plum} upon the circumference of a circle more than a mile and a half;
and finally Neptune, a \textit{good-sized plum} on a circle about two miles and a half
in diameter\ldots. To imitate the motions of the planets in the above-mentioned
orbits, Mercury must describe its own diameter in 41~seconds; Venus,
in $4^{\text{m}}\: 14^{\text{s}}$; the Earth, in 7~minutes; Mars, in $4^{\text{m}}\: 48^{\text{s}}$; Jupiter, in $2^{\text{h}}\: 56^{\text{m}}$;
Saturn, in $3^{\text{h}}\: 13^{\text{m}}$; Uranus, in $2^{\text{h}}\: 16^{\text{m}}$; and Neptune, in $3^{\text{h}}\: 30^{\text{m}}$.'' We may add
that on this scale the nearest star would be on the opposite side of the globe,
at the antipodes, 8000 miles away.
\end{fineprint}
\chelabel{CHAPTERXV}

%% -----File: 386.png---Folio 375-------

\Chapter{XVI}{Determination of Sun's Parallax and Distance}
\nbchapterhang{\stretchyspace
THE DETERMINATION OF THE SUN'S HORIZONTAL PARALLAX
AND DISTANCE.---TRANSITS OF VENUS AND OPPOSITIONS OF
MARS.---GRAVITATIONAL METHODS.---DETERMINATION BY
MEANS OF THE VELOCITY OF LIGHT.}

\nbarticle{665.} \textsc{This} problem, from some points of view, is the most fundamental
of all that are encountered by the astronomer. It is true that
it is possible to deal with many of the subjects that present themselves
in the science without the necessity of employing any units of length
and mass but these that are purely astronomical, leaving for subsequent
determination the relation between these units and the more
familiar ones of ordinary life: we can get, so to speak, a map of the
solar system, \textit{correct in proportion, though without a scale of miles}.
But to give the map its real meaning and use, we must find the scale
finally, if not at first, and until this is done we can form no true conceptions
of the actual dimensions, masses, and distances of the heavenly
bodies.

The problem of finding the true value of the astronomical unit is
difficult, because of the great disproportion between the size of the
earth and the distance of the sun. The relative smallness of the
earth limits the length of our available ``base line,'' which is less
than $\frac{1}{12000}$ part of the distance to be determined by it. It is as if
a person confined in an upper room with a wide prospect were set to
determine the distance of objects ten miles or more away, without
going outside the limits of his single window. It is hopeless to look
for accurate results by \textit{direct} methods, such as answer well enough in
the moon's case, and astronomers are driven to indirect ones.

\begin{fineprint}
\nbarticle{666.} \nbparatext{Historical.}---Until nearly 1700 no even reasonably accurate
knowledge of the sun's distance had been obtained. Aristarchus, by a very
ingenious though inaccurate method, had found, as he thought, that the
distance of the sun was nineteen times as great as that of the moon (it is
really 390 times as great), and Hipparchus, combining this determination of
Aristarchus with his own knowledge of the moon's distance, estimated the
%% -----File: 387.png---Folio 376-------
sun's parallax at $3'$, which is more than twenty times too large. This value
was accepted by Ptolemy, and remained undisputed for twelve centuries, until
Kepler, from Tycho's observations of Mars, satisfied himself that the sun's
parallax could not exceed $1'$; \textit{i.e.}, that the sun's distance must be at least as
great as twelve or fifteen millions of miles. Between 1670 and 1680 Cassini
proposed to determine the solar parallax by observations of Mars; for by that
time the distance of Mars from the earth at any moment could be very accurately
computed in astronomical units, so that the determination of the parallax
of Mars would make known that of the sun. Observations in France,
compared with observations made in South America by astronomers sent out
for the purpose, showed that the parallax of Mars could not exceed $25''$, or
that of the sun, $10''$. Cassini set it at $9''.5$, corresponding to a distance of
86,000000 of miles,---giving the first reasonable approach to the true dimensions
of the solar system.

In 1677, and more fully in 1716, Halley explained how transits of Venus
might be utilized to furnish a far more accurate determination of the solar
parallax than was possible by any method before used. He died before the
transits of 1761 and 1769 occurred, but they were both observed, the first
not very satisfactorily, but the second with perfect success, and in the most
widely separated parts of the globe. The results, however, were by no
means as accordant as had been expected. Various values of the sun's parallax
were deduced, ranging all the way from $8 \frac{1}{2}''$ to $9''$, according to the
observations used, and the way they were treated in the discussion.
Towards the end of the century, La~Place adopted and used for a while the
value $8''.81$, while Delambre preferred $8''.6$.
\end{fineprint}

\nbarticle{667.} In 1822--24 Encke collected all the transit observations that
had been published, and discussed them as a whole in an extremely
thorough manner, deducing as a final result from the two transits of
1761 and 1769, $8''.5776$, corresponding to a distance of $95 \frac{1}{2}$ millions
of miles. The decimal is very imposing, and the discussion by
which it was obtained was unquestionably thorough and laborious,
so that his value stood unquestioned and classical for many years.

The first note of dissent was heard in 1854, when Hansen, in
publishing certain researches upon the motion of the moon, announced
that Encke's parallax was certainly too small; he afterwards
fixed the figure at $8''.97$, but the correction of certain numerical
errors in his work reduced this result to $8''.92$.

\begin{fineprint}
Three or four years later Leverrier found a value of $8''.95$ from the so-called
\textit{lunar equation} of the sun's motion; and in 1862 Foucault, from a new
determination of the velocity of light, combined with the constant of aberration,
got the value $8''.86$. A re-discussion of the old transit of Venus
observations was then made by Stone, of England, who deduced from them
%% -----File: 388.png---Folio 377-------
a value of $8''.943$. The value of $8''.95$ was adopted by the British Nautical
Almanac, and used in it until the issue of 1882. The corresponding distance
of $91\frac{1}{2}$ millions of miles found its way into numerous text-books, and,
though known to be erroneous, still holds its place in some of them.
\end{fineprint}

In 1867 Newcomb made a discussion of all the data then available,
and obtained the value $8''.848$ (or $8''.85$ practically), which
value is now (1888) used in all the Nautical Almanacs except the
French. They use $8''.86$, the result of an investigation published by
Leverrier in 1872.

\nbarticle{668.} The observations of Gill on the planet Mars in 1877, and
the new determinations of the velocity of light by Michelson and
Newcomb in this country, as well as the investigations of Neison
and others upon the so-called ``parallactic inequality'' of the moon,
all point, however, to a somewhat smaller value. Professor Newcomb
says (in 1885), ``All we can say at present is that the solar
parallax is probably between $8''.78$ and $8''.83$, or if outside these
limits, that it can be very little outside.''

\begin{fineprint}
It is not, however, thought worth while to change the constant used in
the almanacs until the final reduction of the transits of 1874 and 1882 has
been made, and until certain experiments and investigations now in progress
have been finished. The difference between $8''.80$ and $8''.85$ is of no practical
account for \textit{almanac} purposes, and the change would involve alterations
in a number of the tables.

Accepting Clarke's value of the earth's equatorial radius (\artref{Art.}{145}), viz.,
$6,\!378,\!206.4^{\text{m}}$ or 3963.3 miles, we find that a solar parallax of
\begin{center}
\begin{tabular}{c@{ }c@{ }c@{ } c@{ } c@{ } c@{ } c@{ }c@{ = } l@{ }c@{ }}
$8''.75 $&corresponds&to&23,573&radii&of&the&earth&93,428000&miles.\\
$8''.80 $&   ``  & ``&23,439& `` &  ``& `` &``&92,897000& ``\\
$8''.85 $&   ``  &``&23,307& `` &  ``& `` &  `` &92,372000& ``\\
$8''.90 $&   ``  &``&23,196& `` &  ``& `` &  `` &91,852000& ``
\end{tabular}
\end{center}
\end{fineprint}

\newenvironment{nbitemize}%
  {\begin{list}{}{\setlength{\itemsep}{0pt}%
                  \setlength{\parsep}{0pt}%
                  \setlength{\topsep}{2pt}}}%
  {\end{list}}

\nbarticle{669.} \nbparatext{Methods of finding the Solar Parallax and Distance.}---We
may classify them as follows:---
\pagebreak
\begin{nbitemize}

\item[I.] Ancient Methods.

\begin{nbitemize}
\item[(A)] Method of Aristarchus [0].

\item[(B)] Method of Hipparchus [0].
\end{nbitemize}

\item[II.] Geometrical and Trigonometrical Methods, in which we attempt
to find by angular measurements the parallax, either of the
sun itself or of one of the nearer planets.
%% -----File: 389.png---Folio 378-------

  \begin{nbitemize}
  \item[(\textit{A})] The direct method [0].

  \item[(\textit{B})] Observations of the displacement of Mars among the stars
at the time of opposition.

    \begin{nbitemize}
    \item[(\textit{a})] {\small Declination observations from two or more stations in widely
different latitudes made with meridian circles or micrometer
[25].}

    \item[(\textit{b})] {\small Observations made at a single station near the equator, by
measuring the distance of the planet east or west from
neighboring stars, using the heliometer [90].}
  \end{nbitemize} % small letters

  \item[(\textit{C})] Declination observations of Venus [20].

  \item[(\textit{D})] Observations of one of the nearer asteroids in the same
way as Mars.

    \begin{nbitemize}
    \item[(\textit{a})] {\small Meridian observations at two stations in widely different
latitudes [20].}

    \item[(\textit{b})] {\small Heliometer observations at an equatorial station [75].}
    \end{nbitemize}

  \item[(\textit{E})] Observations of the transits of Venus at widely separated
stations.

    \begin{nbitemize}
    \item[(\textit{a})] {\small Observations of the contacts.}

      (1)\hspace{0.3em}{\small Halley's method---the ``method of \textit{durations}'' [40].}

      (2)\hspace{0.3em}{\small Delisle's method---observation of \textit{absolute times} [50].}

    \item[(\textit{b})] {\small Heliometer measurements of the position of the planet on
the sun [75].}

    \item[(\textit{c})] {\small Photographic methods---various [20 to 75].}
    \end{nbitemize}
  \end{nbitemize}

\item[III.] Gravitational methods.

  \begin{nbitemize}
  \item[(\textit{A})] By the moon's parallactic inequality [70].

  \item[(\textit{B})] By the lunar equation of the sun's motion [40].

  \item[(\textit{C})] By the perturbations produced by the earth on Venus and
Mars [70]; (ultimately [95]).
  \end{nbitemize}

\item[IV.] By the Velocity of Light, combined with

  \begin{nbitemize}
  \item[(\textit{A})] The light equation [80].

  \item[(\textit{B})] The constant of aberration [90].
  \end{nbitemize}
\end{nbitemize}

\begin{fineprint}
The figures in brackets at the right are intended to express roughly the
relative value of the different methods, on the scale of 100 for a method
which would insure absolute accuracy.
\end{fineprint}

\nbarticle{670.} \nbparatext{Of the Ancient methods}, that of Aristarchus is so ingenious
and simple that it really deserved to be successful. When the moon
is exactly at the half phase, the angle at $M$ (\figref{illo184}{Fig.~184}) must be just
%% -----File: 390.png---Folio 379-------
$90°$, and the angle $AEM$ must equal $MSE$. If, then, we can find
how much shorter the arc $NM$ (from new to half moon) is than $MF$
(from half moon to full), \textit{half the difference will measure $AM$, and
give the angle at $S$.} Aristarchus concluded that the first quarter of
the month was just about \textit{twelve hours} shorter than the second, so
that $AM$ was measured by
six hours' motion of the
moon, or a little less than
$4°$. Hence he found $SE$,
the distance of the sun,
to be about nineteen times
$EM$---a value absurdly
wrong, Since $SE$ is in fact
nearly 390 times $EM$. The
real difference between the two quarters of the month is only about
half an hour, instead of twelve hours.

\includegraphicsouter{illo184}{\textsc{Fig.~184.}\\
Aristarchus' Method of Determining the Sun's Distance.}

\sloppy
The difficulty with the method is that, owing to the ragged and
broken character of the lunar surface, it is impossible to observe
the moment of half moon with sufficient accuracy.

\begin{fineprint}
\nbarticle{671.}  The estimate of Hipparchus was based upon the erroneous calculation
of Aristarchus that the sun's distance is 19~times the moon's, and the
solar parallax, therefore, $\frac{1}{19}$ of the moon's parallax.

The ``radius of the earth's shadow,'' where the moon cuts it at a lunar
eclipse, is given, as Hipparchus knew, by the formula $\rho = P + p - S$ (\artref{Art.}{372}),
or $P + p = \rho + S$. Assuming that $P = 19p$, we have $20p = \rho + S$. Now
$S$, the sun's semi-diameter, is about $15'$; and from the duration of lunar
eclipses Hipparchus found $\rho$ to be about $40'$; hence he obtained for $p$, the
solar parallax, a value a little less than $3'$, which, as has been already mentioned,
was accepted by Ptolemy, and by succeeding astronomers for more
than 1500 years. (Wolf's ``History of Astronomy,'' p.~175.)
\end{fineprint}

\fussy
\nbarticle{672.} \nbparatext{Of the Geometrical Methods}, $A$, the ``direct method'' consists
in observing the sun's apparent declination with the meridian
circle at two stations widely differing in latitude, just as we observe
the moon when finding its parallax (\artref{Art.}{239}). Theoretically,
observations of this sort might give the value of the sun's parallax
within $\frac{1}{2}''$ or so, but the method is practically worthless, because
the errors of observation are large as compared with the quantity
to be determined. The sun's limb is a very bad object to point on,
and besides, its heat disturbs the adjustments of the instrument, thus
rendering the observations still more inaccurate.
%% -----File: 391.png---Folio 380-------

\nbarticle{673.} The first of the two methods of observing the planet Mars
is precisely the same as this direct method of observing the sun;
but the distance of Mars at a ``near opposition'' is only a little
more than $\frac{1}{3}$ that of the sun, so that any error of observation
affects the final result by only about $\frac{1}{3}$ as much; and, moreover,
Mars is a very good object to observe, so that the errors of observation
themselves are much lessened. The planet's distance from
the earth having been found in astronomical units by the method
of \artref{Art.}{515}, the determination of its distance in miles will fix the
value of this unit, and so give us directly the sun's distance and
parallax.

The method requires two observers working at a distance from each
other with different instruments, which is a serious disadvantage.

\begin{fineprint}
For some unexplained reason, observations of this sort seem almost invariably
to give too large a result for the solar parallax, averaging between
$8''.90$ and $8''.98$. The red color of the planet may possibly have something
to do with this by affecting the astronomical refraction. This method, in
1680, was the first to give a reasonable approximation to the sun's true
distance, as has been mentioned before.

The planet Venus can be observed in the same way, and has been once
so observed by Gillis, 1849--52, at Santiago, Chili, in co-operation with the
Washington observers, but the result was not very satisfactory.
\end{fineprint}

\includegraphicsmid{illo185}{\textsc{Fig.~185.}---Effect of Parallax on the Right Ascension of Mars.}

\nbarticle{674.} \nbparatext{Heliometer Observations of Mars} (Method \textit{b}).---It is possible,
however, for a \text{single} observer to obtain better results than
can be got by two or more using the preceding method. Suppose
that the orbital motion of Mars is suspended for a while at opposition,
and that the planet is on or near the celestial equator;
and also that the observer is at a station, $O$, on the earth's
equator. When Mars is rising at $M_e$, \figref{illo185}{Fig.~185}, the horizontal
parallax $OM_eC$ depresses the planet; that is, he appears from $O$ to
be further east than he would if seen from $C$, the centre of the
earth; so that the parallax then \textit{increases} the planet's right ascension.
Twelve hours later, when he is setting, the parallax will
%% -----File: 392.png---Folio 381-------
throw him towards the west, \textit{diminishing} his right ascension by the
same amount. If, then, when the planet is rising, we measure carefully
its distance west of a star $S$, which is supposed to be just east
of it (the distance $M_{e}S$ in \figref{illo186}{Fig.~186}), and then measure the distance
$M_{w}S$ from the same star again when it is setting, the difference will
give us twice the horizontal parallax. The earth's rotation will
have performed for the observer the function of a long journey in
transporting him from one station to another 8000 miles away in a
straight line.

\includegraphicsouter{illo186}{\textsc{Fig.~186.}\\
Micrometric Comparison of Mars with Neighboring Stars.}

\nbarticle{675.} Of course the observations are not practically limited to the
moment when the planet is just rising, nor is it necessary that the
star measured from should be exactly east or west of the planet.
Measures from a \textit{number} of the
neighboring stars, $S_{1}$, $S_{2}$, $S_{3}$, and
$S_{4}$ would fix the positions $M_{e}$ and
$M_{w}$ with more accuracy than measures
from $S$ alone. Nor will the
planet stop in its orbit to be observed,
nor will it have a declination
of zero, nor can the observer
command a station exactly on the
earth's equator. But these variations
from the ideal conditions do
not at all affect the principles involved;
they simply complicate the
calculations slightly without compromising
their accuracy.

The method has the very great
advantage that all the observations
are made by one person, and with
one instrument, so that, as far as can be seen, all errors that could
affect the result are very thoroughly eliminated.

\begin{fineprint}
\nbarticle{676.}  The most elaborate determination of the solar parallax yet made
by this method is that of Mr.~Gill, who was sent out for the purpose by the
Royal Astronomical Society in 1877 to Ascension Island in the Atlantic
Ocean. His result, from 350 sets of measurements, gives a solar parallax
of $8''.783 + 0''.015$,---a result probably very close to the truth, though possibly
a little small. In 1892 and 1894 favorable oppositions of Mars will
occur again, and it is quite likely that the observations will be repeated on
a scale even more extensive.
%% -----File: 393.png---Folio 382-------

Venus cannot be observed in this way, since either her rising or setting
is in the daytime, when the small stars cannot be seen near her; but the
nearer asteroids can be utilized by this method. As they are more distant,
however, than Mars, a given error in observation produces a larger final
error in the result.

\sloppy
\nbarticle{677.} \nbparatext{The Heliometer.}---The heliometer, the instrument employed in
these measures, is one of the most important of the modern instruments of
precision. As its name implies, it was first designed to measure the diameter
of the sun, but it is now used to measure any distance ranging from
a few minutes up to one or two degrees, which it does with the same accuracy
as that with which the filar micrometer measures distances of a few
seconds. It is a ``double image'' micrometer, made by dividing the object-glass
of a telescope along its diameter, as shown
in \figref{illo187}{Fig.~187}. The two halves are so mounted that
they can slide by each other for a distance of three
or four inches, the separation of the centres being
accurately measured by a delicate scale, or by a
micrometer screw operated and read by a suitable
arrangement from the eye-end. The instrument
is mounted equatorially with clock-work, and the
tube can be turned in its cradle so as to make
the line of division of the lenses lie in any desired
direction. When the centres of the two halves of
the object-glass coincide, the whole acts as a single
lens, giving but one image of each object or star
in the field of view. As soon as the centres are separated, each half of
the object-glass forms its own image.

\fussy
\includegraphicsouter{illo187}{\textsc{Fig.~187.}---The Heliometer.}

To measure the distance from Mars to a star, the telescope tube is turned
so that the line of centres points in the right direction, and then the semi-lenses
are separated until one of the two images of the star comes exactly in
the centre of one of the images of Mars; this can be done in two positions
of the semi-lens $A$ with respect to $B$, as shown by the \figref{illo187}{figure}. We may
either make $S_{0}$ (the image of the star formed by semi-lens $B$) coincide with
$M_{1}$ formed by $A$, or make $S_{2}$ coincide with $M_{0}$. The whole distance from
1 to 2 then measures twice the distance between $M$ and $S$.
\end{fineprint}

\nbarticle{678.} \nbparatext{Transit of Venus Observations.}---At the time when Venus
passes between us and the sun, her distance from the earth is only
some 26,000000 of miles, so that her horizontal parallax is nearly
four times as great as that of the sun itself. At this time her apparent
displacement upon the sun's disc, due to a change of the observer's
station upon the earth, is the \textit{difference} between her own parallax
due to this displacement, and that of the sun itself; and this difference
is greater than the sun's parallax nearly in the ratio of 3 to
%% -----File: 394.png---Folio 383-------
1, or, more exactly, of 723 to 277. The object, then, of the observations
of a transit is to obtain in some way a measure of the angular
displacement of Venus on the sun's disc, corresponding to the known
distance between the observer's stations upon the earth.

\includegraphicsouter{illo188}{\textsc{Fig.~188.}---Contacts in a Transit of Venus.}

\nbarticle{679.} \nbparatext{Halley's Method.}---The method proposed by Halley, who in
1677 brought to notice the great advantages presented by a transit of
Venus for determining the sun's parallax, was as follows: Two stations
are chosen upon the earth's surface, as far separated \textit{in latitude}
as possible. From them we observe the \textit{duration} of the transit;
that is, the interval of time between its beginning and end, both of
which must be visible at both stations. If the clock runs correctly
during the few hours during which the
transit lasts, this is all that is necessary.
We do not need to know its error in
reference to Greenwich time, nor even in
respect to the local time, except roughly.
This was a great advantage of the method
in those days, before the era of chronometers,
when the determination of the
longitude of a place was a very difficult
and uncertain operation. The observation
to be made is simply to note the
clock time at which ``\textit{contact}'' occurs,
there being four of these contacts,---two exterior and two internal,
at the points marked 1, 2, 3, 4, in \figref{illo188}{Fig.~188}. Halley depended
mainly on the two internal contacts, which he supposed could be
observed with an error not exceeding one second of time.

\nbarticle{680.} \nbparatext{Computation of the Parallax.}---Having the durations of the
transits at the two stations, and knowing the hourly angular motion
of Venus, we have at once and very accurately the length of the two
chords described by Venus upon the sun, expressed in seconds of
arc. We also know the sun's semi-diameter in seconds, and hence
%% -----File: 395.png---Folio 384-------
in the triangles (\figref{illo189}{Fig.~189}) $Sab$ and $Sde$, we can compute the length
(in seconds still) of $Sb$ and $Se$, the difference of which, $be$, is the
displacement sought, due to the distance between the stations on the
earth.\footnote
  {In order that the method may be practically successful, it is necessary that
  the transit track should lie near the edge of the sun's disc, for two reasons. It
  is desirable that the duration should not be more than three or four hours,
  while for a central transit it lasts eight hours (\artref{Art.}{575}). Moreover, if the two
  chords were near the centre of the disc, any small error in the length of either
  chord would produce a great error in the computed distance between them.
  When they lie as in the \figref{illo189}{figure} (which has been the case in all recent transits),
  the reverse is true: a considerable error in the observed length of one of the
  chords affects their computed distance only slightly.}
The virtual base line is, of course, not the distance between
$B$ and $E$ as a straight line, because that line is not perpendicular to
the line of sight from the earth to Venus, nor to the plane of the
planet's orbit, but the true value to be used is easily found. Calling
this base line $\beta$, we have
\[
p'' = (be)'' × \left( \frac{277}{723} \right) \left( \frac{r}{ \beta} \right),
\]
$r$ being the radius of the earth.

The rotation of the earth, of course, comes in to shift the places
of $E$ and $B$ during the transit, but this can easily be allowed for.

\includegraphicsmid{illo189}{\textsc{Fig.~189.}---Halley's Method.}

\nbarticle{681.} \nbparatext{The Black Drop.}---Halley expected, as has been said, that
it would be possible to observe the instant of internal contact within
a single second of time, but he reckoned without
his host. At the transits of 1761 and
1769, at most of the stations the planet at
the time of internal contact showed a ``ligament''
or ``black drop,'' like \figref{illo190}{Fig.~190}, instead
of presenting the appearance of a round disc
neatly touching the edge of the sun; and the
time of real contact was thus made doubtful
by $10^{\text{s}}$ or $15^{\text{s}}$.

\includegraphicsouter{illo190}{\textsc{Fig.~190.}---The Black Drop.}

\begin{fineprint}
This ``ligament'' depends upon the fact that the optical edge of the
image of a bright body is not, and in the nature of things cannot be, absolutely
sharp in the eye or in the telescope. In the eye itself we have
irradiation. In the telescope we have the difficulty that even in a \textit{perfect}
instrument the image of a luminous point or line has a certain width (which
%% -----File: 396.png---Folio 385-------
with a given magnifying power is less for a large instrument). Moreover
a telescope is usually more or less imperfect, and practically adds other
defects of definition, so that whenever the limbs of two objects approach
each other in the field of view
of a telescope we have more or
less distortion due to the overlapping
of the two ``penumbras
of imperfect definition,''---the
same sort of effect that
is obtained by putting the
thumb and finger in contact,
holding them up within two
or three inches of the eye and
then separating them: as they
separate, a ``black ligament''
will be seen between them.

\includegraphicsouter{illo191}{\textsc{Fig.~191.}---Atmosphere of Venus as seen during a Transit. (Vogel, 1882.)}

With modern telescopes, and
by great care in preventing the
sun's image from being too
bright, so as to diminish \textit{irradiation
in the eye} as far as possible,
the black drop was reduced
to reasonably small proportions
in 1874 and 1882, and
practice beforehand with an ``artificial transit'' enabled the observer in some
degree to allow for its effect. But a new difficulty appeared, from which
there seems to be absolutely no way of escape,---the \textit{planet's atmosphere
causes it to be surrounded by a luminous ring} as it enters upon the sun's
disc, and thus renders the time of the contact uncertain by at least five
or six seconds. In both the transits of 1874 and 1882, differences of that
amount continually appeared among the results of the best observers. \figref{illo191}{Fig.~191}
shows the appearances due to this cause as observed by Vogel in 1882.
\end{fineprint}

\nbarticle{682.} \nbparatext{Delisle's Method.}---Halley's method requires the use of \textit{polar}
stations, uncomfortable and hard to reach, and also that the weather
should permit the observer to see \textit{both} the beginning and end of the
transit.

Delisle's method, on the other hand, utilizes two stations \textit{near the
equator}, taken on a line roughly parallel to the planet's motion. It
requires also that the observers \textit{should know their longitude accurately},
so as to be able to determine the Greenwich time at any moment;
but it does not require that they should see \textit{both} the beginning and
end of the transit; observations of \textit{either} phase can be utilized: and
this is a great advantage. Suppose, then (\figref{illo192}{Fig.~192}), that the
%% -----File: 397.png---Folio 386-------
observer $W$ on one side of the earth notes the moment of internal
contact in Greenwich time, the planet then being at $V_{1}$. When $E$
notes the contact (also in Greenwich time) the planet will be at $V_{2}$,
and the angle at $D$ will be the angular diameter of the earth as seen
from $D$; \textit{i.e., simply twice the sun's parallax}. Now the angle $D$ is at
once determined by the time occupied by Venus in passing from $V_{1}$
to $V_{2}$, since in 584 days (the synodic period) she moves completely
round from the line $DW$ to the same line again. If the time from
$V_{1}$ to $V_{2}$ were twelve minutes, we should find the angle at $D$ to be
about $18''$.

\includegraphicsmid{illo192}{\textsc{Fig.~192.}---Delisle's Method.}

\nbarticle{683.} \nbparatext{Heliometer Observations.}---Instead of observing simply the
times of contact, and leaving the rest of the transit unutilized, as in
the two preceding methods, it is possible to make a continuous series
of measurements of the distance and direction of the planet from the
nearest point of the sun's limb. These measurements are best made
with the heliometer (\artref{Art.}{677}), and give the means of determining
the planet's apparent position upon the sun's disc at any moment
with extreme precision. Such sets of measurements, made at widely
separated stations, will thus furnish accurate determinations of the
apparent displacement of the planet on the sun's disc, corresponding
to known distances on the earth, and so will give the solar parallax.

During the transit of 1882 extensive series of observations of this
sort were made by the German parties, two of which were in the
United States,---one at Hartford, Conn., and the other at San
Antonio, Texas. The results have not yet (August, 1888) been
published, but they will soon appear, and it is understood that they
are considerably more accordant than these obtained by any other
method of observation.

\sloppy
\nbarticle{684.} \nbparatext{Photographic Observations.}---The heliometer measurements
cannot be made very rapidly. Under the most favorable circumstances
a complete set requires at least fifteen minutes, so that the
%% -----File: 398.png---Folio 387-------
whole number obtainable during the seven or eight hours of the
transit is quite limited. Photographs, on the other hand, can be
made with great rapidity (if necessary, at the rate of two a minute),
and then after the transit we can measure at leisure the position of
the planet on the sun's disc as shown upon the plate. At first sight
this method appears extremely promising, and in 1874 great reliance
was placed upon it. Nearly all the parties, some fifty in number,
were provided with elaborate photographic apparatus of various
kinds. On the whole, however, the results, upon discussion, appear
to be no more accordant than those obtained by other methods, so
that in 1882 the method was generally abandoned, and used only by
the American parties, who employed an apparatus having some
peculiar advantages of its own.

\fussy
\begin{fineprint}
\nbarticle{685.} \nbparatext{English, German, and French Methods.}---In 1874, the English
parties used telescopes of six or seven inches aperture, and magnified the
image of the sun formed by the object-glass by a combination of lenses
applied at the eye-end. There were no special appliances for eliminating the
distortion produced by the enlarging lenses, nor for ascertaining the exact
orientation of the picture (that is, the direction of the image upon the plate
with reference to north and south), nor for determining its scale.

\nbenlargepage
The Germans and Russians employed a nearly similar apparatus, but
with the important difference that at the principal focus of the object-glass
they inserted a plate of glass ruled with squares. These, squares are photographed
upon the image of the sun, and furnish a very satisfactory means
of determining the scale and distortion, if any, of the image. The object-glasses
used by the English and the Germans had a focal length of seven
or eight feet. The French employed object-glasses with a focal length of
some fourteen feet, the telescope being horizontal, while the rays of the sun
were reflected into it by a plane mirror; instead of glass plates they used the
old-fashioned metallic daguerreotype plates, in order to avoid any possible
``creeping'' of the collodion film, which was feared in the more modern wet-plate
process. The French plates furnish, however, no accurate orientation
of the picture.

\includegraphicsmid{illo193}{\textsc{Fig.~193.}---American Apparatus for Photographing the Transit of Venus.}

\nbarticle{686.} \nbparatext{The American Apparatus.}---The Americans used a similar plan,
with some modifications and additions. The telescope lenses employed were
five inches in diameter and forty feet in focal length, so that the image
directly formed upon the plate was about $4 \frac{1}{2}$~inches in diameter, and needed
no enlargement. The telescope was placed horizontal and in the meridian,
its exact direction being determinable by a small transit instrument which
was mounted in such a manner that it could look info the photograph telescope,
as into a collimator, when the reflector was removed. The reflector
itself was a plane mirror of unsilvered glass driven by clock-work. \figref{illo193}{Fig.~193}
%% -----File: 399.png---Folio 388-------
shows the arrangement of the apparatus. In front of the photographic
plate, and close to it, was supported a glass plate ruled with squares called
the ``reticle plate,'' and in the narrow space between this and the photograph
plate was suspended a plumb-line of fine silver wire, the image of which
appeared upon the plate, and gave the means of determining the orientation
of the image with extreme
precision. If the reflector
were, and would continue
to be, \textit{perfectly plane}
through the whole operation,
the method could
not fail to give extremely
accurate results; but the
measurements and discussion
of the observations
seem to show that this
mirror was actually distorted
to a considerable
extent by the rays of the
sun. On the whole the
American plates do not
appear to be much more
trustworthy than those
obtained by other methods
Fig.~194 is a reduced
copy of one of the photographs made at Princeton during the transit
of 1882. The black disc near the middle, with a bright spot in the centre,
%% -----File: 400.png---Folio 389-------
is the image of a metal disc cemented to the reticle to mark the centre lines
of the reticle plate; 192 plates were taken during the transit, and at some of
the stations where the weather was good the number was much greater---nearly
300 in some cases.

\includegraphicsouter{illo194}{\textsc{Fig.~194.}---Photograph of Transit of Venus.}

The difficulties to be encountered are numerous. Photographic irradiation,
or the spread of the image on the plate, slight distortion of the image by the
lenses or mirrors employed, irregularities of atmospheric refraction, uncertainty
as to the precise scale of the picture,---all these present themselves
in a very formidable manner. It is obvious why this should be so, when we
recall that on a four-inch picture of the sun's disc, $\frac{1}{10\,000}$ of an inch corresponds
to about $\frac{1}{20}$ of a second of arc, and the whole uncertainty as to the
solar parallax does not amount to as much as that. An image of the sun,
therefore, in which the position of Venus upon the sun's disc cannot be
determined accurately without an error exceeding $\frac{1}{10\,000}$ of an inch, is of
very little value. Imperfections that would be of no account whatever in
plates taken for any other purpose make them practically worthless for
this.

Still there is reason to hope that considering the enormous number of
photographs made in 1874 and 1882 (certainly not less than 5000 in all),
the result to be obtained from such a mass of material will prove to be
worth something.
\end{fineprint}

\textbf{Gravitational Methods.}---These are too recondite to permit any
full explanation here. We can only indicate briefly the principles
involved.

\nbarticle{687.}  (1) The first of these methods is by the \textit{moon's parallactic
inequality}, an irregularity in the moon's motion which has received
this name, because by means of it the sun's parallax can be determined.
It depends upon the fact that the sun's disturbing action
upon the moon differs sensibly from what it would be if its distance,
instead of being less than 400 times that of the moon from the earth,
were \textit{infinitely} great.

\textit{The disturbing action upon the half of the moon's orbit which lies
nearest the sun is greater than on the opposite half of the orbit}. The
retarding action of the tangential force, therefore, during the first
quarter after new moon, is perceptibly greater than the acceleration
produced during the second quarter (\artref{Art.}{447}), so that at the first and
third quarters respectively, the moon is a little more than $2'$ behind
and ahead of the place she would occupy if the tangential forces were
equal in all four quadrants of the orbit---as they would be if the sun's
distance were infinite. This puts the moon about \textit{four minutes of time
behindhand} at the first quarter, and as much \textit{ahead} at the third; and
\textit{if the centre of the moon could be observed within a fraction of a second
%% -----File: 401.png---Folio 390-------
of arc} (as it could if she were a mere point of light like a star), the
observations would give a very accurate determination of the sun's
distance. The irregularities of the moon's limb, however, and the
worse fact, that at the first quarter we observe the \textit{western} limb, while
at the third quarter it is the \textit{eastern} one which alone is observable,
make the result somewhat uncertain, though the method certainly
ranks high.

\nbenlargepage
\begin{fineprint}
\nbarticle{688.} (2) The ``\textit{lunar equation of the sun's motion}'' is, it will be remembered,
an apparent slight monthly displacement of the sun, amounting to
about $6''.3$, and due to the fact that both earth and moon revolve around their
common centre of gravity. It is generally made use of (\artref{Art.}{243}) to determine
the mass of the moon as compared with that of the earth, using as a
datum the assumed known distance of the sun; but if we consider the mass
of the moon as known (determined by the tides, for instance), then we can
find the sun's parallax\footnote
  {Putting $L$ for the maximum value of the lunar equation (about $6''.3$ of arc),
  $P$ for the sun's parallax, and $R$ and $r$ for the distance of the moon and the semi-diameter
  of the earth respectively, we have the equation
\[
P = L \left( \frac{r}{R} \right) \left( \frac{E + m}{m} \right)
  = L \left( \frac{1}{60} \right) (81)\  \text{nearly}.
\]
} in terms of the lunar equation.

The method is not a good one, since the solar parallax, $8''.8$, is greater
than the quantity by means of which it is determined.
\end{fineprint}

\sloppy
\nbarticle{689.} (3) The third method (\textit{by the earth's perturbations of Venus
and Mars}) is one of the most important of the whole list. It depends
upon the principle that if the \textit{mass} of the earth, as compared
with that of the sun, be accurately known, then the \textit{distance} of the
sun can be found at once. The reader will remember that in \artref{Art.}{278}
the mass of the sun was found by comparing the distance
which the earth falls towards the sun in a second (as measured by
the curvature of her orbit) with the force of gravity at the earth's
surface; and in the calculation the sun's distance enters as a necessary
datum. Now, if we know independently the \textit{sun's mass compared
with the earth's}, the distance becomes the only unknown quantity,
and can be found from the other data.

\fussy
\begin{fineprint}
In the same way as in \artref{Art.}{536} we have
\[
(S + E) = 4 \pi^2 \left( \frac{D^3}{T^2} \right),
\]
in which $S$ and $E$ are the masses of the sun and earth, $D$ is the mean
%% -----File: 402.png---Folio 391-------
distance of the earth from the sun, and $T$ the number of seconds in a year.
Also we have for the force of gravity at the earth's surface,
\[
  g = \frac{E}{r^2}, \quad \mathrm{or} \quad E = gr^2,
\]
in which $r$ is the earth's radius.

Dividing the preceding equation by this we, get
\[
\frac{S+E}{E} = \frac{4\pi^2}{gT^2} \left ( \frac{D^3}{r^2} \right );
\]
whence
\hfill$\displaystyle
D^3 = \left (\frac{S+E}{E}\right ) \left ( \frac{gT^2r^2}{4\pi^2}\right ).
$\hfill \phantom{whence}

If we put $\dfrac{S}{E} = M$, this becomes
\[
D^3 = \left( \frac{M+1}{4\pi^2}\right) gT^2r^2.
\]

In this equation everything in the second term is known when we have
once found $M$, or the ratio between the masses of the sun and earth; $g$ is
found by pendulum observations on the earth, $T$ is the length of the year
in seconds, and $r$ is the earth's radius.
\end{fineprint}

Now, the disturbing force of the earth upon its next neighbors,
Mars and Venus, depends directly upon its mass as compared with
the sun's mass, and the ratio of the masses can be determined when
the perturbations have been accurately ascertained; though the calculation
is, of course, anything but simple. But the great beauty of
the method lies in this, that as time goes on, and the effect of the
earth upon the revolution of the nodes and apsides of the neighboring
orbits accumulates, the \textit{determination of the earth's mass in terms
of the sun's becomes continually and cumulatively more precise}. Even
at present the method ranks high for accuracy,---so high that Leverrier,
who first developed it, would have nothing to do with the transit
of Venus observations in 1874, declaring that all such old-fashioned
ways of getting at the sun's parallax were relatively of no value.
The method is the \textit{``method of the future,''} and two or three hundred
years hence will have superseded all the others,---unless indeed it
should appear that bodies at present unknown are interfering with
the movements of our neighboring planets, or unless it should turn
out that the law of gravitation is not quite so simple as it is now
supposed to be.

\nbarticle{690.} \nbparatext{The Physical Method.}---The physical, or \textit{``photo-tachy-metrical''}
method, as it has been dubbed, depends upon the fundamental
%% -----File: 403.png---Folio 392-------
assumption that light travels in interplanetary space with the same
velocity as \textit{in vacuo}. This is certainly very nearly true, and probably
exactly so, though we cannot yet prove it.

By the recent experiments of Michelson and Newcomb in this
country, following the general method of Foucault, the velocity of
light has been ascertained with very great precision and may be
taken as 299,860 kilometres, or 186,330 miles, with a probable error
which cannot well be as great as twenty-five miles either way.

\nbarticle{691.} \nbparatext{Sun's Distance from the Equation of Light.}---(1) ``\textit{The
equation of light}'' is the time occupied by light in travelling between
the sun and earth, and is determined by observation of the eclipses
of Jupiter's satellites (\artref{Art.}{629}). By simply multiplying the velocity
of light by this time ($499^{\text{s}} \pm 2^{\text{s}}$) we have at once the sun's distance;
and that independent of all knowledge as to the earth's dimensions.
The reader will remember, however, that the determination
of this ``light-equation'' is not yet so satisfactory as desirable on
account of the indefinite nature of the eclipse observations involved.

\nbarticle{692.} \nbparatext{From the Constant of Aberration.}---(2) When we know the
velocity of light we can also derive the sun's distance from the
``\textit{constant of aberration},'' and this constant, $20''.492$, derived from
star observations (\artref{Art.}{225}), is known with a considerably higher
percentage of accuracy than the light-equation.

Calling the constant $\alpha$, we have
\[
\tan \alpha = \frac{U}{V},
\]
where $U$ is the velocity of the earth in its orbit, and $V$ the velocity
of light. Now $U$ equals the circumference of the earth's orbit
divided by the length of the year; \textit{i.e.},
\begin{flalign*}
&&  U &= \frac{2 \pi D}{T}; &&\phantom{hence }
\\
&\text{hence }&  \tan \alpha &= \frac{2 \pi D}{VT},  &&
\\
&\text{and }& D &=  \frac{\tan \alpha}{2 \pi} \left( VT \right).  &&
\end{flalign*}
On the whole it seems likely at present that the value of the sun's distance
thus derived is the most accurate of all. Using $\alpha = 20''.492$ and
%% -----File: 404.png---Folio 393-------
$V = 186,330$ miles, we have $D = 92,975,500$ miles, and taking the
earth's equatorial radius as 3963.296 miles (Clarke, 1878), we get
$8''.793$ as the sun's equatorial horizontal parallax.

\nbarticle{693.} The reader will notice that the \textit{geometrical} methods give the
distance of the sun \textit{directly}, apart from all hypothesis or assumption,
except as to the accuracy of the observations themselves, and of their
necessary corrections for refraction, etc.: the \textit{gravitational} methods,
on the other hand, assume the exactness of the law of gravitation;
and the \textit{physical} method assumes that light travels in space with
the same velocity as in our terrestrial experiments. The near accordance
of the results obtained by the different methods shows that
these assumptions must be very nearly correct, if not absolutely so.
\chelabel{CHAPTERXVI}

%% -----File: 405.png---Folio 394-------
\Chapter{XVII}{Comets}
\nbchapterhang{\stretchyspace
COMETS: THEIR NUMBER, MOTIONS, AND ORBITS.---THEIR CONSTITUENT
PARTS AND APPEARANCE.---THEIR SPEC\-TRA.---THEIR
PHYSICAL CONSTITUTION, AND ORIGIN.}

\nbarticle{694.}  \textsc{From} time to time bodies of a very different character from
the planets make their appearance in the heavens, remain visible
for some weeks or months, move over a longer or shorter path among
the stars, and then vanish. These are the \textsc{Comets}, or ``\textit{hairy stars},''
as the word means, since the appearance of such as are bright
enough to be visible to the naked eye is that of a star surrounded
by a hazy cloud, and usually carrying with it a streaming trail of
light. Some of them have been magnificent objects,---the nucleus,
or central star, as brilliant as Venus and visible even by day, while
the cloudy head was nearly as large as the sun itself, and the tail
extended from the horizon to the zenith,---a train of shining substances
long enough to reach from the earth to the sun. The majority
of comets, however, are faint, and visible only with a telescope.

\sloppy
\begin{fineprint}
\nbarticle{695.} \nbparatext{Superstitions.}---In ancient times these bodies were regarded
with great alarm and aversion, being considered from the astrological point
of view as always ominous of evil. Their appearance was supposed to
presage war, or pestilence, or the death of princes. These notions have
survived until very recent times with more or less vigor, but, it is hardly
necessary to say, without the least reason. The most careful research fails
to show any effect upon the earth produced by a comet, even of the largest
size. There is no observable change of temperature or of any meteorological
condition, nor any effect upon vegetable or animal life.
\end{fineprint}

\fussy
\nbarticle{696.} \nbparatext{Number of Comets.}---Thus far we have on our lists about
650 different comets. About 400 of these were recorded previous to
1600, before the invention of the telescope, and must, of course,
have been bright enough to attract the attention of the naked eye.
Since that time the number annually observed has increased very
greatly, for only a few of these bodies, perhaps one in ten, are
visible without telescopic aid. Their total number must be enormous.
Not unfrequently from five to eight are discovered in a
%% -----File: 406.png---Folio 395-------
single year, and there is seldom a day when one or more is not in
sight.

While telescopic comets, however, are thus numerous, brilliant
ones are comparatively rare. Between 1500 and 1800 there were,
according to Newcomb, 79 visible to the naked eye, or about one
in three and three-fourths years. Humboldt enumerates 43 within
the same period as \textit{conspicuous}; during the first half of the present
century there were 9 such, and since 1850 there have been 11. Since,
and including, 1880 we have had 7,---a remarkable number for so
short a time,---and two of them, the principal comet of 1881 and
the great comet of 1882, were unusually fine ones. In August, 1881,
for a little time two comets were conspicuously visible to the naked
eye at once and near together in the sky, a thing almost if not quite
unprecedented.

\nbarticle{697.} \nbparatext{Designation of Comets.}---The more remarkable ones generally
bear the name of their discoverer, or of some astronomer who
made important investigations relating to them,---as for instance,
Halley's, Encke's, and Donati's comets. They are also designated
by the year of discovery, with a Roman number indicating the \textit{order
of discovery} in the year. A third method of designation is by year
and letter, the letters denoting the order in which the comets of a
given year \textit{pass the perihelion}. Thus Donati's comet was both
comet~F and comet~VI, 1858. Comet~I is, however, not \textit{necessarily}
comet~A, though it usually is so. In some cases the comet bears the
name of two discoverers. Thus the Pons--Brooks comet of 1883 is
a comet which was discovered by Pons in 1812, and at its return in
1883 was discovered by Brooks.

\begin{fineprint}
\nbarticle{698.} \nbparatext{The Discovery of Comets.}---As a rule these bodies are first seen
by comet-hunters, who make a business of searching for them. For such
purposes they are usually provided with a telescope known as a ``comet-seeker,''
having an aperture of from four to six inches, with an eye-piece of
low power, and a large field of view. When first seen, a comet is usually
a mere roundish patch of faintly luminous cloud, which, if really a comet,
will reveal its true character within an hour or two by its motion. Some
observers have found a great number of these bodies. Messier discovered
twelve between 1794 and 1798, and Pons twenty-seven between 1800 and
1827.
\end{fineprint}

\nbarticle{699.} \nbparatext{Duration of Visibility, and Brightness.}---The time during
which they are visible differs very much. The great comet of 1811
%% -----File: 407.png---Folio 396-------
was observed for seventeen months, the longest time on record. The
comet of 1861 was observed for a year, and the great comet of 1882
for five months. In some cases, when a comet does not happen to be
discovered until it is receding from the sun, it is seen only for a week
or two.

As to their \textit{brightness} they also differ widely. The great majority
can be seen only with a telescope, although a considerable number
reach the limit of naked-eye vision at that part of their orbit where
they are most favorably situated. A few, as has been said above,
become \textit{conspicuous}; and a \textit{very few}, perhaps four or five in a
century, are so brilliant that they can be seen by the naked eye
in full sunlight, as was the case with the great comets of 1843 and
1882.

\begin{fineprint}
\nbarticle{700.} \nbparatext{Their Orbits.}---The ideas of the ancients as to the motions of
these bodies were very vague. Aristotle and his school believed them to be
nothing but earthly exhalations inflamed in the upper regions of the air,
and therefore \textit{meteorological} phenomena rather than astronomical. Ptolemy
accordingly omits all notice of them in the Almagest.

Tycho Brahe was the first to show that they are more distant than the
moon by comparing observations of the comet of 1577 made in different
parts of Europe. Its position among the stars at any moment, as seen from
his observatory at Uranienburg, was sensibly the same as that observed at
Prague, more than 400 miles to the south. It followed that its distance
must be much greater than that of the moon, and that its real orbit must be
of enormous size, cutting through interplanetary space in a manner absolutely
incompatible with the old doctrine of the crystalline spheres. He
supposed the path to be circular, however, as befitted the motion of a
celestial body.

Kepler supposed that comets moved in straight lines; and he seems to
have been half disposed to consider them as living creatures, travelling
through space with will and purpose, ``like fishes in the sea.''

Hevelius first, nearly a hundred years later, suggested that the orbits are
probably \textit{parabolas}, and his pupil Doerfel \textit{proved} this to be the case in 1681
for the comet of that year. The theory of gravitation had now appeared,
and Newton soon worked out and published a method by which the elements
of a comet's orbit can be determined from the observations. Immediately
afterwards Halley, using this method and computing the parabolic
orbits of all the comets for which he could find the needed observations,
ascertained that a series of brilliant comets having nearly the same orbit
had appeared at intervals of about seventy-five years. He concluded that
these were different appearances of one and the same comet, the orbit not
being really parabolic but \textit{elliptical}, and he predicted its return, which
actually occurred in 1759---the first of the ``periodic comets.''
\end{fineprint}
%% -----File: 408.png---Folio 397-------

\includegraphicsmid{illo195}{\textsc{Fig.~195.}\\The Close Coincidence of Different Species of Cometary Orbits within the Earth's Orbit.}

\nbarticle{701.} \nbparatext{Determination of a Comet's Orbit.}---Strictly speaking, the
orbit of a comet being always a conic section, like that of a planet,
requires only three perfect observations for its determination; but it
seldom happens that the observations\footnote
  {Observations for the determination of a comet's place are usually made with
  an equatorial, by measuring the apparent distance between the comet and some
  neighboring ``comparison star'' with some form of micrometer, as indicated in
  \artref{Art.}{129}. If the star's place is not already accurately known, it is afterwards
  specially observed with the meridian circle of some standard observatory: this
  observation of comparison stars forms quite an item in the regular work of such
  an institution.}
can be made so accurately
as to enable us to distinguish an orbit truly parabolic from one
slightly hyperbolic, or from an ellipse of long period. The \textit{plane
of the orbit} and \textit{its perihelion distance}, can be made out with reasonable
accuracy from such observations as are practically obtainable,
but the \textit{eccentricity}, and the \textit{major axis} with its corresponding \textit{period},
can seldom be determined with much precision from the data obtained
at a single appearance of a comet.

The reason is that a comet is visible only in that very small
%% -----File: 409.png---Folio 398-------
portion of its orbit which lies near the earth and sun, and, as the
figure shows (\figref{illo195}{Fig.~195}), in this portion of the orbit, the long ellipse,
the parabola, and the hyperbola almost coincide. Moreover, from the
diffuse nature of a comet it is not possible to observe it with the
same accuracy as a planet.

Comets which really move in parabolas or hyperbolas visit the sun
but once, and then recede, never to return; while these that move in
ellipses return in regular periods, unless disturbed.

It will be understood, that in a catalogue of comets' orbits, those
which are indicated as parabolic are not \textit{strictly} so. All that can be
said is that during the time while the comet was visible, its position
did not deviate from the parabola given \textit{by an amount sensible to
observation}. The chances are infinity to one against a comet's
moving exactly in a parabola, since the least \textit{retardation} of its
velocity would render the orbit \textit{elliptical}, and the least \textit{acceleration,
hyperbolic}, according to the principles explained in \artref{Article}{430}.

\nbarticle{702.} \nbparatext{Relative Numbers of Parabolic, Elliptical, and Hyperbolic
Comets.}---The orbits of about 270 comets have been thus far computed.
Of this number about 200 are sensibly parabolic, and \textit{six}
appear to be hyperbolic, although the eccentricity exceeds unity by
so small a quantity as to leave the matter somewhat doubtful. There
are also a number of comets which, according to the best computations,
appear to have orbits really elliptical, but with periods so long
that their elliptical character cannot be positively asserted. About
\textit{fifty} have orbits which are certainly and distinctly oval; and \textit{twenty-six}
of these have periods which are less than one hundred years.
\textit{Thirteen} of these periodic comets have already been actually observed
at more than one return.

\begin{fineprint}
As to the rest of the twenty-six, some of them are expected to return
again within a few years, and some of them have been lost,---either in the
same way as the comet of Biela, of which we shall soon speak, or by having
their orbits so changed by perturbations that they no longer come near
enough to the earth to be observed. \hyperref[app:III]{Table III.}\ of the Appendix gives
the elements of these thirteen comets taken from the ``Annuaire du Bureau
des Longitudes'' for 1888. It will be observed that the shortest period
is that of Encke's comet, which is only three and one-half years, while the
period of Halley's comet exceeds seventy-six.

There are three comets with computed periods ranging between seventy
and eighty years, whose returns are looked for within the next forty years.
There is also one comet with a period of thirty-three years which is due to
return in 1899.
\end{fineprint}
%% -----File: 410.png---Folio 399-------

\includegraphicsouter{illo196}{\textsc{Fig.~196.}---Orbits of Short-period Comets.}

\nbarticle{703.} \figref{illo196}{Fig.~196} shows the orbits of several of the comets of short
period,---from three to eight years. (It would cause confusion to
insert all of them.) It will be seen that in every case the comet's
orbit comes very near to the orbit of Jupiter, and when the orbit
crosses that of Jupiter, one
of the nodes is always near
the place of apparent intersection
(the node being
marked on the comet's orbit
by a short cross-line). If
Jupiter were at that point
of its orbit at the time when
the comet was passing, the
two bodies would really be
very near to each other.
The fact, as we shall see,
is a very significant one,
pointing to a connection between
these bodies and the
planet. It is true for \textit{all}
the comets whose periods
are less than eight years---for
those not inserted in
the diagram as well as those that are. The orbits of the seventy-five-year
comets are similarly related to the orbit of Neptune, and the
thirty-three-year comet passes very close to the orbit of Uranus.

\nbarticle{704.} \nbparatext{Recognition of Elliptic Comets.}---Modern observations are
so much more accurate than those made two centuries ago that it is
now sometimes possible to determine the eccentricity and period of
an elliptic comet by means of the observations made at a single
appearance. Still, as a general rule, it is not safe to pronounce upon
the ellipticity of a comet's orbit until it has been observed at least
twice, nor always then. A comet possesses no ``\textit{personal identity},''
so to speak, by which it can be recognized merely by looking at it,---no
personal peculiarities like those of the planets Jupiter and Saturn.
It is identifiable only by its path.

\begin{fineprint}
When the approximate parabolic elements of a new comet's orbit have
been computed, we examine the catalogue of preceding comets to see if we
can find others which resemble it; that is, which have nearly the same \textit{inclination}
%% -----File: 411.png---Folio 400-------
and \textit{longitude of the node} with the same \textit{perihelion distance} and \textit{perihelion
longitude}. If so, it is \textit{probable} that we have to do with the same comet
in both cases. But it is not \textit{certain}, and investigations, often very long and
intricate, must be made to see whether an elliptical orbit of the necessary
period can be reconciled with the observations, after taking into account
the perturbations produced by planetary action. These perturbations are
extremely troublesome to compute, and are often very great, since the comets
not unfrequently pass near to the larger planets. In some such cases the orbit
is completely altered. Even if the result of this investigation appears to
show that the comets are probably identical, we are not yet absolutely safe
in the conclusion, for we have what are known as---
\end{fineprint}

\nbarticle{705.} \nbparatext{Cometary Groups.}---These are groups of comets which pursue
nearly the same orbits, following along one after another at a greater
or smaller interval, as if they had once been united, or had come
from some common source. The existence of such groups was first
pointed out by Hoek of Utrecht in 1865. The most remarkable
group of this sort is the one composed of the great comets of 1668,
1843, 1880, and 1882, and there is some reason to suspect that the
little comet visible on the picture of the corona of the Egyptian
eclipse (\artref{Art.}{328}) also belongs to it. The bodies of this group have
orbits very peculiar in their extremely small perihelion distance (they
actually go within half a million miles of the sun's surface), and yet,
although their elements are almost identical, they cannot possibly all
be different appearances of one and the same comet.

\begin{fineprint}
So far as regards the comets of 1668 and 1843, considered alone, there
is nothing absolutely forbidding the idea of their identity: perturbations
might account for the differences between \textit{their} two orbits. But the comets
of 1880 and 1882 cannot possibly be one and the same; they were both
observed for a considerable time and accurately, and the observations of
both are absolutely inconsistent with a period of two years or anything like
it. In fact, for the comet of 1882 all of the different computers found
periods ranging between 600 and 900 years.

There are about half a dozen other such comet-groups now known.
\end{fineprint}

\sloppy
\nbarticle{706.} \nbparatext{Perihelion Distances.}---These vary greatly. Eight comets
have a perihelion distance less than six millions of miles; about
seventy-four per cent of all that have been observed lie within the
earth's orbit; about twenty-four per cent lie outside, but within twice
the earth's distance from the sun; and six comets have been observed
with a perihelion distance exceeding that limit.
%% -----File: 412.png---Folio 401-------

\fussy
\begin{fineprint}
A single one, the comet of 1729, had a perihelion distance exceeding four
astronomical units,---as great as the mean distance of the remoter asteroids.
It must have been an enormous comet to be visible from such a distance.
It is one of the six \textit{hyperbolic} comets.

Obviously, however, the distribution of comets as determined by observation,
depends not merely on the existence of the comets themselves, but upon
their visibility from the earth. Those comets which approach near the orbit
of the earth have the best chance of being seen, because their conspicuousness
increases as they approach us, so that we must not lay too much stress
on the apparent crowding of the perihelia within the earth's orbit.
\end{fineprint}

The perihelia are not distributed equally in all \textit{directions} from the
sun, but more than sixty per cent are within $45°$ of what is called
``the sun's way''; \textit{i.e.}, the line in space along which the sun is travelling,
carrying with it its attendant systems.

\nbarticle{707.} \nbparatext{Orbit Planes.}---The \textit{inclinations} of the comets' orbits range
all the way from $0°$ to $90°$. The ascending nodes are distributed all
around the ecliptic, with a decided tendency, however, to cluster in
two regions having a longitude of about $80°$ and $270°$.

\nbarticle{708.} \nbparatext{Direction of Motion.}---With the two exceptions of Halley's
comet, and the comet of the Leonid meteors (\artref{Art.}{786}), the elliptical
comets which have periods less than one hundred years all move in
the direction of the planets; and the same is true of the six hyperbolic
comets. Of the other comets, a few more move retrograde than
direct, but there is no decided preponderance either way.

\nbarticle{709.} It is hardly necessary to point out that the fact that the
comets move for the most part in parabolas, and that the planes of
their orbits have no evident relation to the plane of the planetary
motions, tends to indicate (though it falls short of demonstrating)
that \textit{they do not in any proper sense belong to the solar system
itself, but are merely visitors from interstellar space}. They come
towards the sun with almost precisely the velocity they would have
if they had simply dropped towards it from an infinite distance, and
they leave it with a velocity which, if no force but the sun's attraction
operates upon them, will carry them back to an unlimited
distance, or until they encounter the attraction of some other sun.
With one remarkable exception, their motions appear to be just what
might be expected of ponderable masses moving in empty space
under the law of gravitation.
%% -----File: 413.png---Folio 402-------

\nbarticle{710.} \nbparatext{Acceleration of Encke's Comet.}---The one exception referred
to is in the case of Encke's comet which, since its first discovery in
the last century (it was not, however, discovered to be a \textit{periodic}
comet until 1819), has been continually quickening its speed and
shortening its period at the rate of about two hours and a half in
each revolution; as if it were under the action of some resistance
to its motion. No perturbation of any known body will account for
such an acceleration, and thus far no reasonable explanation has
been suggested as even possible, except the one mentioned---the
resistance of an interplanetary medium which retards its motion just
as air retards a rifle bullet. At first sight it seems almost paradoxical
that a \textit{resistance} should \textit{accelerate} a comet's speed; but referring
to \artref{Article}{429} we see that since the semi-major axis of a comet's orbit
is given by the equation
\[
a = \frac{r}{2} \left( \frac {U^2}{U^2 - V^2} \right),
\]
any diminution of $V$ will also diminish $a$; and it can be shown that
this reduction in the \textit{size} of the orbit will be followed by an increase
of velocity above that which the body had in the larger orbit. It is
accelerated by being thus allowed to drop nearer to the sun, and
gains its speed in moving inwards under the sun's attraction.

\begin{fineprint}
\nbarticle{711.} Another action of such a retarding force is to diminish the eccentricity
of the body's orbit, making it more nearly circular. If the action were
to go on without intermission, the result would be a spiral path winding inward
towards the sun, upon which the comet would ultimately fall. For many
years the behavior of Encke's comet was quoted as an absolute demonstration
of the existence of the ``luminiferous ether.'' Since, however, no other
comets show any such action (unless perhaps Winnecke's\footnote
  {Oppolzer, in 1880, found that according to his computations Winnecke's
  comet was accelerated precisely in the same way as Encke's, but by less than half
  the amount. His result, however, is not confirmed by the recent work of Härdtl,
  who finds no acceleration at all.}
comet---No.~5 in
the \hyperref[app:III]{table} in the appendix), and moreover, since according to the investigations
of Von Asten and Backlund the acceleration of Encke's comet itself
seems suddenly to have diminished by nearly one-half in 1868, there remains
much doubt as to the theory of a resisting medium. It looks rather more
probable that this acceleration is due to something else than the luminiferous
ether---perhaps to some regularly recurring encounter of the comet with a
cloud of meteoric matter. The fact that the \textit{planets} show no such effect is
not surprising, since, as we shall see, they are enormously more dense than
any comet, so that the resistance that would bring a comet to rest within a
%% -----File: 414.png---Folio 403-------
single year would not sensibly affect a body like our earth in centuries. The
``resisting medium,'' if it exists at all, must have much less retarding power
than the residual gas in one of Crookes's best vacuum tubes.
\end{fineprint}

\nbarticle{712.} \nbparatext{Physical Characteristics of Comets.}---The orbits of these
bodies are now thoroughly understood, and their \textit{motions} are calculable
with as much accuracy as the nature of the observations permit;
but we find in their physical constitution some of the most perplexing
and baffling problems in the whole range of astronomy,---apparent
paradoxes which as yet have received no satisfactory explanation.
While comets are evidently subject to gravitational attraction, as
shown by their orbits, they also exhibit evidence of being acted upon
by powerful \textit{repulsive} forces emanating from the sun. While they
shine, in part at least, by reflected light, they are also certainly \textit{self-luminous},
their light being developed in a way not yet satisfactorily
explained. They are the \textbf{bulkiest} bodies known, in some cases
thousands of times larger than the sun or stars; but they are ``airy
nothings,'' and the smallest asteroid probably rivals the largest of
them in actual mass.

\nbarticle{713.} \nbparatext{Constituent Parts of a Comet.}---(\textit{a}) The essential part of a
comet---that which is always present and gives it its name---is the
\textit{coma} or nebulosity, a hazy cloud of faintly shining matter, which
is usually nearly spherical or oval in shape, though not always so.

(\textit{b}) Next we have the \textit{nucleus}, which, however, is not found in all
comets, but commonly makes its appearance as the comet approaches
the sun. It is a bright, more or less star-like point near the centre
of the coma, and is the object usually pointed on in determining the
comet's place by observation. In some cases the nucleus is double
or even multiple; that is, instead of a single nucleus there may be
two or more near the centre of a comet. Perhaps three comets out of
four present a nucleus during some portion of their visibility.

(\textit{c}) The \textit{tail} or \textit{train}, is a streamer of light which ordinarily accompanies
a bright comet, and is often found even in connection
with a telescopic comet. As the comet \textit{approaches} the sun, the tail
follows it much as the smoke and steam from the locomotive trail after
it. But that the tail does not really consist of matter simply \textit{left
behind} in that way, is obvious from the fact that as the comet recedes
from the sun, the train \textit{precedes} it instead of following. It is always
\textit{directed away from the sun}, though its precise position and form is to
some extent determined by the comet's motion. There is abundant
evidence that it is a material substance in an exceedingly tenuous
%% -----File: 415.png---Folio 404-------
condition, which in some way is driven off from the comet and then
repelled by some solar action. (See also \artref{Art.}{736}.)

(\textit{d}) \textit{Envelopes and Jets}.---In the case of a very brilliant comet, its
head is often veined by short jets of light which appear to be continually
emitted by the nucleus; and sometimes instead of jets the
nucleus throws off a series of concentric envelopes, like hollow shells,
one within the other. These phenomena, however, are not usually
observed in telescopic comets to any marked extent.
%% -----File: 416.png---Folio 405-------

\nbarticle{714.} \nbparatext{Dimensions of Comets.}---The volume of a comet is often
enormous---sometimes almost beyond conception, if the tail be included
in the estimate of bulk.

As a general rule the \textit{head or coma} of a telescopic comet is from
40,000 to 100,000 miles in diameter. A comet less than 10,000
miles in diameter is very unusual; in fact, such a comet would be
almost sure to escape observation. Many, however, are much larger
than 100,000 miles. The head of the comet of 1811 at one time
measured nearly 1,200000 miles,---more than forty per cent larger
than the diameter of the sun itself. The comet of 1680 had a head
600,000 miles across. The head of Donati's comet of 1858 was
250,000 miles in diameter. The head of the great comet of 1882
was not so bulky as many others, having had a diameter of only
150,000 miles; but its tail was at one time 100,000000 miles in
length.

\includegraphicsmid{illo197}{\textsc{Fig.~197.}---Naked-eye View of Donati's Comet, Oct.~4, 1858. (Bond.)}

\nbarticle{715.} \nbparatext{Contraction of a Comet's Head as it approaches the Sun.}---It
is a very singular fact that the head of a comet continually and regularly
changes its diameter as it approaches to and recedes from the
sun; and what is more singular yet, it \textit{contracts when it approaches
the sun}, instead of expanding, as one would naturally expect it to
do under the action of the solar heat. No satisfactory explanation is
known. Perhaps the one suggested by Sir John Herschel is as
plausible as any,---that the change is optical rather than real; that
near the sun a part of the cometary matter becomes invisible, having
been \textit{evaporated}, perhaps, by the solar heat, just as a cloud of fog
might be.

\begin{fineprint}
The change is especially conspicuous in Encke's comet. When this body
first comes into sight, at a distance of about 130,000000 miles from the
sun, it has a diameter of nearly 300,000 miles. When it is near the perihelion,
at a distance from the sun of only 33,000000 miles, its diameter
shrinks to 12,000 or 14,000 miles, the volume then being less than $\frac{1}{10\,000}$ of
what it was when first seen. As it recedes it expands, and resumes its
original dimensions. Other comets show a similar, but usually less striking,
change.
\end{fineprint}

\sloppy
\nbarticle{716.} \nbparatext{Dimensions of the Nucleus.}---This has a diameter ranging in
different comets from 6000 or 8000 miles in diameter (Comet~III,
1845) to a mere point not exceeding 100 miles. Like the head, it
also undergoes considerable and rapid changes in diameter, though its
changes do not appear to depend in any regular way upon the comet's
%% -----File: 417.png---Folio 406-------
distance from the sun, but rather upon its activity at the time. They
are usually associated with the development of jets and envelopes.

\fussy
\nbarticle{717.} \nbparatext{Dimensions of a Comet's Tail.}---The tail of a large comet, as
regards simple magnitude, is by far its most imposing feature. The
length is seldom less than 10,000000 to 15,000000 miles; it frequently
reaches from 30,000000 to 50,000000, and in several cases has been
known to exceed 100,000000. It is usually more or less fan-shaped, so
that at the outer extremity it is millions of miles across, being shaped
roughly like a cone projecting behind the comet from the sun, and more
or less bent like a horn. The volume of such a train as that of the
comet of 1882, 100,000000 miles in length, and some 200,000 miles in
diameter at the comet's head, with a diameter of 10,000000 at its extremity,
exceeds the bulk of the sun itself by more than 8000 times.

\nbarticle{718.} \nbparatext{The Mass of Comets.}---While the volume of comets is
enormous, their \textit{masses} appear to be insignificant. Our knowledge
in this respect is, however, thus far entirely \textit{negative}; that is, while in
many cases we are able to say positively that the mass of a particular
comet \textit{cannot have exceeded} a limit which can be named, we have
never been able to fix a lower limit which we know it must have
reached; it has in no case been possible to perceive any action whatever
produced by a comet on the earth or any other body of the
planetary system, from which we can deduce its mass; and this,
although they have frequently come so near the earth and other
planets that their own orbits have been entirely transformed, and if
their masses had been as much as $\frac{1}{100\,000}$ of the earth's, they would
have produced very appreciable effects upon the motion of the planet
which disturbed them.

\begin{fineprint}
Lexell's comet of 1770, Biela's comet on more than one occasion, and several
others, have come so near the earth that the length of their periods of
revolution have been changed by the earth's attraction to the extent of
several weeks, but in no instance has the length of the year been altered
by a single second. One might be tempted to think that comets were possessed
of matter without attracting power; but attraction is always \textit{mutual},
and since the comets move according to the law of gravitation, and themselves
suffer perturbation from attraction, there is no escape from the conclusion
that, enormous as they are in volume, they contain very little matter.
Some have gone so far as to say that a comet properly packed could be carried
about in a hat-box or a man's pocket, which, of course, is an extravagant
assertion. The probability is that the total amount of matter in a comet of
any size, though very small as compared with its bulk, is yet to be estimated
%% -----File: 418.png---Folio 407-------
at many millions of tons. The earth's mass (\artref{Art.}{132}, {\scriptsize 4})
is expressed in tons
by 6 with twenty-one ciphers following (6000 millions, of millions, of millions
of tons). A body, therefore, weighing only one-millionth as much as the
earth would contain 6000 millions of millions of tons. The atmosphere of
the earth alone constitutes about $\frac{1}{250\,000}$ the earth's mass, and contains
more than twenty-four millions of millions of tons.
\end{fineprint}

\nbarticle{719.} The late Professor Peirce based his estimate of a comet's
mass upon the extent of the nebulous envelope which it carries with
it, assuming (what may be doubted, however) that this envelope is
gaseous, and is held \textit{in equilibrium} by the attraction of solid matter in
and near the nucleus; and on this assumption he came to the conclusion
that the matter in and near the nucleus of an average comet must be
equivalent in mass to an \textit{iron ball as much as $100$ miles in diameter}.
This would be about $\frac{1}{300\,000}$ of the earth's mass. While this estimate
is not intrinsically improbable, it cannot, however, be relied
upon. We simply do not know anything about a comet's mass, except
that it is exceedingly small as compared with that of the earth.

\nbarticle{720.} \nbparatext{Density.}---This must necessarily be almost inconceivably
small. If a comet 40,000 miles in diameter has a mass equal to $\frac{1}{250\,000}$
of the earth's mass, its mean density is a little less than $\frac{1}{7000}$ of that
of the air at the earth's surface,---much lower than that of the best air-pump
vacuum. Near the centre of the comet the density would probably
be greater than the mean; but near its exterior very much less. As
for the density of its tail, when such a comet has one, that, of course,
must be far lower yet, and much below the density of the residual gas
left in the best vacuum we can make by any means known to science.

This estimate of the density of a comet is borne out by the fact
that small stars can be seen through the head of a comet 100,000
miles in diameter, and even very near its nucleus, with hardly any
perceptible diminution of their lustre. In such cases the writer has
noticed that the image of a star is rendered a little indistinct; and
recent observations of several astronomers have shown a very small
apparent displacement of the star, such as might be ascribed to a
slight refraction produced by the gaseous matter of the comet.

\begin{fineprint}
Students often find difficulty in conceiving how bodies of so infinitesimal
density as comets can move in orbits like solid masses, and with such
enormous velocities. They forget that \textit{in a vacuum} a feather falls as freely
and as swiftly as a stone. Interplanetary space is a vacuum far more perfect
than any air-pump could produce, and in it the rarest and most tenuous
bodies move as freely as the densest.
\end{fineprint}
%% -----File: 419.png---Folio 408-------

\nbarticle{721.} The reader, however, must bear in mind that, although the
\textit{mean} density of a comet (that is, the quantity of matter in a cubic
mile) is small, \textit{the density of the constituent particles of a comet need
not necessarily be so}. The comet may be composed of small, heavy
bodies, \textit{widely separated}, and there is some reason for thinking that this
is the case; that, in fact, the head of a comet is a swarm of meteoric
stones; though whether these stones are many feet in diameter, or only
a few inches, or only a few thousandths of an inch, like particles of
dust, no one can say. In fact, it now seems quite likely that the
greatest portion of a comet's mass is made up of such particles of
solid matter, carrying with them a certain quantity of enveloping gas.

\sloppy
\nbarticle{722.} \nbparatext{Light of Comets.}---There has been much discussion whether
these bodies shine by light reflected or intrinsic. The fact that they
become less brilliant as they recede from the sun, and finally disappear
while they are in full sight simply on account of faintness,\footnote
  {If a comet shone with its own independent light, like a star or a nebula,
  then, so long as it continued to show a disc of sensible diameter, the \textit{intrinsic
  brightness} of this disc would remain unchanged: it would only grow \textit{smaller} as it
  receded from the earth, not \textit{fainter}.}
and not by becoming too small to be seen, shows that their light is in
some way derived from the sun. The further fact that the light
shows traces of polarization also indicates the presence of reflected
sunlight. But while the light of a comet is thus in some way attributable
to the sun's action, the spectroscope shows that it does not
consist, to any considerable extent, of mere reflected sunlight, like
that of the moon or a planet.

\fussy
\begin{fineprint}
\nbarticle{723.} If a comet shone by mere reflected light, or by any light the
intensity of which is proportional inversely to the square of the sun's
distance (as would naturally be the case if the light were excited directly
by the sun's radiation, and proportional to it), we should have its apparent
brightness at any time equal to the quantity $\dfrac{1}{ D^2\Delta^2} $, in which $D$ and $\Delta$ are
the comet's distances from the sun and from the earth respectively. The
brightness of a comet does, in fact, generally follow this law roughly, but
with many and striking exceptions. The light of a comet often \textit{varies
greatly and almost capriciously}, shining out for a few hours with a splendor
seven or eight fold multiplied, and then falling hack to the normal state or
even below it. The Pons--Brooks comet in 1883 furnished several remarkable
instances of this sort (Clerke, p. 418).
\end{fineprint}

\nbenlargepage
\sloppy
\nbarticle{724.} \nbparatext{The Spectra of Comets.}---The spectrum of most comets
consists of a more or less faint continuous spectrum (which may be
%% -----File: 420.png---Folio 409-------
due to reflected sunlight, though it is usually too faint to show the
Fraunhofer lines) overlaid by three bright bands,---one in the yellow,
one in the green, and the third in the blue. These bands are sharply
defined on the lower, or less refrangible, edge, and fade out towards
the blue end of the spectrum. A fourth band is sometimes visible
in the violet. The green band, which is much the brightest of the
three, in some cases is crossed by a number of fine, bright lines, and
there are traces of similar lines in the yellow and blue bands. This
spectrum is \textit{absolutely identical with that given by the blue base of an
ordinary gas or candle flame}; or better, by the blue flame of a Bunsen
burner consuming ordinary illuminating gas. Almost beyond
question it \textit{indicates the presence in the comet of some gaseous hydrocarbon},
which in some way is made to shine; either by a \textit{general}
heating of the whole body to the point of luminosity (which is hardly
probable), or by electric discharges within it, or by \textit{local} heatings due
to collisions between the solid masses disseminated through the gaseous
envelope; or possibly \textit{phosphorescence} due to the action of
sunlight: or none of these surmises may be correct, and we may
have to seek some other explanation not yet suggested.

\fussy
It is not at all certain that the temperature of the comet, considered
as a whole, is very much elevated. Nor will it do to suppose that
because the spectrum reveals the presence in the comet of gaseous
hydrocarbon, this substance, therefore, composes the greater part of
the comet's mass. The probability is that the gaseous portion of the
comet is only a small percentage of the whole quantity of matter
contained in it.

\sloppy
\nbarticle{725.} \nbparatext{Metallic Lines in Spectrum.}---When a comet approaches
very near to the sun, as did Wells's comet in 1882, and a few weeks
later the great comet of that year, the spectrum shows bright metallic
lines in addition to the hydrocarbon bands. The lines of sodium
and magnesium are most easily and certainly recognizable. As for
the other lines---a multitude of which were seen by Ricco (of Palermo)
for a few hours, in the spectrum of the great comet of 1882---they
are probably due to iron; though that is not certain, for they were
not seen long enough to be studied thoroughly.

\includegraphicsmid{illo198}{\textsc{Fig.~198.}---Comet Spectra.\\[2ex]
(For convenience in engraving, the \textit{dark} lines of the solar spectrum in the lowest strip of the
figure are represented as \textit{bright}.)}

\fussy
\nbarticle{726.} \nbparatext{Anomalous Spectra.}---While most comets show the hydrocarbon
spectrum, occasionally a different spectrum of bands appears.
Fig.~198 shows the spectra of three comets compared with the solar
spectrum and with that of hydrocarbon gas.
%% -----File: 421.png---Folio 410-------

\begin{fineprint}
The first, the spectrum of Tebbutt's comet of 1881, is the usual one.
The other two are unique. Brorsen's comet, at its later returns, showed the
\textit{ordinary} comet spectrum, and it might perhaps be considered possible that
an error was made in fixing the position of the bands at the first observation.
But the peculiar spectrum of comet~C, 1877, hardly permits such
an explanation. It was observed at Dunecht on the same night, by the
same observers and with the same spectroscope, as another comet which
gave the usual spectrum; so that in this case it hardly seems possible that
the anomalous result can be a mistake, though the spectrum itself as yet
remains unidentified and unexplained.

Indeed, from some points of view, it is strange that comets, coming as
they do from such widely separated regions of space, do not show an almost
infinite variety of spectra; \textit{a priori} we should expect differences rather than
uniformity.

It is maintained by Mr.~Lockyer that the spectrum of a comet \textit{changes} as
it varies its distance from the sun, the bands altering in appearance and
shifting their position. But the evidence of this is not yet conclusive.
\end{fineprint}

\nbenlargepage
\nbarticle{727.} \nbparatext{Development of Jets and Envelopes.}---When a comet is
first seen at a great distance from the sun it is ordinarily a mere
roundish, hazy patch of faint nebulosity, a little brighter near the
centre.
%% -----File: 422.png---Folio 411-------

\includegraphicsmid{illo199}{\textsc{Fig.~199.}---Head of Donati's Comet, Oct. 6, 1858. (Bond.)}

As the comet draws near the sun it brightens, and the central condensation
becomes more conspicuous and sharply defined, or star-like.
Then, on the side next the sun, the newly formed nucleus begins to
emit jets and streamers of light, or to throw off more or less symmetrical
envelopes, which follow
each other concentrically
at intervals of some hours, expanding
and growing fainter
as they ascend, until they are
lost in the general nebulosity
which forms the head. During
these processes the nucleus
continually changes in brilliancy
and magnitude, usually
growing smaller and brighter
just before the liberation of
each envelope. When jets are
thrown off, the nucleus seems
to oscillate, moving slightly
from side to side; but no evidences of a continuous rotation have
ever been discovered. The two figures, \figref{illo199}{199} and \figref{illo200}{200}, represent the
%% -----File: 423.png---Folio 412-------
heads of two comets which behaved quite differently. \figref{illo199}{Fig.~199} is
the head of Donati's comet as seen on Oct.~5, 1858. This comet
was characterized by the quiet, orderly vigor of its action. It did
very little that was anomalous or erratic, but behaved in all respects
with perfect propriety. The system of envelopes in the head of this
comet was probably the most symmetrical and beautiful ever seen.
Fig.~200 is from a drawing by Common of the head of Tebbutt's
comet in 1881. This comet, on the other hand, was always doing
something \textit{outré}, throwing off jets, breaking into fragments, and, in
fact, continually exhibiting unexpected phenomena.

\includegraphicsouter{illo200}{\textsc{Fig.~200.}---Tebbutt's Comet, 1881. (Common.)}

\nbenlargepage
\nbarticle{728.} \nbparatext{Formation of the Tail.}---The material which is projected
from the nucleus of the comet, as if repelled by it, is also \textit{repelled
by the sun}, and driven backward, still luminous, to form the train. (At least,
this is the appearance.) \figref{illo201}{Fig.~201}
shows the manner in which the
tail is thus supposed to be formed.\footnote
  {Other theories of comets' tails have been presented, and have had a certain
  currency,---theories according to which the tail is a mere ``luminous shadow'' of
  the comet, so to speak, or a swarm of meteors. But all these theories break
  down in the details. They fail to account for the phenomena of jets, envelopes,
  etc., in the head of the comet, and they furnish no mathematical determination
  of the outlines and curvature of the tail.}

The researches of Bessel, Norton,
and especially the late investigations
of the Russian Bredichin, have
shown that this theory---that the
tail is composed of matter repelled
by both the comet and the sun---not
only accounts for the phenomena
in a \textit{general} way, but for almost all
the details, and agrees mathematically
with the observed position
and magnitude of the tail on different
dates.

\includegraphicsouter{illo201}{\textsc{Fig.~201.}---Formation of a Comet's Tail by Matter
expelled from the Head.}

\begin{fineprint}
The repelled particles are still subject to the sun's gravitational attraction,
and the \textit{effective} force acting upon them is therefore the difference between
the gravitational attraction and the electrical $(?)$ repulsion. This \textit{difference}
may or may not be in favor of the attraction, but in any case, the sun's
attracting force is, at least, lessened. The consequence is that those repelled
%% -----File: 424.png---Folio 413-------
particles, as soon as they get a little away from the comet, begin to move
around the sun in \textit{hyperbolic}\footnote
  {Referring to the formula for the semi-major axis of an orbit, viz.,
\[
a = \frac{r}{ 2} \left( \frac{U^2}{U^2-V^2} \right),
\]
  we see that a repulsive force acting from the sun diminishes $U$ (which measures
  the sun's attraction), and the consequence is that if the unrepelled particles are
  describing a parabola (in which case $U^2=V^2$) then for the \textit{repelled} particles the
  denominator will become negative ($U$ having been made smaller than $V$ by the
  repulsive action), and thus $a$ will also become negative, so that the orbit for a
  repelled particle will be a hyperbola.}
orbits which
lie in the plane of the comet's orbit, or
nearly so, and are perfectly amenable to
calculation.

The tail is simply an assemblage of these
repelled particles, and, according to theory,
ought, therefore, to be a sort of flat, hollow,
horn-shaped cone, as represented by \figref{illo202}{Fig.~202},
open at the large end, and rounded
and closed at the smaller one, which contains
the nucleus.
\end{fineprint}

\includegraphicsouter{illo202}{\textsc{Fig.~202.}\\
A Comet's Tail as a Hollow Cone.}

\nbenlargepage
\nbarticle{729.} \nbparatext{Curvature of the Tail.}---The cone is curved as shown,
because the particles repelled still retain their original orbital motion,
so that they will not be arranged along a straight line drawn from
the sun through the comet, but along a curve convex to the direction
of the comet's motion; but the stronger the repulsion, the less will
be the curvature. \figref{illo203}{Fig.~203} shows how the tail ought to lie as the
%% -----File: 425.png---Folio 414-------
comet rounds the perihelion of its orbit. According to this theory,
the tail should be \textit{hollow}, and in the case of comets when at their
brightest it usually seems to be so, the centre being darker than the
edges.

\includegraphicsmid{illo203}{\textsc{Fig.~203.}---A Comet's Tail at Different Points in its Orbit near Perihelion.}

\includegraphicsouter{illo204}{\textsc{Fig.~204.}\\
Bright Centred Tail of Coggia's Comet. June, 1874.}

\nbarticle{730.} \nbparatext{The Central Stripe in a Comet's Tail.}---Very often, there
is a peculiar straight, dark stripe through the axis of the tail as
shown in Figs.~199 and 200 of the head of Donati's and Tebbutt's
comets. It might be mistaken for the shadow of the nucleus if it
were pointed exactly away from the sun; but it is not, usually making
an angle of several degrees with the direction of a true shadow.
Sometimes, however, and not very unfrequently, the tail has a \textit{bright}
centre instead of a dark one, perhaps on
account of the feebleness of the comet's
own repulsive action; in fact, this seems
to be \textit{usually} the case when the comet
has reached a considerable distance from
the sun in receding from it, and often it
is so when the comet is approaching the
sun, but is still remote, as in the case
of Coggia's comet shown in \figref{illo204}{Fig.~204}.
In such cases the tail is generally faint
and ill-defined at the edge, with a central spine of light, and in some
cases it becomes apparently a mere slender ray, of less diameter
than the head of the comet itself. This, however, is unusual.
The explanation of this kind of tail requires a slight modification of
the theory, so far as to admit that the particles at first repelled by
the front of the comet are afterwards attracted by it, though still
repelled by the sun.

\sloppy
\nbarticle{731.} \nbparatext{Tails of Three Different Types.}---Bredichin has found that
the tails of comets may be grouped under three types:---

\fussy
1. The long, straight rays. They are formed of matter upon which
the sun's repulsive action is from twelve to fifteen times as great as
the gravitational attraction, so that the particles leave the comet with
a relative velocity of at least four or five miles a second; and this
velocity is continually increased as they recede, until at last it becomes
enormous, the particles travelling several millions of miles in a day.
The straight rays which are seen in the \figref{illo205}{figure} of the tail of Donati's
comet, tangential to the tail, are streamers of this first type; as also
was the enormous tail of the comet of 1861.
%% -----File: 426.png---Folio 415-------

\includegraphicsouter{illo205}{\textsc{Fig.~205.}---Bredichin's Three Types of Cometary Tails.}

2. The second type is the curved, plume-like train, like the
principal tail of Donati's comet. In this type the repulsive force
varies from 2.2 times gravity (for the particles on the fixed edge of
the tail) to half that amount for those which form the inner edge.
This is by far the most
common type of cometary train.

3. A few comets show
tails of the third type,---short,
stubby brushes
violently curved, and
due to matter of which
the repulsive force is
only a fraction of gravity,---from
$2\frac{1}{10}$ to $2\frac{1}{2}$.

\sloppy
\begin{fineprint}
\nbarticle{732.} According to
Bredichin, the tails of the
first type are probably
composed of \textit{hydrogen},
those of the second type
of some \textit{hydrocarbon gas},
and those of the third of
\textit{iron vapor}, with probably
an admixture of sodium
and other materials.

There has been no opportunity
since Bredichin
published this result to
test the matter spectroscopically
for tails of the
first and third types, by
looking for the lines of
hydrogen and iron. The
hydrogen tails are almost
always very faint, and the
tails of the third class are uncommon. Tails of the second type, which are
brightest and most usual, do show a hydrocarbon spectrum throughout their
entire length, and so far confirm his view.

The reason for this conclusion of Bredichin is that he supposes the
repulsive force to be a \textit{surface action}, the same for equal surfaces of any kind
of matter; the \textit{effective} accelerating force, therefore, measured by the velocity
it would produce, would depend upon the \textit{ratio of surface to mass} in the
particles acted upon, and so, in his view, should be inversely proportional
%% -----File: 427.png---Folio 416-------
to their molecular weights. Now the molecular weights of hydrogen, of
hydrocarbon gases, and of the vapor of iron, bear to each other just about
the required proportion.
\end{fineprint}

\fussy
\nbarticle{733.} \nbparatext{Nature of the Repulsive Force.}---As to this we have so far
no absolute knowledge. While the old ``corpuscular theory of
light'' held its ground, many thought that the apparent repulsion was
due to the actual impact of the light corpuscles. Since the abandonment
of this theory, others have tried to find it in the ``impulse'' of
the light and heat waves of the ether, without, however, explaining
how such waves could produce any such impulsive action. No
experiments show any such carrying power of light or any pressure
produced by its impact, although when Crookes first invented his
radiometer he seems to have thought he had found it. On the whole,
opinion at present strongly inclines to the view long ago suggested
by numerous speculators, but specially worked out and enforced by
Zöllner; namely, that the force is \textit{electrical}; and some authorities of
such eminence as Dr.~Huggins and the late Professor Peirce have
asserted it positively. The difficulty is that we have no evidence that
the sun is electrically charged, nor do we know how it could acquire
a charge. At the same time, the unquestionable \textit{magnetic} effects
produced upon the earth by solar disturbances rather favor the belief
that there must be also a purely electric reaction between the sun and
its attendant bodies.

\begin{fineprint}
A singular theory has been proposed by Zenker, that the repulsion is
due to the reaction produced by rapid evaporation on the surface of the
little solid and liquid particles of which he supposed a comet to consist: this
evaporation would, of course, be most rapid on the side of the particles next
the sun, and would cause \textit{a recoil} in a manner analogous to that by which
the so-called spheroidal state of liquids is produced on a heated surface.
Ranyard has suggested that the cometary particles may consist principally
of minute liquid drops or frozen ``hail-stones'' of certain hydrocarbons
which evaporate rapidly at a very low temperature (such as rhigoline
and its congeners).
\end{fineprint}

\nbarticle{734.} \nbparatext{State of the Matter composing the Tail.}---This also is a subject
of speculation rather than of knowledge. Perhaps the simplest
supposition is that we have to do with gaseous matter rarefied even
beyond the limits of the gas contained in Crookes's tubes,---so rarefied
that since its molecules no longer suffer frequent collisions with
each other, it has thus lost all the peculiar \textit{mechanical} characteristics
of a gaseous mass, and become a mere cloud of separate particles
%% -----File: 428.png---Folio 417-------
each particle consisting, however, of but a single molecule.
Spectroscopically such a cloud would still be \textit{gaseous}, but from a
mechanical point of view extremes would have met, and this most
tenuous gas would have become a cloud of finely powdered solid.

\sloppy
\nbarticle{735.} \nbparatext{What becomes of the Matter thrown off in Comets' Tails.}---To
this we have no certain answer at present; but if the theory
which has been stated is true, it is clear that most of the matter so
repelled from comets can never be re-gathered by the nucleus, but
must be dissipated in space.

\fussy
\begin{fineprint}
Whenever a planet meets any of the particles, it picks them up, of course,
as it picks up meteors; and Newton long ago suggested, what has of late
been forcibly dwelt upon by Dr.\ Sterry Hunt, that in this way the atmospheres
of the planets may be supplied with material to take the place of the
carbon which has been absorbed and fixed by the processes of crystallization
and of life. Otherwise it would seem that the processes now going on upon
the earth's surface must necessarily in the course of time deprive the atmosphere
of all its carbonic acid.

If this view is correct, it follows that such comets as have tails lose a
portion of their substance every time that they visit the sun. It is quite conceivable,
also, that the processes by which light is excited in the head of a
comet may use up and render unfit for future shining, a portion of its
material; so that, as a periodic comet grows old, it may become both smaller
and less luminous, until finally it ceases to be observable.
\end{fineprint}

\nbarticle{736.} \nbparatext{Anomalous Tails and Streamers.}---It is not very unusual for
comets to show tails of two different types at the same time, as, for
instance, Donati's comet. But occasionally stranger things happen,
and the great comet of 1744 is reported to have had six tails diverging
like a fan. Winnecke's comet of 1877 threw out a tail \textit{laterally},
making an angle of about $60°$ with the normal tail, and having the
same length,---about $1°$. Pechüle's comet of 1880 (a small one),
besides the normal tail, had another of about the same dimensions
directed straight \textit{towards} the sun: streamers of considerable length
so directed are not very infrequent. The great comet of 1882 presented
a number of peculiarities, which will be mentioned in the
more particular description of that body, which is to follow. Most
of these anomalies are as yet entirely unexplained.

\nbarticle{737.} \nbparatext{Nature of Comets.}---It is obvious from what has been said
that we have little certain knowledge on this subject; but perhaps on
the whole the most probable hypothesis is the one which has been
%% -----File: 429.png---Folio 418-------
hinted at repeatedly,---that a comet is, as Professor Newton expresses
it, nothing but a ``\textit{sand-bank}''; \textit{i.e.}, a swarm of solid particles
of unknown size and widely separated (say pin-heads several
hundred feet apart), each particle carrying with it an envelope of
gas, largely hydrocarbon, in which gas light is produced, either by
electric discharges between the particles, or by some other light-evolving
action\footnote
  {Some have ascribed the light to the \textit{collisions} between the little stones of
  which they assume the comet to be made up, forgetting that, although the \textit{absolute}
  velocity of the comet is extremely great, the \textit{relative} velocities of its constituent
  masses with reference to each other must be very slight---far too small
  apparently to account for any considerable rise of temperature or evolution of
  light in that way. It is perhaps worth considering whether gases \textit{in the mass} may
  not become sensibly luminous at a much lower temperature than has usually
  been supposed. It would seem not improbable \textit{a priori} that at every temperature,
  radiations of every wave-length must be omitted in some degree; \textit{i.e.}, that at \textit{any
  temperature above the absolute zero no body is absolutely non-luminous.}}
due to the sun's influence.

This hypothesis derives its chief plausibility from the modern discovery
of the close relationship between meteors and comets, to be
discussed in the \chapref{CHAPTERXVIII}{next chapter}.

\nbarticle{738.} \nbparatext{Origin of Periodic Comets.}---It is obvious that the comets
which move in parabolic orbits are, as has been said already, mere
visitors to the solar system, and not citizens of it: but as to those
which now move in elliptical orbits around the sun, returning as
regularly as planets, it is a question whether we are to regard them
as \textit{native-born}, or only as \textit{naturalized}. Did they originate in the
system, or are they captives?

\nbarticle{739.} \nbparatext{Planets' Families of Comets.}---It is quite clear that in some
way or other many of them owe their present status in the system to
Jupiter, Saturn, and the other planets. In \artref{Article}{703} we called attention
to the fact that, without exception, all the short-period comets
(\textit{i.e.}, those having periods ranging from three to eight years), pass
very near to Jupiter's orbit at some point in their paths; and they
are now recognized and spoken of as Jupiter's family of comets,---sixteen
of them in all, at present known.

\begin{fineprint}
Nine of the sixteen are in the table of comets whose returns have been
actually observed more than once. One of the others was Lexell's comet,
which was removed from the range of observation by being thrown into a
new and larger orbit by Jupiter in 1779; and two are comparatively recent
discoveries, whose returns are soon expected. The other four have failed to
%% -----File: 430.png---Folio 419-------
be observed at second and subsequent returns for unknown reasons; quite
possibly for the same reason, whatever that may be, that has deprived us
of Biela's comet.
\end{fineprint}

Similarly, Saturn is at present credited with two comets, one of
which is Tuttle's comet, given in the catalogue of periodic comets.
Uranus stands sponsor for three,---one of them Tempel's comet,
which is very interesting in its relation to the November meteors, and
is expected back in 1900. Finally, Neptune has a family of six.
Halley's comet is one of them, and two of the others have been
observed for a second time since 1880; the other three are not due
on their second return for some years to come.

\nbarticle{740.} \nbparatext{Origin of Comets: the ``Capture'' Theory.}---The generally
accepted theory as to the \textit{origin} of these comet families is that the
comets which compose them have been \textit{captured} by the planets to
which they now belong.

A comet entering the system from an infinite distance, and moving
in a parabolic orbit, when it comes near a planet will be either
accelerated or retarded. If \textit{accelerated}, its orbit becomes \textit{hyperbolic},
and that is the end of that comet so far as the solar system is concerned;
it never returns for a second observation. If, on the other
hand, it is \textit{retarded}, the orbit becomes \textit{elliptical}, and the comet will
return at regular intervals, moving in a path which, of course, always
passes through the point where the disturbance took place.

It is true, as Mr.\ Proctor has pointed out, that the attraction of
Jupiter, huge as is his mass, could not \textit{at one effort} transform a parabolic
orbit into an orbit so small as that, say, of Biela's comet. But
it is not necessary that the thing should be done at one effort. The
comet's orbit lies near to Jupiter's, and after a lapse of time, Jupiter
and the comet will be sure to come alongside again: the comet may
then be sent into a hyperbolic or parabolic orbit,---the chances for
such a result are nearly even;---but it \textit{may} also have its velocity \textit{a
second time diminished, and its orbit made still smaller}; and this may
be done over and over again unlimitedly, until the aphelion of the
comet falls at such a distance within the orbit of Jupiter that the
planet is no longer able to disturb it seriously. Given time enough,
and comets enough, for Jupiter to work upon, and the final result
would necessarily be a comet-family such as really exists, with the
aphelia of their orbits near to the orbit of Jupiter, and periods
roughly half his own. But it must be frankly admitted that the
extent of time, and the quantity of cometary material demanded, are
enormous.
%% -----File: 431.png---Folio 420-------

\begin{fineprint}
It may be added that the results of the recent investigations of Professor
Newton of New Haven upon the nature and distribution of cometary orbits
are favorable to the hypothesis that comets come into the solar system from
outer space, and do not originate within it.

\nbarticle{741.} \nbparatext{The ``Ejection'' Theory.}---Mr.\ Proctor has suggested, and
vigorously defended, a very different theory,---\textit{that comets are masses of matter
which have been thrown off from the heavenly bodies by eruptions of some sort};
that the comets of Jupiter's family, for instance, once formed a portion of
its mass, and were at some time \textit{ejected} with a velocity sufficient to set them
free in space; and that many of the parabolic comets may have been similarly
ejected from our own, or from other suns. The main difficulty with
this theory is that there is no evidence of the necessary eruptive energy in
Jupiter, or in any of the planets. A body would have to leave the upper
surface of Jupiter's atmosphere with a velocity exceeding thirty-five miles
a second, in order to fulfil the conditions of the problem, and become
independent of the parent planet.

It cannot be said, however, that there is any special \textit{mechanical} difficulty
in supposing that some of the \textit{parabolic} comets may owe their origin to
eruptions from distant suns. Our own sun unquestionably sometimes ejects
clouds of matter (in the form of the solar prominences) with enormous
velocity, perhaps in some cases sufficient to send them off into space. But
so far as we can make out from the spectroscopic evidence, the material of
comets is not identical with that of the prominences.
\end{fineprint}

\nbarticle{742.} \nbparatext{Remarkable Comets.}---(1) \textit{Halley's Comet.} This was the
first periodic comet whose return was predicted. Halley based his
prediction upon the fact that he found its orbit in 1682 to be nearly
identical with those of the comets of 1607 and 1531, which had been
carefully observed by Kepler and Apian; and he also found records
of the appearance of great comets in 1456, in 1301, in 1145 and
1066, which would correspond as regards the time-intervals concerned,
though data were wanting for an accurate calculation of their orbits.
He noticed, of course, that the two intervals between 1531 and 1607,
and between 1607 and 1682 were not quite equal; but he had sagacity
enough to see that the differences were no greater than might be
accounted for by the attractions of Jupiter and Saturn.

\begin{fineprint}
The theory of perturbation was not then sufficiently developed to make it
possible to compute with precision just what the effect would be upon the
next return of the comet, but he saw that the action of Jupiter would
\textit{retard} it, and he accordingly fixed upon the early part of 1759 as the time
at which the comet might be expected. Before that date, however, mathematics
had so advanced that the necessary calculations could be made.
Clairaut, as the result of a most laborious investigation, fixed upon April~13
%% -----File: 432.png---Folio 421-------
for the perihelion passage; but in publishing his result, he remarked that
it might easily be a month out of the way owing to the uncertainty as to
the masses of the planets, and the possible action of undiscovered planets
beyond Saturn (Uranus and Neptune were then unknown). The comet
actually came to perihelion on March~13. At this return it was best seen
in the southern hemisphere, and at one time had a tail nearly $50°$ long. At
its next return, in 1835, it came to the predicted time within two days.
It did not appear on this occasion as an extremely brilliant comet, but was
reasonably conspicuous, with a tail of the first type (hydrogen) about $15°$ in
length.

Its next return will occur in or about $1911$, but the necessary calculations
have not yet been made to determine the date with accuracy.

The most remarkable of its earlier appearances were in 1066 and 1456.
The comet of 1066 figures on the Bayeux tapestry as a propitious omen for
William the Conqueror (of England). In 1455 the comet, according to
popular belief, was formally excommunicated by Pope Calixtus III. in a
bull directed mainly against the Turks, who were then threatening eastern
Europe. It is doubtful, however, whether such a formal bull was ever really
promulgated.
\end{fineprint}

\nbarticle{743.} (2) \textit{Encke's Comet.} This is interesting us the first of the
short-period comets, and also as the comet having the shortest known
time of revolution,---only about three years and a half. Encke
first detected its \textit{periodicity} in 1819, but it had been frequently
observed during the preceding fifty years, and has been observed at
almost every return since then. It is usually visible only in the
telescope, though sometimes, under very favorable circumstances,
it can be seen by the naked eye, with a tail a degree or two
long. It is often irregular in form, and ``lumpy,'' and seldom
shows a well-defined nucleus; nor does it exhibit very much that
is interesting in the way of jets, envelopes, and other cometary
freaks. We have already mentioned its remarkable contraction in
volume on approaching the sun (\artref{Art.}{715}), and the progressive
shortening of its period, which has been ascribed to a resisting
medium (\artref{Art.}{710}).

\nbarticle{744.} (3) \textit{Biella's Comet.} This is also, or rather \textit{was}, a small
comet with a period of 6.6 years,---the second comet of short period
in order of discovery. Its history is very interesting. It was discovered
in 1826 by Biela, an Austrian officer, and its periodic character
was soon detected by Gambart, whose name is connected
with it by many French authorities. Its orbit comes within a
very few thousand miles of the earth's orbit, the nearness varying,
%% -----File: 433.png---Folio 422-------
of course, from time to time, on account of perturbations. The
approach is often so close, however, that if the comet and the earth
were to arrive at the nearest point at the same time there would be a
collision, and the earth would pass through the outer portions of the
comet's head. At the return of the comet in 1832, some one started
the report that such an encounter would occur, and in consequence
there was something hardly short of a panic in southern France, the
first of the since numerous ``comet-scares.'' At this time the comet
passed the critical point about a month before the earth reached it,
so that the two bodies were never really within 15,000000 miles of
each other.

\nbarticle{745.} At the comet's next return in 1839 it failed to be observed on
account of its unfavorable position in the sky; but in 1846 it duly
reappeared, and did a very strange thing, so far unprecedented. It
\textit{divided into two!} When first seen on November~28, it presented the
ordinary appearance of any newly discovered comet. On Dec.~19
it had become rather pear-shaped, and ten days later it had
divided, the duplication being first seen in New Haven, and soon
after at Washington, some days before any European astronomer
had noticed it.

\begin{fineprint}
The twin comets travelled along side by side for more than four months,
at an almost unvarying distance of about 160,000 miles, without showing
the least sign of mutual attraction or disturbance; but internally both
comets were intensely active, each developing a nucleus very bright for a
telescopic comet, with a tail some half a degree in length, and showing
curious fluctuations of light, which seemed as a general rule to \textit{alternate}; \textit{i.e.},
whenever comet~$A$ brightened up, comet~$B$ grew fainter, and \textit{vice versa}.
During a part of the time the two comets were connected by a faint arc of
light.

When next the comet returned in August, 1852, it was under rather
unfavorable circumstances for observation, but the twins were both seen,
now separated by about 1,500000 miles, and travelling quietly in their
appointed orbits. Neither of them has ever been seen again, although they
ought to have returned five times, and more than once under favorable
conditions for visibility.
\end{fineprint}

\nbarticle{746.} But the story is not yet ended, though the remainder perhaps
belongs more properly in the next \chapref{CHAPTERXVIII}{chapter} of our book.

On the night of Nov.~27, 1872, just as the earth was passing the
old track of the lost comet, she encountered a wonderful meteoric
shower. As Miss Clerke expresses it, perhaps a little too positively,
%% -----File: 434.png---Folio 423-------
``It became evident that Biela's comet was shedding over us the
pulverized products of its disintegration.''\footnote
  {``Hist.\ Ast.\ in 19th Century,'' p.~384. It is probable enough that the meteors
  were really the product of the comet's \textit{``disintegration''}; still it is by no means
  certain. It is, of course, beyond question that they bear \textit{some} relation to the lost
  comet.}

The same thing happened again in November, 1886, when the
earth once more passed the comet's path.

The meteors of this so-called Bielid swarm, in their motion through
the sky, all appear to come from a point in the constellation of
Andromeda, and are therefore sometimes called the ``Andromedes,''
and their motion is parallel to the comet's orbit, at the point where
it intersects our own.

\nbarticle{747.} (4) \textit{Donati's Comet of $1858$.} This, on the whole, was perhaps
the finest (though not the largest or the most extraordinary) of
the comets of the present century, having been very favorably situated
for observation in the October sky.

\begin{fineprint}
It was discovered at Florence as a telescopic object on June 2. It did
not, however, become visible to the naked eye until near the end of August,
when it began to exhibit the beautiful phenomena which have made it, so
to speak, the normal and typical comet. The comet had an apparently
well-defined nucleus, which varied in diameter at different times from 500
miles to 3000. For several weeks the coma exhibited in unrivalled perfection
the development and structure of concentric envelopes. Its tail was of the
second or hydrocarbon type, with faint tangential streamers which belong
to the first or hydrogen type; it had a maximum apparent length of about
50°, and was some 5° or 6° wide at the extremity, and its real length was
about 45,000000 miles, with a width of 10,000000. The object was kept
under accurate observation for fully nine months, so that its orbit is unusually
well determined as a very long ellipse, with a periodic time of nearly
2000 years. Figs.~197 and 199 show its principal features.
\end{fineprint}

Our space permits us to cite in detail only one other comet:---

\nbarticle{748.} (5) \textit{The Great Comet of $1882$}, which will always be remembered,
not only for its beauty, but for the great variety of unusual
phenomena it presented.

\textit{Discovery and Brightness.} The comet seems to have been first
seen as a naked-eye object by some one whose name is not given, at
Auckland, New Zealand, on Sept.~3. By the 7th or 8th it had
%% -----File: 435.png---Folio 424-------
become somewhat conspicuous, and was observed both at Cordova
(South America) and at the Cape of Good Hope. It was telegraphed
to the northern hemisphere by Cruls, of Rio Janeiro, on Sept.~11,
but was not seen in the north until the day when it passed its
perihelion, Sept.~17. It was then independently discovered by
Common, in England (who had not heard of Cruls's telegram), in
broad daylight, within $2°$ of the sun; and the next day it was similarly
discovered by a number of observers, especially by Thollon, at
Nice, who observed its spectrum in full sunlight, and measured the
displacement of the sodium lines produced by its motion. It was so
bright that there was not the slightest difficulty in seeing it by simply
shutting off the sun with the hand held at arm's length.

\begin{fineprint}
\nbarticle{749.} \nbparatext{Transit across the Sun's Disc.}---On the afternoon of the 17th,
its approach to the sun's disc and actual contact with it was observed by two
persons at the Cape of Good Hope, with the same telescope and dark glasses
ordinarily used for observing sun spots. The comet seemed to be as bright
as the sun's surface itself, and was followed right up to the sun's limb,
where it vanished so utterly that the observers supposed that it had gone
\textit{behind} the sun. On the contrary, however, it passed directly across the sun's
disc, but absolutely invisible; there was not the least trace of it upon the
sun's surface, so that we must suppose it to have been sensibly transparent.
It must have traversed the disc in less than fifteen minutes, though unfriendly
clouds prevented the observation of its exit. For four days after the perihelion
passage it remained still visible to the eye by daylight. On Sept.~22d,
a French observer in Paris ascended in a balloon to observe it, and succeeded
in seeing it, but not in getting any valuable results. A few days more
carried it so far from the sun that it became a magnificent object for the
hours before sunrise.
\end{fineprint}

\nbarticle{750.} \nbparatext{Member of a Comet Group.}---As has been stated before
(\artref{Art.}{705}), its orbit---at least, that portion of it within the earth's
orbit---coincides almost exactly with the orbits of three other comets
belonging to the same group; viz., the comets of 1668, 1843, and
1880. The salient peculiarity of these orbits lies in the \textit{closeness of
their approach to the sun}, the perihelion distance of each of them
being less than 750,000 miles, so that they all passed within 300,000
miles of the sun's surface, and with a velocity which at perihelion
exceeded $300$ miles per second, and carried them through $180°$ of
their orbit in less than three hours. And yet, this passage through
the sun's coronal regions did not disturb their motion in the least, as
is shown by the fact that the orbit of the comet of 1882, deduced
from the observations made before the perihelion passage, agrees
%% -----File: 436.png---Folio 425-------
exactly with that deduced from those made after it. The inference
as to the extreme rarity of the sun's corona is obvious. Only one
other comet---Newton's comet of 1680---has ever approached even
nearly as close to the sun as the four comets of this group.

\includegraphicsmid{illo206}{\textsc{Fig.~206.}---Orbit of the Great Comet of 1882.}

\begin{fineprint}
The comet continued visible until March, and this long period of observation
enabled the computers to determine the orbit with a greater degree of
accuracy than is usual. They all agree in making it a very elongated
ellipse, with a period ranging from 652~years (according to Morrison) to
843~years (according to Kreutz). \figref{illo206}{Fig.~206} represents the form of the orbit
and the way in which its plane is related to the orbit of the earth.
\end{fineprint}

\sloppy
\nbarticle{751.} \nbparatext{Telescopic Features.}---When the comet first became telescopically
observable in the morning sky it presented very nearly the
normal appearance. The nucleus was sensibly circular, and there
were a number of clearly developed, concentric envelopes in the
head; the dark, shadow-like stripe behind the nucleus was also well
marked. In a few days the nucleus became elongated, and finally
stretched out into a lengthened, luminous streak some 50,000 miles
in extent, upon which there were six or eight star-like knots of condensation.
The largest and brightest of these knots was the third
from the forward end of the line, and was some 5000 miles in
%% -----File: 437.png---Folio 426-------
diameter. This  ``string of pearls'' continued to lengthen as long as
the comet was visible, until at last the length exceeded 100,000 miles.
This fact, of course, made it difficult to fix upon the precise point to
be observed in determining the comet's position.

\fussy
\includegraphicsmid{illo207}{\textsc{Fig.~207.}---The Head of the Great Comet of 1882.}

When the nucleus first broke up in this way, the dark stripe behind
it was still conspicuous, making, however, a slight angle with the line
of nuclei. A bright streak followed the line of nuclei, and in a few
days this seemed to encroach on and to obliterate the dark stripe, so
that after that time the backbone of the comet's tail became bright
instead of dark, as it had been previously. The engraving (\figref{illo207}{Fig.~207})
represents the telescopic appearance at Princeton on Oct.~9 and 15.
The comet continued to be visible with the telescope until it was
more than 470,000000 miles from the earth, a distance to which no
other such object has ever been followed, except the single comet of
1729,---the one whose perihelion distance exceeded four times the
earth's distance from the sun.

\nbarticle{752.} \nbparatext{Tail.}---The comet was so situated that its tail was not seen
to the best advantage, being directed nearly away from the earth, and
never having an apparent length much exceeding $35°$. In this respect
it has, therefore, been often surpassed by much inferior comets. The
%% -----File: 438.png---Folio 427-------
actual length of the tail, however, at one time exceeded 100,000000
miles,---more than the distance of the earth from the sun. It was
of the second or hydrocarbon type.

\includegraphicsmid{illo208}{\textsc{Fig.~208.}---The ``Sheath,'' and the Attendants of the Comet of 1882.}

A unique and so far utterly unexplained phenomenon was a faint,
straight-edged beam of light, or ``\textit{sheath},'' that accompanied the
comet, enveloping the head and projecting three or four degrees in
front of it, as shown in the figure (\figref{illo208}{Fig.~208}). Besides this, at different
times, three or four irregular shreds of cometary matter were
detected by Schmidt, of Athens, and other observers, accompanying
the comet at a distance of three or four degrees when first seen, but
gradually receding from it, and at the same time growing fainter.

\begin{fineprint}
Possibly they may have been fragments of the tail which belonged to the
comet before passing the perihelion, or of the matter repelled from the
comet when near perihelion. Since the comet, in passing the perihelion,
changed the direction of its motion by nearly $180°$ in less than three hours,
it was, of course, physically impossible that the tail it had before the perihelion
passage could have made the circuit of the sun in that time. Its tail
after the perihelion passage was in no sense the same that it had before; the
comet ran away from the old one, and developed a new one. Visible or invisible,
the particles of the old train must have kept on their way under the combined
%% -----File: 439.png---Folio 428-------
action of the sun's gravitation and repulsion, and calculation can show
whether any of its fragments could have occupied the position and had the
motion possessed by these companion-comets.
\end{fineprint}

We close the chapter with a few remarks upon a subject which has
been much discussed.

\nbarticle{753.} \nbparatext{The Earth's Danger from Comets.}---It has been supposed
that comets might do us harm in two ways,---either by actually striking
the earth, or by falling into the sun, and thus producing such an
increase of solar heat as to burn us up.

As regards the possibility of a collision with a comet, it is to be
admitted that such an event \textit{is} possible. In fact, if the earth lasts
long enough, it is practically sure to happen; for there are several
comets' orbits which pass nearer to the earth's orbit than the semi-diameter
of the comet's head, and at some time the earth and comet
will certainly come together. Such encounters will, however, be very
rare. If we accept the estimate of Babinet, they will occur about
once in 15,000000 years in the long run.

As to the consequence of such a collision it is impossible to speak
with confidence, for want of sure knowledge of the state of aggregation
of the matter composing a comet. If the theory presented in this
chapter is true, everything depends on the size of the separate solid
particles which form the main portion of the comet's mass. If they
weigh \textit{tons}, the bombardment experienced by the earth when struck
by a comet would be a very serious matter: if, as seems more
likely, they are for the most part smaller than pin-heads, the result
would be simply a meteoric shower.

\nbarticle{754.} \nbparatext{Effect of the Fall of a Comet into the Sun.}---As regards the
effect of the fall of a comet into the sun, it may be stated that, except
in the case of Encke's comet, there is no evidence of any action going
on that would cause a now existing \textit{periodic} comet to strike the sun's
surface; it is, however, undoubtedly possible that a comet may enter
the system from without, so accurately aimed that it will hit the sun.

But, if a comet actually strikes the sun, it is not likely that the least
harm will be done. If a comet, having a mass equal to $\frac{1}{100\,000}$ of
the earth's mass, were to strike the sun's surface with the parabolic
velocity of nearly 400~miles a second, it would generate about as
much heat as the sun radiates in eight or nine hours. If this were all
instantly effective in producing increased radiation at the sun's surface
%% -----File: 440.png---Folio 429-------
(increasing it, say eightfold, for even a single hour), mischief
would follow, of course. But it is almost certain that nothing of the
sort would happen. The cometary particles would pierce the photosphere,
and liberate their heat mostly \textit{below the solar surface}, simply
expanding, by some slight amount, the sun's diameter, and so adding
to its store of potential energy about as much as it ordinarily expends
in a few hours. There might, and very likely would be, a flash of
some kind at the solar surface, as the shower of cometary particles
struck it, but probably nothing that the astronomer would not take
delight in watching.
\chelabel{CHAPTERXVII}
%% -----File: 441.png---Folio 430-------

\Chapter{XVIII}{Meteors}

\nbarticle{755.} \nbparatext{Meteors.}---Occasionally bodies fall on the earth from the
sky,---masses of stone or iron which sometimes weigh several tons.
During its flight through the sky such a body is called \textit{a meteor}, and
the pieces which fall from it are called \textit{meteorites}, or \textit{aerolites} (air-stones),
or \textit{uranoliths} (heaven-stones), or simply \textit{meteoric stones}.

\sloppy
\nbarticle{756.} \nbparatext{Circumstances of their Fall.}---The circumstances which
attend the fall of a meteorite are in most cases substantially as
follows. If it occurs at night a ball of fire is seen, which moves
with an apparent speed depending both on its real velocity and on the
observer's position. If the body is coming ``head on,'' so to speak,
the motion will be comparatively slow; so also if it is very distant.
The fire-ball is generally followed by a luminous train, which marks
out the path of the body, and often continues visible for a long time
after the meteor itself has disappeared. The motion is seldom exactly
straight, but is more or less irregular, and every here and there along
its path the meteor seems to throw off fragments, and to change its
course more or less abruptly. If the observer is near enough, the
flight is accompanied by a heavy, continuous roar, accentuated
by sharp detonations which accompany the visible explosions by
which fragments are burst off from the principal body. The noise
is sometimes tremendous, and heard for distances of forty or fifty
miles.

\fussy
If the fall occurs by day the luminous appearances are, of course,
principally wanting, and white clouds take the place of the fire-ball
and the train.

\sloppy
\nbarticle{757.} \nbparatext{The Aerolites themselves.}---The mass that falls is sometimes
a single piece, but more usually there are many separate fragments,
sometimes numbering many thousands. In such cases the stones are
mostly small, and sometimes they are mere grains of sand. Nearly
all the aerolites that are actually seen to fall, and are found at the time,
are masses of \textit{stone}; but a very few, perhaps three or four per cent
of the whole number, consist of \textit{nearly pure iron}, more or less alloyed
%% -----File: 442.png---Folio 431-------
with nickel. There are also a good many cases of uranoliths, which
are mainly stony, but have a considerable portion of iron disseminated
through the mass in grains and globules; and nearly all the
stony uranoliths contain as much as twenty or thirty per cent of iron
in the form of sulphides or analogous compounds.

\fussy
\begin{fineprint}
\sloppy
\nbarticle{758.}\hspace{0em} The only \textit{iron meteors which have been actually seen} to fall so far,
and are represented by specimens in our museums, are the following:---
\fussy
\begin{center}
\begin{tabular}{r l@{\ldots\ldots\ldots} r}
(1) & Agram, Bohemia,\dotfill               &\dotfill 1751.\\
(2) & Dickson Co., Tennessee,\dotfill       &\dotfill 1835.\\
(3) & Braunau, Bohemia,\dotfill             &\dotfill 1847.\\
(4) & Tabarz, Saxony,\dotfill               &\dotfill 1854.\\
(5) & Nejed, Arabia,\dotfill                &\dotfill 1865.\\
(6) & Nedagollah, India,\dotfill            &\dotfill 1870.\\
(7) & Maysville, California,\dotfill        &\dotfill 1873.\\
(8) & Rowton, Shropshire, England,\dotfill  &\dotfill 1876.\\
(9) & Emmett Co., Iowa,\dotfill             &\dotfill 1879.\\
(10)& Mazapil, Mexico,\dotfill              &Nov.~27, 1885.\\
(11)& Johnson Co., Arkansas,\dotfill        &\dotfill 1886.\\
\end{tabular}
\end{center}
The Emmett County iron was mostly in small fragments, and along with
them there were many large stones with quantities of iron included. The
separate fragments of pure iron which reached the earth probably came by
the breaking up of the stony masses.

Besides these iron meteors which have been seen to fall, our cabinets
contain a very large number of so-called meteoric irons; \textit{i.e.}, masses of iron
found under such circumstances that they cannot easily be accounted for in
any way except by supposing them to be of meteoric origin.

\nbarticle{759.} The number of meteorites which have fallen since 1800 and
been gathered into our cabinets\footnote
  {In this country the cabinets of Amherst College and Harvard and Yale
  Universities are especially rich in meteorites. The finest collection in the world,
  however, is that at Vienna. The collection of the British Museum is also noteworthy,
  as well as that at Paris.}
is about 250. The most remarkable
falls in the United States have been the six following: namely, Weston,
Connecticut, 1807; Bishopsville, So.~Carolina, 1843; Cabarrus Co., No.~Carolina,
1849; New Concord, Ohio, 1860; Amana, Iowa, 1875; and Emmett
Co., Iowa, 1879. In the first case and the three last, several hundred
fragments fell at the same time, ranging in size from five hundred pounds
to half an ounce.

\includegraphicsouter{illo209}{\textsc{Fig.~209.}\\Fragment of one of the Amana Meteoric Stones.}

\nbarticle{760.} \textit{Twenty-four} of the sixty-seven known chemical \textit{elements} have
been found in meteors, and not a single new one. The \textit{minerals} of
%% -----File: 443.png---Folio 432-------
which meteorites are composed present a great resemblance to terrestrial
minerals of volcanic origin, but
many of them are peculiar, and
found in meteors only. (The
study of these meteoric minerals
is a very curious and important
branch of mineralogy,
though naturally it has not
many votaries.) The occasional
presence of carbon is to be specially
noted, and in a meteor
which recently fell in Russia the
carbon appeared to be in a crystalline
form, identical with the
black diamond, though in exceedingly
minute particles. \figref{illo209}{Fig.~209}
is from a photograph of a
fragment of one of the meteoric
stones which fell at Amana, Iowa,
in 1875. The picture is taken
by the permission of the publishers
from Professor Langley's
``New Astronomy,'' where the
body is designated as ``part of a
comet.''
\end{fineprint}

\nbarticle{761.} \nbparatext{The Crust.}---The most characteristic external feature of an
aerolite is the thin, black \textit{crust} that covers it, usually, but not always,
glossy like a varnish. It is formed by the fusion of the surface in the
meteor's swift motion through the air, and in some cases penetrates
deeply into the mass of the meteor through fissures and veins. It is
largely composed of oxide of iron, and is always strongly magnetic.
The crusted surface usually exhibits pits and hollows, such as would
be produced by thrusting the thumb into a mass of putty. These
cavities are explained by the burning out of certain more fusible substances
during the meteor's flight.

\nbarticle{762.} \nbparatext{Magnitude.}---Of the meteors actually seen to fall the largest
pieces found thus far weigh about 500 pounds, though the whole mass
of the body when it first entered the atmosphere has sometimes been
much larger, perhaps, in a few cases, amounting to two or three tons.\footnote
  {Some of the masses of iron supposed to be of meteoric origin, but not actually
  seen to fall, are very much larger. The iron mass from Otumpa in Mexico is said
  to weigh fully sixteen tons. As regards some of these hypothetical meteorites,
  however, their meteoric origin is more than questionable; such, for instance, is
  the case with the Ovifak iron found by Nordenskiold on the coast of Greenland,
  and exhibited at the Philadelphia Centennial Exhibition.}
%% -----File: 444.png---Folio 433-------

As seen from a distance of many miles, the meteoric fire-ball sometimes
\textit{appears} to have a diameter as large as the moon, which would
indicate a real diameter of several hundred feet. The great apparent
size, however, is an illusion, partly due to irradiation, and partly,
undoubtedly, to the fact that the meteor itself is surrounded by an
extensive envelope of heated air and smoke which becomes luminous
throughout. No meteor ever yet investigated would make a mass as
large as ten feet in diameter.

\nbarticle{763.} \nbparatext{Path.}---When a meteor has been observed by a number of
persons at \textit{different points}, who have noted any data which will give
its altitude and bearing at identified moments, the path can be computed.
Observations from a single point are worthless for the purpose,
since they can give no information as to the meteor's \textit{distance}.

The meteor is generally first seen at an altitude of between eighty
and 100 miles, and disappears at an altitude of between five and ten
miles. The length of the path may be anywhere from 50 miles to
500, according to its inclination to the earth's surface. The velocity
is rather difficult to ascertain, but is found to range from ten to forty
miles per second at the moment when the meteor first becomes visible,
and diminishes to one or two miles per second, at the time when it
disappears. The \textit{average} velocity with which meteors enter the
atmosphere appears not to vary much from the ``parabolic velocity''
of twenty-six miles per second, due to the sun's attraction at the
earth's distance---a fact which, of course, indicates that these bodies,
whatever their origin may be, are now moving in space, like the comets,
under the sun's attraction.

\sloppy
\begin{fineprint}
With possibly a very few exceptions in cases where the meteor \textit{glances},
so to speak, on the earth's atmosphere, like a skipping-stone on water, a
body which has once entered the air is sure to be brought to the ground:
it is hardly possible that one meteor in a million should escape after becoming
involved in the atmosphere. We mention this especially, because some
authorities erroneously speak of it as a usual thing for the meteor to keep
on its course, and leave the earth, after throwing off a few fragments.
\end{fineprint}

\fussy
\nbarticle{764.} \nbparatext{Observation of Meteors.}---The object of the observation
should be to obtain accurate estimates of the altitude and azimuth of
the body at moments which can be identified. At night this is best
%% -----File: 445.png---Folio 434-------
done by noting the position of the meteor with reference to neighboring
stars at the moments of its appearance and disappearance, or of
the intervening explosions. In the daytime it can often be done by
noting the position of the object with reference to trees or buildings.
The observer should then mark the exact position where he is standing,
so that by going there afterwards with proper instruments he can
determine the data desired.

\begin{fineprint}
Of course, all such measurements must be given \textit{in angular units}. To
speak of a meteor as having an altitude of twenty \textit{feet}, and pursuing a path
100 \textit{feet} long, is meaningless, unless the size of the ``foot'' is somehow
defined.
\end{fineprint}

The determination of the meteor's \textit{velocity} is more difficult, as it
is seldom possible to look at a watch-face quickly enough, even in
the daytime. The usual course is for the observer to repeat some
familiar piece of doggerel as rapidly as possible, beginning when the
object first becomes visible and stopping when it explodes or disappears,
noting also the precise syllable where he stops. By repeating
the same sentence over again before a clock it is possible to determine
within a few tenths of a second the time occupied by the
meteor's flight.

\nbarticle{765.} \nbparatext{Explanation of the Heat and Light of a Meteor.}---These
are due simply to the destruction of the body's velocity; its kinetic
mass-energy of motion is transformed into heat by the friction of the
air. If a moving body whose mass is $M$ kilograms, and its velocity
$V$ metres per second, is stopped, the number of calories of heat
developed is given by the equation
\[
Q = \frac{MV^2}{8339} \  \text{(\artref{Art.}{354}).}
\]

The quantity of heat evolved in bringing to rest a body which has a
velocity of forty-two kilometres, or twenty-six miles a second, is enormous,
vastly more than sufficient to fuse it even if it were made of
the most refractory material, and hundreds of times more than would
be produced by its combustion in oxygen if it were a mass of coal.

This heat is developed all along the meteor's course, and mostly just
upon its surface. As Sir William Thomson has shown, the thermal
effect of the rush through the air is the same as if the meteor \textit{were
immersed in a blow-pipe flame} having a temperature of many thousand
degrees; and it is to be noted that this \textit{temperature is independent
%% -----File: 446.png---Folio 435-------
of the density of the air} through which the meteor may be passing.
The \textit{quantity of heat} developed in a given time is greater, of course,
where the air is dense; but the \textit{temperature} produced in the air itself,
at the surface where it rubs against the moving body, is the same
whether the gas be dense or rare.

\begin{fineprint}
When a moving body has a velocity of about 1500 metres per second,
the virtual temperature of the surrounding air is about that of red heat; \textit{i.e.},
the body becomes heated as fast as it would if it were at rest and the air
about it were heated to that temperature. When the velocity reaches twenty
or thirty miles per second, it is acted upon as if the surrounding gas were
heated to the liveliest incandescence at a temperature of several thousand
degrees. The surface is fused, and the liquefied portion is continually
swept off by the rush of the air, condensing as it cools into the luminous
powder that forms the train. The fused surface itself is continually
renewed until the velocity falls below two miles a second or thereabouts,
when it solidifies and forms the characteristic crust. As a general rule,
therefore, the fragments are hot if found soon after their fall; but if the
stone is a large one and falls nearly vertically, so as to have but a short path
through the air, the heating effect will be mainly confined to its surface; and
owing to the low conducting power of stone, the \textit{centre} may still remain
intensely \textit{cold}, retaining nearly the temperature which it had in interplanetary
space. It is recorded that one of the large fragments of the Dhurmsala
(India) meteorite, which fell in 1860, was found in moist earth half an hour
or so after the fall, \textit{coated with ice}.
\end{fineprint}

\nbarticle{766.} \nbparatext{Train.}---One unexplained feature of meteoric trains deserves
notice. They often remain luminous for a long time, sometimes
as much as half an hour, and are carried by the wind like
clouds. It is impossible to suppose that such a cloud of dust remains
\textit{incandescent from heat} for so long a time in the cold upper regions of
the atmosphere; and the question of its enduring luminosity or phosphorescence
is an interesting and puzzling one.

\sloppy
\nbarticle{767.} \nbparatext{Origin.}---We may at once dismiss the theories which make
meteors to be the \textit{immediate} product of volcanic eruption on the
earth or on the moon. They come to us for the most part, as has
been said, from the depths of space, with the velocity of planets and
comets, and there is no certain reason for assuming that they originated
in any manner different from the larger heavenly bodies.

\fussy
\begin{fineprint}
At the same time, many of them so closely resemble each other as almost
to compel the idea of some common source; and though lunar volcanoes are
now extinct, and no terrestrial volcano, not even Krakatćo, is \textit{now} competent
%% -----File: 447.png---Folio 436-------
to send off its ejected missiles through the earth's atmosphere into space,
it is not certain that this was always so. Some still maintain that these
bodies may be fragments which were shot off millions of years ago when the
moon's volcanoes were in full vigor and the earth was young. Since then,
according to this view, these masses have been travelling around the sun in
long ellipses which intersect the orbit of the earth, until at last they happen
to come along at the right time and encounter her surface, and so return to
the old home.

As to the \textit{iron} meteors, some maintain that they are masses which have
been ejected from our sun, or from some other star; and they fortify their
opinion by the remarkable but unquestionable fact that these meteoric irons
are full of ``\textit{occluded}'' hydrogen and carbon oxides which can be extracted
from them by heating them in a proper receptacle connected with a mercurial
pump. They argue that these gases could have been absorbed by
the iron only when it was in the liquid state and overlaid by a dense, hot
atmosphere containing them; and that, they say, is just the condition of the
minute drops of iron which are supposed to form part of the photosphere of
the sun: upon \textit{our} sun or in some other sun only could such conditions be
found. There is no question of the sun's ability to project masses to planetary
distances, as shown by the chromospheric eruptions which have been
repeatedly observed by students of solar physics. And if our sun can do
this, it is natural to suppose that other suns can do it also.\\
\end{fineprint}

However these bodies originated, it is quite certain that before they
reach the earth they have been moving independently in space for a long
time, just as planets and comets do. But a recent important research by
Professor Newton has shown that more than 90 per cent of some 200
aerolites, for the approximate determination of whose paths we have the
data, were moving before their fall in orbits, not parabolic, but analogous
to those of the short-period comets; and \textit{direct}, not retrograde.

\nbarticle{768.} \nbparatext{Detonating Meteors, or ``Bolides,'' of which Fragments are
not known to reach the Earth.}---Some writers discriminate between
these meteors and aerolites, but the distinction does not seem to be
well founded. The phenomena appear to be precisely the same, except
that in the one case the fragments are actually found, and in the other
they fall into the sea, the forest, or the desert; or sometimes when the
path is nearly horizontal, and therefore long, they may be consumed
and dissipated in the dust and vapor of the train, without reaching
the earth's surface at all, except ultimately as impalpable dust.

\nbarticle{769.} \nbparatext{Number.}---As to the number of aerolites which strike the
earth, it is difficult to make a trustworthy estimate. Since the
beginning of the century, at least two or three have been seen to
%% -----File: 448.png---Folio 437-------
fall every year, and have been added to our cabinets (see \artref{Art.}{759}),---in
some years as many as half a dozen. This, of course,
implies a vastly greater number which are not seen, or are not
found. Schreibers, some years ago, estimated the number as high
as 700 a year, and Reichenbach sets it still higher---not less than
3000 or 4000.

\section*{SHOOTING STARS.}

\nbarticle{770.} A few minutes' watching on any clear, moonless night will
be sure to reveal one or more of the swiftly moving, evanescent
points of light that are known as ``shooting stars.'' No sound is
ever heard from them, nor (with a single exception to be mentioned
further on) has anything ever been known to reach the earth's surface
from them, not even when the sky was ``as full of them as of
snow-flakes,'' as sometimes has happened in a great meteoric shower.
For this reason it is perhaps justifiable to allow the old distinction to
remain between them and the bodies we have been discussing, at
least provisionally. The difference \textit{may be}, and according to opinion
at present prevalent very probably is, merely one of size, like that
between boulders and grains of sand. Still there are some reasons
for supposing that there is also a difference of constitution,---that
while the aerolite is a solid, compact mass, the shooting star is a
little cloud of dust and intermingled gas, like a puff of smoke.

\nbarticle{771.} \nbparatext{Numbers.}---The number of these bodies is very great. A
single watcher sees on the average from four to eight hourly. If
observers enough are employed to guard the whole sky, so that
none can escape unnoticed, the visible number becomes from thirty
to sixty an hour. Since ordinarily only these are seen which are
within two or three hundred miles of the observer, the estimated
total daily number of these which enter the earth's atmosphere, and
are large enough to be visible to the naked eye, rises into the millions.
Professor Newton sets it at from 10,000000 to 15,000000, the
average distance between them being about 200 miles.

\begin{fineprint}
There is a still larger number too small to be seen with the naked eye.
One hardly ever works many hours with a telescope carrying a low power,
and having a field of view as large as $15'$ in diameter, without seeing several
of them flash across the field. In a few instances observers have reported
\textit{dark meteors} crossing the moon's disc while they were watching it. There
may be some question, however, as to the real nature of the objects seen in
such a case. Birds~(?).
\end{fineprint}
%% -----File: 449.png---Folio 438-------

\nbarticle{772.} \nbparatext{Comparative Number in Morning and Evening.}---The hourly
number about six o'clock in the morning is fully double the hourly
number in the evening. The obvious reason is simply that in the
morning we are on the \textit{front of the earth} as regards its orbital
motion, while in the evening we are in the rear; in the evening we
only see such meteors as \textit{overtake us}; in the morning we see all that
we either meet or overtake. If they are really moving in all directions
alike, with the parabolic velocity corresponding to the earth's
distance from the sun (twenty-six miles per second), theory indicates
that the relative hourly numbers for morning and evening ought to
be in just the observed proportion.

\nbarticle{773.} \nbparatext{Brightness.}---For the most part these bodies are much like
the stars in brightness,---a few are as brilliant as Venus or Jupiter;
more are like stars of the first magnitude; and the majority are like
the smaller stars. The bright ones not unfrequently show trains
which sometimes last from five to ten minutes, when they are folded
up and wafted away by the winds of the upper air.\footnote
  {These air currents, at an elevation of forty miles above the earth's surface,
  are thus observed to have, ordinarily, velocities from fifty to seventy five miles
  per hour.}

\sloppy
\nbarticle{774.} \nbparatext{Elevation, Path, and Velocity.}---By observations made by
two or more observers thirty or forty miles apart, it is possible to
determine the height, path, and velocity of these bodies. It is found
as the result of a great number of such observations that they first
appear at an average elevation of about \textit{seventy-four miles}, and disappear
at an average height of about \textit{fifty miles}, after traversing a
distance of \textit{forty or fifty miles}, with an average velocity of about
\textit{twenty-five miles} per second. They do not begin to be visible at so
great an elevation as the aerolitic meteors, and they vanish before
they penetrate so deeply into the atmosphere.

\fussy
\nbarticle{775.} \nbparatext{Materials.}---Occasionally it has been possible to catch a
glimpse of the spectrum of one of the brighter shooting stars, and
the lines of sodium and magnesium (probably) are quite conspicuous,
along with some other lines which cannot be securely identified by
such a hasty glance.

As these bodies are completely burned up before they reach the
earth, all we can ever hope to get of their material is the product
of the combustion. In most places the collection and identification
of this meteoric ashes is, of course, hopeless: but Nordenskiold
%% -----File: 450.png---Folio 439-------
has thought he might find it in polar snows, and others have
thought it might be found in the material dredged up from the
bottom of the ocean. In fact, the Swedish naturalist, by melting
several tons of Spitzbergen snow and filtering the water, \textit{did} find in
it a sediment containing minute globules of oxide and sulphide of
iron: similar globules are also found in the products of deep sea
dredging. These \textit{may be} meteoric ashes; but what we have lately
learned from Krakatćo of the distance to which smoke and fine
volcanic dust is carried by the winds, makes it quite possible that
the suspected material is purely terrestrial in its origin.

\nbarticle{776.} \nbparatext{Probable Mass of Shooting-Stars.}---We have no very certain
means of getting at this. We can, however, fix a provisional value
\textit{by means of the amount of light they give}. Photometric comparisons
between a standard star and a meteor, when we know the meteor's distance
and the duration of its flight, enable us to ascertain how the total
amount of light emitted by it compares with that given by a standard
candle shining for one minute. Now, according to determinations
made some thirty years ago by Thomson at Copenhagen, (which
ought to be repeated,) the \textit{light given by a standard candle in a minute
is equivalent to about twelve foot-pounds of energy}. This excludes
all the energy of the dark, invisible radiation of the candle. Our
observations of the meteor give us, therefore, its total \textit{luminous energy}
in foot-pounds; and if the whole of the meteor's energy appeared as
light, then, since $\text{\textit{Energy}} = \frac{1}{2}MV^{2}$, we could at once get its mass by
dividing twice this luminous energy by the square of the meteor's
velocity. Since, however, only a small portion of the meteor's
whole energy is transformed into light, the mass obtained in this
way would be too small, and must be multiplied by a factor which
expresses the ratio between the \textit{total} energy and that which is \textit{purely
luminous}. It is not likely that this factor exceeds \textit{one hundred}, or is
less than \textit{ten}, though on this point we very much need information.
Assuming the largest value, however, for this factor, the photometric
observations made in 1866 and 1867 by Professors Newcomb and
Harkness (stationed respectively at Washington and Richmond),
showed that the majority of the meteors of those star-showers weighed
\textit{less than a single grain}. The largest of them did not reach 100 grains,
or about a quarter of an ounce.

\begin{fineprint}
\nbarticle{777.} \nbparatext{Growth of the Earth.}---Since the earth (in fact, every planet)
is thus continually receiving meteoric matter, and sending nothing away
from it, \textit{it must be constantly growing larger:} but this growth is extremely
%% -----File: 451.png---Folio 440-------
insignificant. The meteoric matter received daily by the earth, if we accept
one grain as the average weight of a shooting star, would be only about
a ton, after making a reasonable addition for occasional aerolites, if we
multiply this estimate by one hundred, it certainly will be exceedingly
liberal, and at that rate the amount received by the earth in a year would
amount to the very respectable figure of 36,500 tons; and yet, even at this
rate, assuming the specific gravity of the average meteor as three times that
of water, it would take, about 1000,000000 \textit{years to accumulate a layer one
inch thick over the earth's surface.}

\sloppy
\nbarticle{778.} \nbparatext{Effect on the Earth's Orbit.}---Theoretically, the encounter of
the earth with meteors must \textit{shorten the year} in three distinct ways:---

\fussy
First. By acting as a resisting medium, and so diminishing the size
of the earth's orbit, and indirectly accelerating its motion, in the same
manner as is supposed to happen with Encke's comet.

Second. By increasing the attraction between the earth and the sun
through the increase of their masses.

Third. By lengthening the day---the earth's rotation being made slower
by the increase of its diameter, so that the year will contain a smaller number
of days.

The whole effect, however, of the three causes combined, does not amount
to $\frac{1}{1000}$ second in a million of years. The diminution of the earth's
distance from the sun, assuming that one hundred tons of meteoric matter
fall daily, and also assuming that the meteors are moving equally in all
directions with the parabolic velocity of twenty-six miles per second, comes
out about $\frac{1}{20\,000}$ of an inch \textit{per annum}.

Theoretically, also, the same meteoric action should produce a shortening
of the \textit{month}, and Oppolzer investigated the subject a few years ago, to see
what amount of meteoric matter would account for the observed \textit{lunar acceleration}
(\artref{Art.}{459}). He found that it would require an amount immensely
greater than really falls.

\nbarticle{779.} \nbparatext{Meteoric Heat: Effect of Meteors on the Transparency of
Space.}---Of course each meteor brings to the earth a certain amount of \textit{heat}
developed in the destruction of its motion; and at one time if was thought
that a very considerable percentage of the total heat: received by the earth
might be derived from this source (See \artref{Art.}{355}, (2)). Assuming, however,
as before, the fall of one hundred tons of meteoric matter daily with
an average velocity of twenty miles per second relative to the earth, the
whole amount of heat comes out about $\frac{1}{20}$ calorie \textit{per annum} for each square
metre of the earth's surface---as much in a \textit{year} as the sun imparts to the
same surface in about \textit{one-tenth of a second.}

One other effect of meteoric matter in space should be alluded to. It
must necessarily render space imperfectly transparent, like a thin haze.
Less light reaches us from a remote star than if the meteors were absent.
\end{fineprint}
%% -----File: 452.png---Folio 441-------

\sloppy
\nbarticle{780.} \nbparatext{Meteoric Showers.}---At certain times the shooting stars,
instead of appearing here and there in the sky at intervals of several
minutes, and moving in all directions, appear by thousands, and even
hundreds of thousands, for a few hours.

\nbenlargepage
\fussy
\textit{The Radiant.}---At such times they do not move without system;
but they all appear to diverge or ``\textit{radiate}'' from one point in the
sky; that is, their paths produced backward all intersect at a common
point (or nearly so), which is called ``\textit{the radiant.}'' As an old lady
expressed it, in speaking of the meteoric shower of 1833, ``The sky
looked like a great umbrella.'' The meteors which appear near the
\textit{radiant} are stationary, or have paths extremely short, while those
which appear at a distance from it have long courses. The radiant
keeps its place among the stars unchanged during the whole continuance
of the shower, and the shower is named accordingly. Thus we
have the meteor shower of the ``\textit{Leonids},'' whose radiant is in the
constellation of Leo; similarly the ``\textit{Andromedes}'' (or Bielids), the
``\textit{Perseids},'' the ``\textit{Geminids},'' the ``\textit{Lyrids},'' etc. \figref{illo210}{Fig.~210}.\ is a chart
of the tracks of meteors observed on the night of Nov.~13, 1866,
showing the radiant near $\zeta$~Leonis.

\includegraphicsmid{illo210}{\textsc{Fig.~210.}---The Meteoric Radiant in Leo, Nov.~13, 1866.}

The simple explanation is that the radiant is purely an effect of
%% -----File: 453.png---Folio 442-------
perspective. The meteors are really moving, relatively to the observer,
in lines which are sensibly straight and parallel, as are also
the tracks of light which they leave in the air. Hence they all seem
to diverge from one and the same perspective ``vanishing point.'' The
position of the radiant depends entirely upon the \textit{direction} of the
meteor's motion relative to the earth.

On account of the irregular form of the meteoric particles, they are
deflected a little one way or the other by the air, so that their paths
do not intersect at an absolute point; neither is it likely that before
they enter the air their paths are \textit{exactly} parallel. The consequence
is that the radiant, instead of being a point, is an \textit{area} of some little
size, usually less than $2°$ in diameter.

\begin{fineprint}
\nbarticle{781.} Probably the most remarkable of all meteoric showers that ever
occurred was that which appeared in the United States on Nov.~12, 1833,
in the early morning---a shower of Leonids. The number that fell in
the five or six hours during which the shower lasted was estimated at
Boston as fully 250,000. A competent observer declared that ``he never
saw snow-flakes thicker in a storm than were the meteors in the sky at
some moments.'' No sound was heard, nor was any particle known to
reach the earth.
\end{fineprint}

\nbarticle{782.} \nbparatext{Dates of Showers.}---Since the meteor-swarm pursues a regular
orbit around the sun, the earth can only encounter it when she is
at the point where her orbit cuts the path of the meteors; and this,
of course, must always be on the same day of the year, except as, in
the process of time, the meteors' orbits slowly shift their positions on
account of perturbations. The Leonid showers, therefore, always
appear on the 18th of November (within a day or two); the Andromedes
on the 27th or 28th of the same month; and the Perseids early
in August.

\nbarticle{783.} \nbparatext{Meteoric Rings and Swarms.}---If the meteors are scattered
nearly uniformly around their whole orbit, so as to form \textit{a ring}, the
shower will recur \textit{every year}; but if the flock is concentrated, it will occur
only when the meteor group is at the meeting-place at the same time
as the earth. The latter is the case with the Leonids and Andromedes.
The great star-showers from these groups occur only rarely,---for the
Leonids once in thirty-three years, and for the Andromedes (otherwise
known as the Bielids) about once in thirteen. The Perseids are
much more equally and widely distributed, so that they appear in
considerable numbers every year, and are not sharply limited to a
%% -----File: 454.png---Folio 443-------
particular date, but are more or less abundant for a fortnight in the
latter part of July and the first of August.

\begin{fineprint}
The meteors which belong to the same group all have a resemblance to
each other. The Perseids are yellowish, and move with medium velocity.
The Leonids are very swift, for we meet them almost directly, and they are
characterized by a greenish or bluish tint, with vivid and persistent trains.
The Andromedes are sluggish in their movements, because they simply
overtake the earth, instead of meeting it. They are usually decidedly red in
color and have only small trains.

\nbarticle{784.} \nbparatext{The Mazapil Meteorite.}---As has been said, during these
showers no sound is heard, no sensible heat perceived, nor do any masses
reach the ground; with the one exception, however, that on Nov.~27, 1885,
a piece of meteoric iron mentioned in the list given in \artref{Article}{758}, fell at
Mazapil in Northern Mexico during the shower of Andromedes which
occurred that evening. Whether the coincidence is accidental or not, it is
interesting. Many high authorities speak confidently of this particular iron
meteor as being really a piece of Biela's comet itself.
\end{fineprint}

And now we come to one of the most remarkable discoveries of
modern astronomy,---the discovery of---

\nbarticle{785.} \nbparatext{The Connection between Comets and Meteors.}---At the time
of the great meteoric shower of 1833, Professors Olmsted and
Twining, of New Haven, recognized the fact and meaning of the
radiant as pointing to the existence of \textit{swarms} of meteoric particles
revolving in regular orbits around the sun; and Olmsted at the time
went so far as even to call the body or swarm a ``comet.'' In some
respects, however, his views were seriously wrong, and soon received
modification and correction from other astronomers. Erman especially
pointed out that in some cases, at least, it would be necessary
to suppose that the meteors were distributed in \textit{rings}, and he also
developed methods by which the meteoric orbits could be computed
if the necessary data could be secured. Olmsted and Twining,
however, were the first to show that the meteors are not terrestrial
and atmospheric, but bodies truly cosmical.

The subject was taken up later by Professor Newton, of New Haven,
who in 1864 showed by an examination of old records that
there had been a number of great autumnal meteoric star-showers at
intervals of just about thirty-three years, and he predicted confidently
a shower for Nov.~13--14, 1866. As to the orbit of the meteoric
body (or ring, according to Erman's view), he found that it might,
%% -----File: 455.png---Folio 444-------
consistently with what had been so far observed, have either of \textit{five}
different orbits; one with a period of $33 \frac{1}{4}$ years, two with periods of
one year $\pm 11$ days, and two with periods of half a year $\pm 5 \frac{1}{2}$ days.
He considered rather most probable the period of 354 days; but he
pointed out that the slow change that had taken place in the annual
date of the shower\footnote
  {In \textsc{a.d.}~902 (the ``year of the stars'' in the old Arab chronicles), the date
  was what would be Oct.~19, in our ``new style'' reckoning. In 1202 the shower
  occurred five days later, and in 1833 the date was Nov.~12.}
would furnish the means of determining which
of the orbits was the true one.

This change of date indicates a slow motion of the nodes of the
orbit of the meteoric body at the rate of about $52''$ a year. Adams,
of Neptunian fame, made the laborious calculation of the effect of
planetary perturbations upon each of the five different orbits suggested
by Professor Newton, and showed that the true orbit must be
the largest one which has a period of $33 \frac{1}{4}$ years.

\nbenlargepage
The meteoric shower occurred in 1866 as predicted, and was
repeated in 1867, the meteor-swarm being stretched out along its
orbit for such a distance that the procession is nearly three years in
passing any given point.
%% -----File: 456.png---Folio 445-------

\includegraphicsmid{illo211}{\textsc{Fig.~211.}---Orbits of Meteoric Swarms which are known to be associated with Comets.}

\nbarticle{786.} \nbparatext{Identification of Cometary and Meteoric Orbits.}---The researches
of Newton and Adams had awakened lively interest in
the subject, and Schiaparelli, of Milan, a few weeks after the Leonid
shower, published a paper upon the Perseids, or August meteors, in
which he brought out the remarkable fact that they were \textit{moving in
the same path as that of the bright comet of $1862$, known as Tuttle's
Comet}. Shortly after this Leverrier published his orbit of the
Leonid meteors, derived from the observed position of the radiant in
connection with the periodic time assigned by Adams; and almost
simultaneously, but without any idea of a connection between them,
Oppolzer published his orbit of Tempel's comet of 1866; and the
two orbits were at once seen to be \textit{practically identical}. Now a \textit{single}
case of such a coincidence as that pointed out by Schiaparelli, might
possibly be accidental, but hardly \textit{two}. Then five years later, in
1872, came the meteorite shower of the Andromedes, following in the
track of Biela's comet; and among the more than one hundred distinct
meteor-swarms now recognized, Professor Alexander Herschel
finds four or five others which have a ``comet annexed,'' so to speak.
Fig.~211 represents the orbits of four of the meteoric swarms which
are known to be associated with comets.
%% -----File: 457.png---Folio 446-------

\begin{fineprint}
\nbarticle{787.} In the cases of the Leonids and Andromedes the meteor-swarm
\textit{follows} the comet. Many believe, however, that the comet itself is simply
the thickest part of the swarm. Kirkwood and Schiaparelli have both
pointed out that a body constituted as a comet is supposed to be, must
almost necessarily break up in consequence of the ``tide-producing'' perturbations
of the sun, independent of any repulsive action such as is supposed
to be the cause of a comet's tail. They hold that these meteor-swarms are
therefore merely the \textit{product of a comet's disintegration}.

\includegraphicsmid{illo212}{\textsc{Fig.~212.}---Transformation of the Orbit of the Leonids by the Encounter with Uranus, \textsc{a.d.}~126.}

The longer the comet has been in the system, the more widely scattered
will be its particles. The Perseids are supposed, therefore, to be old inhabitants
of the solar system, while the Leonids and Andromedes are comparatively
new-comers. Leverrier has shown that in the year \textsc{a.d.}~126 Tempel's
comet must have been very near to Uranus, and a natural inference is that
it was introduced into the solar system at that time. \figref{illo212}{Fig.~212} illustrates his
hypothesis. However these things may be, it is now certain that the connection
between comets and meteors is a very close one, though it can hardly
be considered certain as yet that every scattered group of meteors is the
result of cometary \textit{disintegration}. We are not sure that when a cometary
mass first enters the solar system from outer space, it comes in as a close-packed
swarm.
\end{fineprint}
\chelabel{CHAPTERXVIII}

%% -----File: 458.png---Folio 447-------
\Chapter{XIX}{The Stars}
{\setlength{\hfuzz}{2pt}
\nbchapterhang{\stretchyspace
THE STARS:\hspace{0em} THEIR NATURE AND NUMBER.---THE
CON\-STELLATIONS.---STAR-CATALOGUES.---DESIGNATION AND
NOMENCLATURE.---PROPER MOTIONS AND THE MOTION
OF THE SUN IN SPACE.---STELLAR PARALLAX AND DISTANCE.}

}%end hfuzz
\nbarticle{788.} \textsc{We} enter now upon a vaster subject. Leaving the confines
of the solar system we cross the void that makes an island\footnote
  {That the solar system is thus isolated by a surrounding void is proved by the
  almost undisturbed movements of Uranus and Neptune; for their perturbations
  would betray the presence of any body, at all comparable with the sun in magnitude,
  within a distance a thousand times as great as that between the earth and
  sun.}
of the
sun's domains, and enter the universe of the stars. The nearest star,
so far as we have yet been able to ascertain, is one whose distance is
more than 200,000 times the radius of the earth's annual orbit; so
remote that, seen from that star, the sun itself would appear only
about as bright as the pole star, and from it no telescope ever yet
constructed could render visible a single one of all the retinue of
planets and comets that make up the solar system.

\nbarticle{789.} \nbparatext{Nature of the Stars.}---As shown by their spectra the stars
are \textit{suns}; that is, they are bodies comparable in magnitude and in
physical condition with our own sun, shining by their own light as the
sun does, and emitting a radiance which in many cases could not be
distinguished from sunlight by any of its spectroscopic characteristics.
Some of them are vastly larger and hotter than our sun, others smaller
and cooler, for, as we shall see, they differ enormously among themselves.

\nbarticle{790.} \nbparatext{Number of the Stars.}---The impression on a dark night is of
absolute countlessness; but, in fact, the number visible to the naked
eye is very limited, as one can easily discover by taking some definite
area in the sky, say the ``bowl of the dipper,'' and counting the stars
which he can see within it. He will find that the number which he
%% -----File: 459.png---Folio 448-------
can fairly count is surprisingly small, though by averted vision he will
get uncertain glimpses of many more. In the whole celestial sphere
the number bright enough to be visible to the naked eye is only from
6000 to 7000 in a clear, moonless sky. A little haze or moonlight
cuts out fully half of them, and of course there is a great difference
in eyes. But the sharpest eyes could probably never fairly see more
than 2000 or 3000 at one time, since near the horizon the smaller stars
are invisible, and they are immensely the most numerous, fully half
of the whole number being those which are just on the verge of visibility.
\textit{The total number that can be seen well enough for observation
with such instruments as were used before the invention of the telescope
is not quite $1100$.}

With even a small telescope the number is enormously increased.
A mere opera-glass an inch a half in diameter brings out at least
100,000. The telescope with which Argelander made his \textit{Durchmusterung}
of more than 300,000 stars---all north of the celestial equator---had
a diameter of only two inches and a half. The number visible
in the great Lick\footnote
  {Neglecting the loss of light in the lenses, the Lick telescope ought, theoretically,
  to show stars so faint that it would take more than 30,000 of them to make
  a star equal to the faintest that can be seen with the naked eye. (See \artref{Art.}{38}.)}
telescope of three feet diameter is probably nearly
100,000000.

\sloppy
\nbarticle{791.} \nbparatext{Constellations.}---In ancient times the stars were grouped by
``constellations,'' or ``asterisms,'' partly as a matter of convenient
reference and partly as superstition. Many of the constellations now
recognized,---all of those in the zodiac and those about the northern
pole,---are of prehistoric antiquity. To these groups were given fanciful
names, mostly of persons or objects conspicuous in the mythological
records of antiquity; a great number of them are connected in some
way or other with the Argonautic expedition.

\fussy
\begin{fineprint}
In some cases the eye can trace in the arrangement of the stars a vague
resemblance to the object which gives name to the constellation; but generally
no reason can be assigned why the constellation should be so named or
so bounded. Of the sixty-seven constellations now usually recognized on
celestial globes, forty-eight have come down from Ptolemy. The others
have been formed by Hevelius, Bayer, Royer, and one or two other astronomers,
to embrace stars not included in Ptolemy's constellations, and especially
to furnish a nomenclature for the stars never seen by Ptolemy on
account of their nearness to the southern pole. A considerable number of
%% -----File: 460.png---Folio 449-------
other constellations, which have been tentatively established at various
times, and are sometimes found on globes and star-maps, have been given
up as useless and impertinent.

\nbarticle{792.} We present a \hyperref[tab:const]{list of the constellations}, omitting, however, some
of the modern ones which are now not usually recognized by astronomers.
The constellations are arranged both vertically and horizontally. The order
in the vertical columns is determined by right ascension, as indicated by
the Roman numbers at the left. Horizontally the arrangement is according
to distance from the north pole, as shown by the headings of the columns.
The number appended to each constellation gives the number of stars it
contains, down to and including the 6th magnitude. The zodiacal constellations
are italicized, and the modern constellations are marked by an
asterisk.
\begin{sidewaystable}
\nblabel{tab:const}
\scriptsize\centering
\renewcommand{\arraystretch}{0.8}
\begin{tabular}{@{}r@{ }r@{\;}| *{5}{@{\,}c@{\,}l@{ }r|} @{\,}c@{\,}l@{ }r@{}}
\multicolumn{20}{c}{LIST OF CONSTELLATIONS, SHOWING THEIR POSITION IN THE HEAVENS.}\\[1ex]

\hline\hline

\multicolumn{2}{@{}c@{}|@{\,}}{\textsc{\scriptsize
   \rlap{\raisebox{-1ex}{\;R. A.}}%
   \raisebox{0.8ex}{\rotatebox[origin=c]{-18}{\rule{6.5em}{0.5pt}}}% Diagonal line across top left cell
   \llap{\raisebox{1ex}{Decl.\;}}}}
& \multicolumn{3}{c|@{\,}}{\textsc{\scriptsize $-90°$ to $+50°$.\rule{0pt}{3.5ex}}}
& \multicolumn{3}{c|@{\,}}{\textsc{\scriptsize $+50°$ to $+25°$.}}
& \multicolumn{3}{c|@{\,}}{\textsc{\scriptsize $+25°$ to $0°$.}}
& \multicolumn{3}{c|@{\,}}{\textsc{\scriptsize $0°$ to $-25°$.}}
& \multicolumn{3}{c|@{\,}}{\textsc{\scriptsize $-25°$ to $-50°$.}}
& \multicolumn{3}{c }{\textsc{\scriptsize $-50°$ to $-90°$.}}\\[1ex]

\hline

\multirow{2}{*}{\scriptsize h.} &
\multirow{2}{*}{\scriptsize h.} &
&& &&&  &&& &&& &&& &&
\\
\multirow{2}{*}{I,}  & \multirow{2}{*}{II,}
& & \multirow{2}{*}{Cassiopeia,} & \multirow{2}{*}{46}
& & Andromeda, & 18
& & \textit{Pisces}, & 18 & & \multirow{2}{*}{Cetus,}  & 32
& & Ph\oe nix,       & 32 & & Ph\oe nix, \textit{bis.}
\\
&
&&& & & Triangulum, &  5
& & \textit{Aries}, & 17 &&&
&*& App.\ Sculp.    & 13 & & Hydrus, & 18
\\[1.5ex]

\multirow{2}{*}{III,} & \multirow{2}{*}{IV,}
& \multicolumn{3}{c|@{\,}}{\multirow{2}{*}{ --- \hspace{3em} --- }}
& & \multirow{2}{*}{Perseus,}  & \multirow{2}{*}{40}
& & \multirow{2}{*}{\textit{Taurus},} & \multirow{2}{*}{58}
& & \multirow{2}{*}{Eridanus,} & \multirow{2}{*}{64}
& & \multirow{2}{*}{(Eridanus, \textit{bis.})} & &*& Horologium, & 11
\\
&
&&& &&&
&&& &&&
&&& &*& Reticulum, & 9
\\[1.5ex]

&
&&& &&&
& & \multirow{2}{*}{Orion,}        & \multirow{2}{*}{37}  &&&
&&& &*& Dorado, & 16
\\
V, & VI,
&*& Camelopardus,    & 36 & & Auriga, & 35
& & \multirow{2}{*}{\textit{Gemini},} & \multirow{2}{*}{28} & & Lepus,  & 18
&*& Columba,         & 15 &*& Pictor, & 14
\\
&
&&& &&&
&&& &&&
&&& &*& Mons Mensę, & 12
\\[1.5ex]

&
&&& &&&
& & \multirow{2}{*}{Canis Minor,} & \multirow{2}{*}{ 6}
& & \multirow{2}{*}{Canis Major,} & \multirow{2}{*}{27}
&&&
& & Argo-Navis, \textit{bis.} &
\\
VII, & VIII,
& \multicolumn{3}{c|@{\,}}{ --- \hspace{3em} --- }
&*& Lynx, & 28
& & \multirow{2}{*}{\textit{Cancer},} & \multirow{2}{*}{15}
&\multirow{2}{*}{*}& \multirow{2}{*}{Monoceros,} & \multirow{2}{*}{12}
& & Argo-Navis, & 133
& & \quad\ (Puppis) &
\\
&
&&& &&&
&&& &&&
&&& &*& Piscis Volans, & 9
\\[1.5ex]

\multirow{2}{*}{IX,} & \multirow{2}{*}{X,}
& \multicolumn{3}{c|@{\,}}{\multirow{2}{*}{ --- \hspace{3em} --- }}
&\multirow{2}{*}{*}& \multirow{2}{*}{Leo Minor,} & \multirow{2}{*}{15}
& & \multirow{2}{*}{\textit{Leo},} & \multirow{2}{*}{47} & & Hydra, & 49
& \multicolumn{3}{c|@{\,}}{\multirow{2}{*}{ --- \hspace{3em} --- }}
& & Argo-Navis &
\\
&
&&& &&&
&&& &*& Sextans, & 3
&&& & & \quad (Vela) &
\\[1ex]

&
&&& &&&
&&& & & \multirow{2}{*}{Crater,}    & \multirow{2}{*}{9}
&&& & & Argo Navis &
\\
XI, & XII,
& & Ursa Major, & 53 & \multicolumn{3}{c|@{\,}}{ --- \hspace{3em} --- }
&*& Coma Ber.   & 20 & & \multirow{2}{*}{Corvus,} & \multirow{2}{*}{8}
& & Centaurus,  & 54 & & \ \ (Carina) &
\\
&
&&& &&&
&&& &&&
&&& &*& Chameleon, & 13
\\[1.5ex]

&
&&& &\multirow{2}{*}{*} & \multirow{2}{*}{Canes Venat.} & \multirow{2}{*}{15}
&&& &&&
&&& & & Centaurus, \textit{bis.} &
\\
XIII, & XIV,
& \multicolumn{3}{c|@{\,}}{ --- \hspace{3em} --- }
& & \multirow{2}{*}{Boötes,} & \multirow{2}{*}{35}
& \multicolumn{3}{c|@{\,}}{ --- \hspace{3em} --- }
& & \textit{Virgo}, & 39
& & Lupus, & 34 &*& Crux, & 13
\\
&
&&& &&&
&&& &&&
&&& &*& Musca, & 15
\\[1.5ex]

\multirow{2}{*}{XV,} & \multirow{2}{*}{XVI,}
& & \multirow{2}{*}{Ursa Minor,} & \multirow{2}{*}{23} & & Corona Bor.     & 19
& & \multirow{2}{*}{Serpens,}    & \multirow{2}{*}{23}
& & \multirow{2}{*}{\textit{Libra},} & \multirow{2}{*}{23}
& & \multirow{2}{*}{Norma,}      & \multirow{2}{*}{14}
&\multirow{2}{*}{*}& \multirow{2}{*}{Circinus,} & \multirow{2}{*}{10}
\\
&
&&& & & Hercules, & 65
&&& &&&
&&& &&&
\\[1.5ex]

\multirow{2}{*}{XVII,} & \multirow{2}{*}{XVIII,}
& & \multirow{2}{*}{Draco,}  & \multirow{2}{*}{80}
& & \multirow{2}{*}{Lyra,}             & \multirow{2}{*}{18}
& & Aquila, & 37 & & \textit{Scorpio}, & 34
& & \multirow{2}{*}{Ara,}    & \multirow{2}{*}{15} &*& Triangul.\ Aust.& 11
\\
&
&&& &&&
& & Sagitta, & 5 & & Ophiuchus, & 46
&&& &*& Apus, & 15
\\[1.5ex]

&
&&& &&&
&\multirow{2}{*}{*}& \multirow{2}{*}{Vulpecula,} & \multirow{2}{*}{23} &&&
&&& &*& Telescopium, & 16
\\
XIX, & XX,
& \multicolumn{3}{c|@{\,}}{ --- \hspace{3em} --- }
& & Cygnus, & 67
& & \multirow{2}{*}{Delphinus,} & \multirow{2}{*}{10}
& & \textit{Sagittarius}, & 38
& & Corona Austr. & 7 & & Pavo. & 37
\\
&
&&& &&&
&&& &&&
&&& &*& Octans, & 22
\\[1.5ex]

\multirow{2}{*}{XXI,} & \multirow{2}{*}{XXII,}
& & \multirow{2}{*}{Cepheus,}      & \multirow{2}{*}{44}
&\multirow{2}{*}{*}& \multirow{2}{*}{Lacerta,} & \multirow{2}{*}{13}
& & \multirow{2}{*}{Equuleus,}     & \multirow{2}{*}{ 5}
& & \multirow{2}{*}{\textit{Capricornus},} & \multirow{2}{*}{22}
& & \multirow{2}{*}{Piscis Austr.} & \multirow{2}{*}{16} &*& Indus, & 15
\\
&
&&& &&&
&&& &&&
&&& &*& Octans &
\\[1.5ex]

\multirow{2}{*}{XXIII,} & \multirow{2}{*}{XXIV,}
& \multicolumn{3}{c|@{\,}}{\multirow{2}{*}{ --- \hspace{3em} --- }}
& \multicolumn{3}{c|@{\,}}{\multirow{2}{*}{ --- \hspace{3em} --- }}
& & \multirow{2}{*}{Pegasus,} & \multirow{2}{*}{43}
& & \multirow{2}{*}{\textit{Aquarius},} & \multirow{2}{*}{25}
&\multirow{2}{*}{*}& \multirow{2}{*}{Grus,} & \multirow{2}{*}{30}
&*& Toucana, & 22
\\
&
&&& &&&
&&& &&&
&&& &*& Octans &
\\[1.5ex]

\hline
\end{tabular}
\end{sidewaystable}

The different groups of constellations are found near the meridian at
half-past eight o'clock, \textsc{p.m.}, on the dates indicated below.

\medskip
{\parindent=1em
\hangindent=4em
Group (I., II.), Dec.~1. These constellations contain no first-magnitude
stars, but Cassiopeia, Andromeda, Aries, and Cetus include enough
stars of the second and third magnitude to be fairly conspicuous.

\hangindent=4em
Group (III., IV.), Jan.~1. Perseus north of the zenith, and the Pleiades
and Aldebaran in Taurus, are characteristic.

\hangindent=4em
Group (V., VI.), Feb.~1. On the whole this is the most brilliant region
of the sky and Orion the finest constellation.

\hangindent=4em
Group (VII., VIII.), March 1. Characterized by Procyon and Sirius, the
latter incomparably the brightest of all the fixed stars.

\hangindent=4em
Group (IX., X.), April 1. Leo is the only conspicuous constellation.

\hangindent=4em
Group (XI., XII.), May 1. A barren region, except for Ursa Major north
of the zenith.

\hangindent=4em
Group (XIII., XIV.), June 1. Marked by Arcturus, the brightest of the
northern stars, with the paler Spica south of the equator.

\hangindent=4em
Group (XV., XVI.), July 1. The Northern Crown and Hercules are the
most characteristic configurations.

\hangindent=4em
Group (XVII., XVIII.), Aug.~1. Vega is nearly overhead, and the red
Antares low down in the south, with Altair near the equator, just
east of Ophiuchus.

\hangindent=4em
Group (XIX., XX.), Sept.~1. Cygnus is in the zenith, and Sagittarius
low down, while the brightest part of the Milky Way lies athwart
the meridian.

\hangindent=4em
Group (XXI., XXII.), Oct.~1. A barren region, relieved only by the
bright star Fomalhaut of the Southern Fish near the southern
horizon.

\hangindent=4em
Group (XXIII., XXIV.), Nov.~1. This region also is rather barren,
though the ``great square'' of Pegasus is a notable configuration
of stars.
\par} %end \parindent=1em
\end{fineprint}
%% -----File: 461.png---Folio 450-------
%% -----File: 462.png---Folio 451-------

\nbarticle{793.} A thorough knowledge of these artificial groups, and of the
names and locations of the stars in them, is not at all essential, even
to an accomplished astronomer; but it is a matter of very great convenience
to know the principal constellations, and perhaps a hundred
of the brightest stars, well enough to be able to recognize them readily
and to use them as points of reference. This amount of knowledge
is easily acquired by three or four evenings' study of the sky in
connection with a good star-map or celestial globe, taking care to
observe on evenings at different seasons of the year, so as to command
the whole sky.

\begin{fineprint}
At present the best star-atlas for reference is probably that of Mr.\
Proctor. The maps of Argelander's ``Uranometria Nova'' and Heis's atlas
(both in German) are handsomer, and for some purposes more convenient.
There are many others, also, which are excellent. The smaller maps which
are found in the text-books on astronomy are not on a scale sufficiently large
to be of much scientific use (as, for instance, in the observation of meteors),
though they answer well enough the purpose for which they were designed,
of introducing the student to the principal star-groups.
\end{fineprint}

\nbarticle{794.} \nbparatext{Designation of Bright Stars.}---(\textit{a}) \textit{Names.} Some fifty or
sixty of the brighter stars have names of their own in common use.
A majority of the names belonging to stars of the first magnitude are
of Greek or Latin origin, and significant, as, for instance, Arcturus,
Sirius, Procyon, Regulus, etc. Some of the brightest stars, however,
have Arabic names, as Aldebaran, Vega, and \DPtypo{Betelguese}{Betelgueze}, and the
names of most of the smaller stars are Arabic, when they have names
at all.

(\textit{b}) \textit{Place in Constellation.} Spica is the star in the handful of
wheat carried by Virgo; Cynosure signifies the star at the end of the
Dog's Tail (in ancient times the constellation we now call Ursa
Minor seems to have been a dog); Capella is the goat which
Auriga, the charioteer, carries in his arms. Hipparchus, Ptolemy,
and, in fact, all the older astronomers, including Tycho Brahe, used
this clumsy method almost entirely in designating particular stars;
speaking, for instance, of the star in the ``head of Hercules,'' or in
the ``right knee of Boötes,'' and so on.

(\textit{c}) \textit{Constellation and Letters.} In 1603 Bayer, in publishing a new
star-map, adopted the excellent plan, ever since in vogue, of designating
the stars in the different constellations by the letters of the Greek
alphabet, assigned usually in order of brightness. Thus Aldebaran is
%% -----File: 463.png---Folio 452-------
$\alpha$~Tauri, the next brightest star in the constellation is $\beta$~Tauri, and
so on, as long as the Greek letters hold out; then the Roman letters
are used as long as they last; and finally, whenever it is found necessary,
we use the numbers which Flamsteed assigned a century later.
At present every naked-eye star can be referred to and identified by
some letter or number in the constellation to which it belongs.

(\textit{d}) \textit{Current Number in a Star-Catalogue.} Of course all the above
methods fail for the hundreds of thousands of smaller stars. In their
case it is usual to refer to them as number so-and-so of some well-known
star-catalogue; as, for instance, 22,500 Ll. (Lalande), or
2573 B.A.C. (British Association Catalogue). At present our various
star-catalogues contain from 600,000 to 800,000 stars, so that, except
in the Milky Way, almost any star visible in a telescope of two or
three inches' aperture can be identified and referred to by means of
some star-catalogue or other.

\textit{Synonyms.} Of course all the brighter stars which have \textit{names} have
also letters, and are sure to be included in every star-catalogue which
covers their part of the sky. A given star, therefore, has often a
large number of aliases, and in dealing with the smaller stars great
pains must be taken to avoid mistakes arising from this cause.

\section*{STAR-CATALOGUES.}

\sloppy
\nbarticle{795.} These are lists of stars arranged in regular order (at present
usually in order of right ascension), and giving the places of the stars
at some given epoch, either by means of their right ascensions and
declinations, or by their (celestial) latitudes and longitudes. The
so-called ``magnitude,'' or brightness of the star, is also ordinarily
indicated. The first of these star-catalogues was that of Hipparchus,
containing 1080 stars (all that are \textit{easily} visible and measurable by
naked-eye instruments), and giving their longitudes and latitudes for
the epoch of 125 \textsc{b.c.}

\fussy
\begin{fineprint}
This catalogue has been preserved for us by Ptolemy in the Almagest,
and from it he formed his own catalogue, reducing the positions of the
stars (\textit{i.e.}, correcting for precession the positions given by Hipparchus) to
his own epoch, about 150~\textsc{a.d.} The next of the old catalogues of any value
is that of Ulugh Beigh made at Samarcand about 1450~\textsc{a.d.} This appears
to have been formed from independent observations. It was followed in
1580 by the catalogue of Tycho Brahe containing 1005 stars, the last which
was constructed before the invention of the telescope.

The modern catalogues are numerous. Some give the places of a great
number of stars rather roughly, merely as a means of \textit{identifying} them when
%% -----File: 464.png---Folio 453-------
used for cometary observations or other similar purposes. To this class
belongs Argelander's \textit{Durchmusterung} of the northern heavens, which contains
over 324,000 stars,---the largest number in any one catalogue thus far
published. Then there are the ``\textit{catalogues of precision},'' like the Pulkowa
and Greenwich catalogues, which give the places of a few hundred stars as
accurately as possible in order to furnish ``fundamental stars,'' or reference
points in the sky. The so-called ``\textit{Zones}'' of Bessel, Argelander, and many
others, are catalogues covering limited portions of the heavens, containing
stars arranged in zones about a degree wide in declination, and running
some hours in right ascension. To the practical astronomer the most useful
catalogue is likely to be the one which is now in process of formation by the
co-operation of various observatories under the auspices of the Astronomische
Gesellschaft (an International Astronomical Society, with its headquarters
in Germany). This catalogue will contain accurate places of all stars above
the ninth magnitude in the northern sky. Most of the necessary observations
have already been made.
\end{fineprint}

\nbenlargepage
\sloppy
\nbarticle{796.} \nbparatext{Determination of Star-Places.}---The observations from which
a star-catalogue is constructed are usually made with the meridian
circle (\artref{Art.}{63}). For the catalogues of precision, comparatively few
stars are observed, but all with the utmost care and during several
years, taking all possible means to eliminate instrumental and observational
errors of every sort.

\fussy
In the more extensive catalogues most of the stars are observed only
once or twice, and everything is made to depend upon the accuracy
of the places of the fundamental stars, which are assumed as correct.
The instrument in this case is used only ``differentially'' to measure
the comparatively small differences between the right ascension and
declination of the fundamental stars and those of the stars to be catalogued.

\begin{includegraphicspage}{213}
\includegraphicsmid{illo213}{\textsc{Fig.~213.}---The Photographic Telescope of the Henry Brothers, Paris.}
\end{includegraphicspage}

\nbarticle{797.} \nbparatext{Method of using a Catalogue.}---The catalogue contains the
\textit{mean} right ascension and declination of its stars for the beginning
of some given year; \textit{i.e.}, the right ascension and declination the star
\textit{would have} at that time if there were no aberration of light and no
irregular motion in the celestial pole to affect the position of the
equator and equinox. To determine the actual \textit{apparent} right ascension
and declination of a star for a given date (which is what we
want in practice), the catalogue place must be ``\textit{reduced}'' to the
date in question; \textit{i.e.}, it must be corrected for precession, nutation,
and aberration.

\begin{fineprint}
The operation with modern tables and formulę is not a very tedious one,
involving perhaps five minutes' work, but without the catalogue places are
%% -----File: 465.png---Folio 454-------
%% -----File: 466.png---Folio 455-------
useless for most purposes. \textit{Vice versa}, the observations of a fixed star with
the meridian circle do not give its \textit{mean} right ascension and declination
ready to go into the catalogue, but the observations must be reduced from
\textit{apparent} place to \textit{mean} before they can be tabulated.
\end{fineprint}

\nbarticle{798.} \nbparatext{Star-Charts.}---For many purposes \textit{charts} of the stars are
more convenient than a catalogue, as, for instance, in searching for
new planets. The old-fashioned way of making such charts was by
plotting the results of zone observations. The modern way, introduced
within the last few years, is to do it by photography. The
plan decided upon at the Paris Astronomical Congress in 1887
contemplates the photographing of the whole sky upon glass plates
about six inches square, each covering an area of $2°$ square (four
square degrees), showing all stars down to the fourteenth magnitude,---a
project which is entirely feasible, and can be accomplished in
five or six years by the co-operation of about a dozen different observatories
in the northern and southern hemispheres. The instruments
are now (1888) in process of construction.

\begin{fineprint}
The figure (\figref{illo213}{Fig.~213}) is a representation of the Paris instrument of the
Henry Brothers, which was adopted as the typical instrument for the operation.
It has an aperture of about fourteen inches, and a length of about
eleven feet, the object-glass being specially corrected for the photographic
rays. A 9-inch visual telescope is enclosed in the same tube so that the observer
can watch the position of the instrument during the whole operation.

It was originally planned to give each plate 20~minutes' exposure, but
improvements in the photographic plates since the meeting of the Congress
now make it possible to cut down the time very materially. It will require
about 11,000 plates of the size named to cover the whole sky, and as each
star is to appear on two plates at least, the whole number of plates, allowing
for overlaps, will be about 22,000. As every plate will contain upon it a
number of well-determined catalogue stars, it will furnish the means of
determining accurately, whenever needed, the place of any other star which
appears upon the same plate.
\end{fineprint}


\section*{STAR MOTIONS.}

\nbarticle{799.} The stars are ordinarily called ``\textit{fixed},'' in distinction from
the planets or ``\textit{wanderers},'' because as compared with the sun and
moon and planets they have no evident motion, but keep their relative
relations and configurations unchanged. Observations made at sufficiently
wide intervals of time, and observations with the spectroscope,
show, however, that they are really moving, and that with velocities
which are comparable to the motion of the earth in her orbit.
%% -----File: 467.png---Folio 456-------

If we compare the right ascension and declination of a star
determined to-day with that determined a hundred years ago, it will
be found different. The difference is \textit{mainly} due to precession and
nutation, which are not motions of the stars at all, but simply
changes in the position of the reference circles used, and due to
alterations in the direction of the earth's axis (Arts.~\arnref{205} and \arnref{214}).
Aberration also comes in, and this also is not a real motion of the
stars, but only an apparent one.

\nbarticle{800.} \nbparatext{Proper Motions.}---But after allowing for all these \textit{apparent}
and \textit{common} motions, which depend upon the stars' places in the
sky, and are sensibly the same for all stars in the same telescopic
field of view, whatever may be their real distance from us, we find
that most of the larger stars have a ``\textit{proper motion}'' of their own,
(``proper'' as opposed to ``common,'') which displaces them slightly
with reference to the stars about them. There are only a few stars
for which this proper motion amounts to as much as $1''$ a year; perhaps
150 such stars are now known, but the number is constantly increasing,
as more and more of the smaller stars come to be accurately
observed.

\nbenlargepage
\sloppy
The maximum proper motion known is that of the seventh magnitude
star 1830 Groombridge (\textit{i.e.}, No.~1830 in Groombridge's catalogue of
circumpolar stars), which has an apparent drift of $7''$ annually,---enough
to carry it completely around the heavens in 185,000 years.
The largest known proper motions are the following:---
\begin{center}
\footnotesize
\begin{tabular}{ll@{ }l|ll@{ }l} \hline \hline
  1830, Groombridge,  & 7th mag.,   & $7''.0$
& $\epsilon$~Indi,    & 5th mag.,   & $4''.5$
\\
  9352, Lacaille,     & 7th \ \ ``  & $6''.9$
& Lalande 21,258      & 8th \ \ ``  & $4''.4$
\\
  32,416, Gould,      & 9th \ \ ``  & $6''.2$
& $o_2$ Eridani       & 6th \ \ ``  & $4''.1$
\\
  61 Cygni,           & 6th \ \ ``  & $5''.2$
& $\mu$~Cassiopeię,   & 5th \ \ ``  & $3''.8$
\\
  Lalande 21,185,     & 7th \ \ ``  & $4''.7$
& $\alpha$~Centauri,  & 1st \ \ ``  & $3''.7$
\\
\hline \hline
\end{tabular}
\end{center}

\fussy
The proper motions of Arcturus ($2''.1$), and of Sirius ($1''.2$),
are considered ``large,'' but are exceeded by a considerable number
of stars besides these given above. Since the time of Ptolemy,
Arcturus has moved more than a degree, and Sirius about half as
much. Those motions were first detected by Halley in 1718.

It is found, as might be expected, that the brighter stars, which as
a class are presumably nearer than the fainter ones, have on the
average a greater proper motion; on the \textit{average} only, however, as
is evident from the list given above. Many smaller stars have larger
proper motions than any bright one, for there are more of them.
%% -----File: 468.png---Folio 457-------

\nbarticle{801.} \nbparatext{Real Motions of Stars.}---The \textit{average} proper motion of the first-magnitude
stars appears to be about $\frac{1}{4}''$ annually, and that of a sixth-magnitude
star,---the smallest visible to the eye,---is about $\frac{1}{25}''$.

\includegraphicsouter{illo214}{\textsc{Fig.~214.}\\
Components of a Star's Proper Motion.}

The proper motion of a star gives comparatively little information
as to its real motion until we know the distance of the star and
the true direction of the motion,
since the proper motion \textit{as determined
from the star-catalogues} is
only the angular value of that part
or component of the star's whole
motion which is perpendicular to
the line of sight, as is clear from
the \figref{illo214}{figure}. When the star really moves from $A$ to $B$ (\figref{illo214}{Fig.~214}),
it will appear, as seen from the earth, to have moved from $A$ to $b$.
The angular value of $Ab$ as seen from the earth is the proper motion
(usually denoted by $\mu$), as determined from the comparison of
star-catalogues. Expressed in seconds of arc, we have
\[
  \mu'' = 206\,265 \left(\frac{Ab}{\text{distance}}\right).
\]
A body moving directly towards or from the earth has, therefore, no
(angular) proper motion at all,---none that can be obtained from the
comparison of star-catalogues.

Since $Ab$ in miles
\[
= \frac{\mu'' × \text{distance}}{206\,265},
\]
these motions cannot be translated into miles without a knowledge
of the star's distance; and this knowledge, as we shall see, is at
present exceedingly limited; nor can the true motion $AB$ be found
until we also know either the angle $BAE$ or else the line $Aa$.

\begin{fineprint}
But since $AB$ is necessarily \textit{greater} than $Ab$, it is possible in some cases
to determine a \textit{minor} limit of velocity, which must certainly be exceeded by
the star. In the case of 1830,~Groombridge, for instance, we have certain
knowledge that its distance is not \textit{less} than 2,000000 times the earth's
distance from the sun. It may be vastly greater; but it cannot be less.
Now at that distance the observed proper motion of $7''$ a year would correspond
to an actual velocity along the line $Ab$ of more than 200 miles a second,
and the star \textit{may} be moving many times more swiftly. This star has sometimes
been called the ``runaway star.''
\end{fineprint}
%% -----File: 469.png---Folio 458-------

\nbarticle{802.} \nbparatext{Motion in the Line of Sight.}---Although the comparison of
star-catalogues gives us no information of the body's motion towards
or from us along the line of sight, the velocity $Aa$ in the \figref{illo214}{figure}, yet
it is possible, when the star is reasonably bright, to determine somewhat
roughly its rate of approach or recession by means of the spectroscope.
If the star is approaching us, the lines in the spectroscope,
according to Doppler's principle (\artref{Art.}{821}, \textit{note}), will be shifted
towards the blue; and \textit{vice versa}, towards the red, if it is receding
from us. Dr.~Huggins was the first actually to use this method of
investigating the star movements in 1868, and by means of it he
arrived at some very interesting results, which, however, must be
admitted to be somewhat uncertain as regards their quantitative
value. He found, for instance, that Sirius was receding from us at
the rate of nineteen miles a second, and that Arcturus was rushing
towards us at the rate of nearly sixty miles a second; and results
of a similar character were found for a considerable number
of other stars.

Of late the investigations of this class have been carried on mainly
at the Greenwich Observatory, usually by comparing the stellar
spectra with hydrogen and sodium. The observations, however,
are extremely difficult to make, for the displacements of the lines
are very small, and in most star spectra the lines are broad and hazy
and not well adapted for accurate measurements. The results of different
days' observations, therefore, for a single star are sometimes
mournfully discrepant.

\includegraphicsouter{illo215}{\textsc{Fig.~215.}\\
Displacement of $H_{\gamma}$ Line in the Spectrum $\beta$~Orionis.}

\begin{fineprint}
Vogel has recently taken up the work \textit{photographically}
at Potsdam, with very encouraging
results. \figref{illo215}{Fig.~215} is from one of his
plates (a \textit{negative}), showing the displacement
towards the red of the $H_{\gamma}$ line in the spectrum
of $\beta$~Orionis, or \textit{Rigel}, and indicating
a \textit{recession} of the star at a very rapid rate.
Much is hoped in this line from the photographic
spectrum-work of the Draper Memorial at Cambridge, of which
more will be said a few \hyperref[pg:few]{pages} farther on.
\end{fineprint}

\sloppy
\nbarticle{803.} \nbparatext{Star-Groups.}---Star-atlases have been constructed by Proctor
and Flammarion, which show by arrows the direction and rate
of the angular proper motion of the stars as far as now known. A
moment's inspection shows that in many cases stars in the same
neighborhood have a proper motion nearly the same in direction and
in amount.
%% -----File: 470.png---Folio 459-------

\fussy
\begin{fineprint}
Thus, Flammarion has pointed out that the stars in the ``dipper'' of \textit{Ursa
Major} have such a community of motion, except $\alpha$ and $\eta$,---the brighter of
the pointers and the star in the end of the handle,---which are moving in
entirely different directions, and refuse to be counted as belonging to the
same group. \figref{illo216}{Fig.~216} shows the proper motions of the stars which compose
this group. The same thing appears when their motion is tested by the
spectroscope. Huggins found that the five associated stars are rapidly
receding from the earth, while $\alpha$ is approaching us, and $\eta$, though receding,
has a widely different rate of motion from the others.

\includegraphicsmid{illo216}{\textsc{Fig.~216.}---Common Proper Motions of Stars in the ``Dipper'' of Ursa Major.}

The brighter stars of the Pleiades are found in the same way to have a
common motion.
\end{fineprint}

In fact, it appears to be the rule rather than the exception that
stars apparently near each other are really connected as comrades,
travelling together in groups of twos and threes, dozens or hundreds.
They show, as Miss Clerke graphically expresses it, a distinctly
``\textit{gregarious} tendency.''

\nbarticle{804.} \nbparatext{The ``Sun's Way.''}---The proper motions of the stars are
due partly to their own real motion, and partly also to the motion of
our sun, which is moving swiftly through space, taking with it the
earth and the planets. Sir William Herschel was the first to investigate
and determine the direction of this motion a little more than 100
years ago. The principle involved is this: that the apparent motion
of each star is made up of its own motion combined with the motion
of the sun \textit{reversed} (\artref{Art.}{492}). The effect must be that \textit{on the whole},
the stars in that part of the sky towards which the sun is moving are
separating from each other,---the \textit{intervals between them widening out},---while
in the opposite part of the heavens \textit{they are closing up}; and
in the intermediate part of the sky the general drift must be \textit{backward}
with reference to the sun's (and earth's) real motion. Just as one
walking in a park filled with people moving indiscriminately in different
directions, would, on the whole, find that those in front of him
%% -----File: 471.png---Folio 460-------
appeared to grow larger,\footnote
  {Theoretically, of course, the stars towards which we are moving must appear
  to \textit{grow brighter} as well as to drift apart; but this change of brightness, though
  real, is entirely imperceptible within a human lifetime.}
and the spaces between them to open out,
while at the sides they would drift backwards, and in the rear close
up.

\begin{fineprint}
The spectroscope, moreover, ought to indicate this motion and undoubtedly
will do so when the apparent motion in the line of sight has been accurately
determined for a considerable number of stars. In fact one or two
attempts have already been made to determine the solar motion in this way;
in the quarter of the sky towards which the sun is moving, the star spectra
should, on the whole, show displacement of their lines indicating \textit{approach},
and \textit{vice versa} in the opposite quarter; and these observations will have the
advantage of showing directly the sun's rate of motion \textit{in miles}, a result
which is not given by investigations founded upon the angular proper
motions of the stars. As yet, however, this spectroscopic method has not
furnished results of any great weight.
\end{fineprint}

\nbarticle{805.} About twenty different determinations of the point in the sky
towards which this motion of the sun is directed have been worked
out by various astronomers, using in their discussions the angular
proper motions of from twenty to twenty-five hundred stars. All the
investigations  present a reasonable accordance of results, differing
from each other only by a few degrees, and show that the \textit{sun is now
moving towards a point in the constellation of Hercules, having a right
ascension of about $\mathit{267}°$ and a declination of about $+\mathit{31}°$}. This point
is known as the ``\textit{apex of the sun's way}.''

\includegraphicsouter{illo217}{\textsc{Fig.~217.}\\
The Earth's Motion in Space an affected
by the Sun's Drift.}

\nbarticle{806.} \nbparatext{Velocity of the Sun's Movement.}---This also is determined
by the discussion, and comes out to be such as would carry the sun
and its system about $5''$ in 100 years as seen from the average sixth
magnitude star (the sixth magnitude is the smallest easily visible to
the naked eye). If we knew with any certainty the distance of this
average sixth magnitude star we could translate this motion into
\textit{miles}; but at present this indispensable datum can be little more than
guessed at. On the reasonable assumption adopted by Ludwig Struve
(who has made the most recent and extensive of all the investigations
upon the motion of the solar system), that this distance is about
20,000000 times the astronomical unit, the velocity of the sun's
motion in space comes out about five units per year, that is, about
five-sixths of the earth's orbital velocity, or nearly sixteen miles per
second; but this result must be considered as still very uncertain.
%% -----File: 472.png---Folio 461-------

\begin{fineprint}
It is to be noted that this swift motion of the solar system, while of
course it affects the real motion of the planets \textit{in space}, converting them into
a sort of corkscrew spiral like the figure (\figref{illo217}{Fig.~217}), does not in the least
affect the \textit{relative} motion of sun and planets,
as some paradoxers have supposed it
must.
\end{fineprint}

\nbarticle{807.} \nbparatext{The Central Sun.}---We mention
this subject simply to say that
there is no real foundation for the belief
in the existence of such a body.
The idea that the motion of our sun
and of the other stars is a revolution
around some great central sun is a
very fascinating one to certain minds,
and one that has been frequently suggested.
It was seriously advocated
some fifty years ago by Mädler, who
placed this centre of the stellar universe
at Alcyone, the principal star
in the Pleiades.

It is certainly within bounds to
deny that any such motion has been demonstrated, and it is still less
probable that the star Alcyone is the centre of such a motion, if the
motion exists. So far as we can judge at present it is most likely that
the stars are moving, not in regular closed orbits around any centre
whatever, but rather as bees do in a swarm, each for itself, under
the action of the predominant attraction of its nearest neighbors.
The \textit{solar} system is an absolute monarchy with the sun supreme. The
great \textit{stellar} system appears to be a republic, without any such central,
unique, and dominant authority.

\section*{THE PARALLAX AND DISTANCE OF THE STARS.}

\nbarticle{808.} When we speak of the ``parallax'' of the moon, the sun, or
a planet, we always mean the \textit{diurnal} parallax, \textit{i.e.,  the angular
semi-diameter of the earth} as seen from the body in question. In
the case of the stars, this kind of parallax is hopelessly insensible,
never reaching an amount of $\frac{1}{20\,000}$ of a second of arc.

The expression ``parallax of a \textit{star}'' always means its \textit{annual} parallax,
that is, the semi-diameter of the earth's \textit{orbit} as seen from the
star. Even this in the case of all stars but a very few is a mere fraction
of a second of arc, too small to be measured.  In a few instances
%% -----File: 473.png---Folio 462-------
it rises to about half a second, and in the one case of our nearest
neighbor (so far as known at present), the star $\alpha$~Centauri, it appears
to be about $0''.9$, according to the earlier observers, or about $0''.75$,
according to the latest determination of Gill and Elkin. In \figref{illo218}{Fig.~218}
the angle at the star is the star's parallax.

\includegraphicsmid{illo218}{\textsc{Fig.~218.}---The Annual Parallax of a Star.}

In accordance with the principle of relative motion (\artref{Art.}{492}), every
star has, superposed upon its own motion and combined with it, an
\textit{apparent} motion equal to that of the earth but reversed. If the star is
really at rest it must seem to travel around each year in a little orbit
180,000000 miles in diameter, the precise counterpart of the earth's
orbit in size and form, and having its plane parallel to the ecliptic.

\begin{fineprint}
If the star is near the pole of the ecliptic this apparent ``parallactic'' orbit
will be viewed perpendicularly and appear as a circle; if the star is on the
ecliptic it will be seen edgewise as a short, straight line, while in intermediate
latitudes the parallactic orbit will appear as an ellipse. In this
respect it is just like the ``aberrational'' orbit of a star (\artref{Art.}{220}); but
while the aberrational orbit is of the same size for every star, having always
a semi-major axis of $20''.492$, the size of the parallactic orbit depends upon
the distance of the star. Moreover, in the parallactic orbit the star is
always opposite to the earth, while in the aberrational orbit it keeps just
$90°$ ahead of her.
\end{fineprint}

\nbarticle{809.} If we can find a way of measuring this parallactic orbit, the
star's distance is at once determined. It equals
\[
  \frac{206\,265 × R}{p''},
\]
in which $p''$ is the parallax in seconds of arc (the apparent semi-major
axis of the parallactic orbit), and $R$ is the earth's distance
from the sun.

The determination of stellar parallax had been attempted over and
over again from the days of Tycho down, but without success until
Bessel, in 1838, succeeded in demonstrating and measuring the parallax
of the star 61~Cygni; and the next year Henderson, of
the Cape of Good Hope, brought out that of $\alpha$~Centauri. It will
%% -----File: 474.png---Folio 463-------
be remembered that it was mainly on account of his failure to detect
stellar parallax that Tycho rejected the Copernican theory and substituted
his own (\artref{Art.}{504}). %[* is 'own' really bold?][F2: I don't think so.]

\begin{fineprint}
Roemer of Copenhagen, in 1690, thought that he had detected the effect of
stellar parallax in his observations of the difference of right ascension between
Sirius and Vega at different times of the year. A few years later, Horrebow,
his successor, from his own discussion of Roemer's observations, made out
the amount to be nearly four seconds of time or $1'$, and published his premature
exultation in a book entitled ``Copernicus Triumphans.'' The discovery
of \textit{aberration} by Bradley explained many abnormal results of the early
astronomers which had been thought to arise from stellar parallax, and
proved that the parallax must be extremely small. About the beginning
of the present century, Brinkley of Dublin and Pond, the Astronomer Royal,
had a lively controversy over their observations of $\alpha$~Lyrę (Vega). Brinkley
considered that his observations indicated a parallax of nearly $3''$. Pond,
on the other hand, from his observations deduced a minute \textit{negative} parallax,
which, as some one has expressed it, would put the star ``somewhere on the
other side of nowhere.'' In fact, as it turns out, Pond was nearer right than
Brinkley, the actual parallax as deduced from the latest observations being
only about $0''.2$. The negative parallax, like the much too large result of
Brinkley, simply indicates the uncertainties and errors incident to the
instruments and methods of observation then used. The periodical changes
of temperature and air pressure continually lead to fallacious results, except
under the most extreme precautions.
\end{fineprint}

\nbarticle{810.} \nbparatext{Methods of determining Parallax.}---The operation of measuring
a stellar parallax is, on the whole, the most delicate in the
whole range of practical astronomy. Two methods have been successfully
employed so far---the \textit{absolute} and the \textit{differential}.

(\textit{a}) The first method consists in making meridian observations of
the right ascension and declination of the star in question at different
seasons of the year, applying till known corrections for precession,
nutation, aberration, and proper motion, and then studying the
resulting star-places. If the star is without parallax, the places
should be identical after the corrections have been duly applied. If
it has parallax, the star will be found to change its right ascension
and declination systematically, though slightly, through the year. But
the changes of the seasons so disturb the constants of the instrument
that the method is treacherous and uncertain. There is no possibility
of getting rid of these temperature effects (in producing changes
of refraction and varying expansions of the instrument itself) by
merely multiplying observations and \textit{taking averages,} since the
%% -----File: 475.png---Folio 464-------
changes of temperature are themselves annually periodic, just as is
the parallax itself.

Still, in a few cases the method has proved successful. Different
observers at different places with different instruments have found
for a few stars fairly accordant results; as, for instance, in the case
of $\alpha$~Centauri, already mentioned as our nearest neighbor.

\nbarticle{811.} (\textit{b}) \nbparatext{The Differential Method.}---This consists in measuring
the change of position of the star whose parallax we are seeking
(which is supposed to be comparatively near to us), with reference to
other small stars, which are in the same telescopic field of view, but
are supposed to be so far beyond the principal star as to have no
sensible parallax of their own. If the comparison stars are near
the large one (say within two or three minutes of arc), the ordinary
wire micrometer answers very well for the necessary measures; but
if they are farther away, the heliometer (\artref{Art.}{677}) represents special
and very great advantages. It was with this instrument that Bessel,
in the case of 61~Cygni, obtained the first success in this line of
research.

The great advantage of the differential method is that it avoids
entirely the difficulties which arise from the uncertainties as to the
exact amount of precession, etc.; and in great measure, though not
entirely, those arising from the effect of the seasons upon refraction
and the condition of the instruments. On the other hand, however,
it gives as the final result, not the absolute parallax of the star, but
only the \textit{difference between its parallax and that of the comparison
star}. If the work is accurate the parallax deduced \textit{cannot be too
great; but it may be sensibly too small}, and so may make the star
apparently too remote. This is because the parallax of the comparison
star can never be quite zero: if the comparison star happens to have
a parallax of its own as large as that of the principal star, there
will be no relative parallax at all; if larger, the parallax sought will
come out \textit{negative}.

\sloppy
\nbarticle{812.} \nbparatext{Determination of Parallax by Photography.}---Recently it
has been attempted to press photography into the service, and
Professor Pritchard has obtained apparently excellent results from an
extensive series of photographs of 61~Cygni and its neighboring stars,
made at Oxford during 1886.

\fussy
\begin{fineprint}
Now and then a plate was found in which the sensitive film appeared to
have slipped a little during the development of the picture, but the measurements
%% -----File: 476.png---Folio 465-------
at once showed up the faulty plates, so that they could be confidently
rejected. The measurements made upon the other plates appeared
to be just as trustworthy as measurements made upon the actual objects in
the sky; and of course the measurement of a photographic plate at leisure
in one's laboratory is a vastly more comfortable operation than that of
making micrometric settings by night.

\nbarticle{813.} \nbparatext{Selection of Stars.}---It is important to select for investigations
of this kind these stars which may reasonably be supposed to be near,
and, therefore, to have a sensible parallax. The most important indication
of proximity is a \textit{large proper motion}, and \textit{brightness} is, of course, confirmatory.
At the same time, while it is probable that a bright star with
large proper motion is comparatively near, it is not certain. The small stars
are so much more numerous than the large ones that it will be nothing surprising
if we should find among them one or more neighbors nearer than
$\alpha$~Centauri itself.
\end{fineprint}
%*this paragraph is perhaps a 'quote' as it appears to be intermediate in point size between the adjacent paragraphs..][F2: It's a quote.]

\nbarticle{814.} \nbparatext{Unit of Stellar Distance.}---\textit{The Light-Year}. The ordinary
``\textit{astronomical unit},'' or distance of the sun from the earth, is not
sufficiently large to be convenient in expressing the distances of the
stars. It is found more satisfactory to take us a unit the distance
that light travels in a year, which is about 63,000 times the distance
of the earth from the sun. A star with a parallax of $1''$ is at a distance
of 3.262 ``\textit{light-years},'' so that the distance of any star in
``light-years'' is expressed by the formula
\[
D_{y} = \frac{3.262}{p''}.
\]

\nbarticle{815.} \hyperref[app:IV]{Table IV.}\ in the Appendix, based upon that given in Houzeau's
``Vade Mecum of Astronomy,'' but brought down to date by some
additions and changes, gives the parallaxes, and the distances in
light-years, of these stars whose parallaxes may be considered as
now fairly determined.

\begin{fineprint}
The student will, of course, see that the tabulated distance in the case of
a remote star is liable to an enormous percentage of error. Considering the
amount of discordance between the results of different observers, it is
extremely charitable to assume that any of the parallaxes are certain
within $\frac{1}{50}$ of a second of arc: but in the case of a star like the pole-star,
which appears to have a parallax of less than $0''.08$, this $\frac{1}{50}$ of a second
is $\frac{1}{4}$ of the whole amount; so that the distance of that star is uncertain by
at least twenty-five per cent. ($\frac{1}{50}$ of a second is the angle subtended
by $\frac{1}{16}$ of an inch at the distance of ten miles.)
%% -----File: 477.png---Folio 466-------

A vigorous campaign has lately been organized for the purpose of obtaining
within a reasonable time the parallaxes of a considerable number of
stars (perhaps one or two hundred),---enough to enable us to deduce some
general laws by statistical methods. The instruments to be used are heliometers
of six or seven inches' aperture, one of which is at the Cape of
Good Hope, another has just been erected at Bamberg in Germany, and a
third is at New Haven in this country, under the charge of Dr.~Elkin.

As regards the distance of stars, the parallax of which has not yet been
measured, very little can be said with certainty. It is \textit{probable} that the
remoter ones are so far away that light in making its journey occupies a
thousand and perhaps many thousand years.

\nbarticle{815*.} Since the above was written Dr.~Elkin has published the result
of his observations upon ten first-magnitude stars, as follows: $\alpha$~Tauri
(\textit{Aldebaran}), $0''.116 \pm .029$; $\alpha$~Aurigę (\textit{Capella}), $0''.107 \pm .047$; $\alpha$~Orionis
(\textit{Betelgueze}), $-0''.009 \pm .049$; $\alpha$~Canis Minoris (\textit{Procyon}), $0''.266 \pm .047$; $\beta$~Geminorum
(\textit{Pollux}), $0''.068 \pm .047$; $\alpha$~Leonis (\textit{Regulus}), $0''.093 \pm .048$; $\alpha$~Bootis
(\textit{Arcturus}), $0''.018 \pm .022$; $\alpha$~Lyrę (Vega), $0''.034 \pm .045$; $\alpha$~Aquillę (\textit{Altair}),
$0''.199 \pm .047$; $\alpha$~Cygni (\textit{Deneb}), $-0''.042 \pm .047$.

Of course the two \textit{negative} results simply indicate that the parallax of the
large star was less than that of the comparison stars employed. The very
small results for Vega and Arcturus are also rather surprising. See \hyperref[app:IV]{Table IV.}\ of Appendix.
\end{fineprint}
\chelabel{CHAPTERXIX}
%% -----File: 478.png---Folio 467-------

\Chapter{XX}{The Light of the Stars}
\nbchapterhang{\stretchyspace
THE LIGHT OF THE STARS.---STAR MAGNITUDES AND PHOTOMETRY.---VARIABLE
STARS.---STELLAR SPECTRA.\hspace{0pt}---SCINTILLATION
OF STARS.}

\nbarticle{816.} \nbparatext{Star Magnitudes.}---The term ``magnitude,'' as applied to a
star, refers simply to its brightness. It has nothing to do with its
apparent angular diameter. Hipparchus and Ptolemy arbitrarily
graded the visible stars, according to their brightness, into six classes,
the stars of the sixth magnitude being the smallest visible to the
eye, while the first class comprises about twenty of the brightest.
There is no assignable reason why \textit{six} classes should have been
constituted, rather than eight or ten.

After the invention of the telescope the same system was extended
to the smaller stars, but without any general agreement or concert,
so that the magnitudes assigned by different observers to telescopic
stars vary enormously. Sir William Herschel, especially, used very
high numbers: his twentieth magnitude being about the same as the
fourteenth on the scale now generally used, which more nearly
corresponds with that of the elder Struve.

\nbarticle{817.} \nbparatext{Fractional Magnitudes.}---Of course, the stars classed together
under one magnitude are not exactly alike in brightness, but
shade from the brighter to the fainter, so that exactness requires the
use of \textit{fractional} magnitudes. It is now usual to employ decimals
giving the brightness of a star to the nearest tenth of a magnitude.
Thus, a star of 4.3 magnitude is a shade brighter than one of 4.4,
and so on.

\begin{fineprint}
A peculiar notation was employed by Ptolemy, and used by Argelander
in his ``Uranometria\footnote
  {The term ``Uranometria'' has come to mean a catalogue of \textit{naked-eye stars};
  like the catalogues of Hipparchus, Ptolemy, and Ulugh Beigh.}
Nova.'' It recognizes \textit{thirds} of a magnitude as the
smallest subdivision. Thus,\enskip2,\enskip2,3,\enskip3,2,\enskip and\enskip3\enskip express the gradations
between second and third magnitude, 2,3 being applied to a star whose
brightness is a little inferior to the second, and 3,2 to one a little brighter
than the third magnitude.
%% -----File: 479.png---Folio 468-------

\nbarticle{818.} \nbparatext{Stars Visible to the Naked Eye.}---Heis enumerates the
stars clearly visible to the naked eye in the part of the sky north of
$35°$ south declination, as follows:---
\begin{center}
\begin{tabularx}{\textwidth}{l@{ }c@{ }X@{ }r | l@{ }c@{ }X@{ }r}
\hline\hline
  1st & magnitude, & \dotfill &   14\rule{0pt}{3ex}
& 4th & magnitude, & \dotfill &  313
\\
  2d  &  ``        & \dotfill &   48
& 5th &  ``        & \dotfill &  854
\\
  3d  &  ``        & \dotfill &  152
& 6th &  ``        & \dotfill & 2010
\\
  & \multicolumn{6}{c}{Total \dotfill\ 3391} &
\\[1ex]
\hline\hline
\end{tabularx}
\end{center}

According to Newcomb, the number of stars of each magnitude is such
that united they would give, roughly speaking, somewhere nearly the same
amount of light as that received from the aggregate of those of the next
brighter magnitude. But the relation is very far from exact, and seems to
fail entirely for the fainter magnitudes below the tenth or eleventh, the
smaller stars being less numerous than they should be. In fact, if the law
held out perfectly, and if light was transmitted through space without loss,
the whole sky would be a blaze of light like the surface of the sun.
\end{fineprint}

\sloppy
\nbarticle{819.} \nbparatext{Light-Ratio and Absolute Scale of Star Magnitudes.}---It
was found by Sir John Herschel, about fifty years ago, that the light
given by the average star of the first magnitude is just about one
hundred times as great as that received from one of the sixth, and
that a corresponding ratio has been pretty nearly maintained throughout
the scale of magnitudes, the star's of each magnitude being about
$2 \frac{1}{2}$ times ($\root 5\of{100}$) brighter than those of the next inferior magnitude.
The number which expresses the ratio of the light of a star to that of
another one magnitude fainter is called the \textit{light-ratio}.

\fussy
In the star magnitudes of the maps by Argelander, Heis, and others,
which are most used at present, the divergence from a strict uniformity
of light-ratio is, however, sometimes serious. Some forty years ago
it was proposed by Pogson to reform the system, by adopting a
scale with the uniform light-ratio of $\root 5\of{100}$, adjusting the first six
magnitudes to correspond as nearly as possible with the magnitudes
hitherto assigned by leading authorities, and then carrying forward
the scale indefinitely among the telescopic stars. Until recently this
``\textit{absolute scale} of magnitude,'' as it has been called, has not been
much used; but in the New Uranometrias lately made at Cambridge
and Oxford it has been adopted, and astronomers generally now
endeavor to conform to it.

\nbarticle{820.} \nbparatext{Relative Brightness of Different Star Magnitudes.}---In this
scale the light-ratio between successive magnitudes is made exactly
%% -----File: 480.png---Folio 469-------
$\root 5\of{100}$, \textit{or the number whose logarithm is} 0.4000, viz., 2.512. Its reciprocal
is the number whose logarithm is 9.6000, viz., 0.3981. If $b_{1}$ is
the brightness of a standard first-magnitude star, expressed either in
candle-power or other convenient unit, and $b_{n}$ be the brightness of a
star of the $n$th magnitude on this scale, we shall therefore have
\[
\log b_n = \log b_1 - \frac{4}{10} (n -1);
\]
$(n - 1)$ in this equation being the number of magnitudes \textit{between} the
star of the first magnitude and the star of the $n$th magnitude: \textit{i.e.},
for a star of the sixth magnitude $(n - 1)$ is 5; so that for a star of
the sixth magnitude, the equation reads,
\[
\log b_6 = \log b_1 - \frac{4}{10} × 5 = \log b_1 - 2.
\]
With this light-ratio, every difference of five magnitudes corresponds
to a multiplication or division of the star's light by 100; \textit{i.e.}, to make
one star as bright as the standard star of the first magnitude it would
require 100 of the sixth, 10,000 of the eleventh, 1,000000 of the sixteenth,
and 100,000000 of the twenty-first magnitude.

As nearly standard stars of the first magnitude on this scale we
have $\alpha$~Aquilę and Aldebaran ($\alpha$~Tauri). The other stars usually
counted as of first magnitude are some of them sensibly brighter, and
others fainter than these. The pole-star and the two ``pointers'' are
very nearly standard stars of the \textit{second} magnitude.

\sloppy
\begin{fineprint}
\nbarticle{821.} \nbparatext{Negative Magnitudes.}---According to this scale, stars that are
one magnitude \textit{brighter} than those of the standard first would be of the \textit{zero}
magnitude, (as is the case with Arcturus), and those that are brighter yet
would be of a \textit{negative} magnitude; \textit{e.g.}, the magnitude of Sirius is $-1.13$;
and Jupiter at opposition, in conformity to this system, is described as a
star of nearly $-2$d magnitude, which means that it is nearly $2.51^3$, or about
16 times \textit{brighter than} a star of the $+1$st magnitude like Aldebaran. According
to Seidel, Jupiter at opposition is about $8\frac{1}{4}$ times as bright as Vega, which
would make its ``magnitude'' $-2.09$, Vega being of magnitude 0.2.
\end{fineprint}

\fussy
\nbarticle{822.} \nbparatext{Relation of Size of Telescope to the Magnitude of the
Smallest Star Visible with it.}---If a telescope just shows a star
of a given magnitude, then to show stars one magnitude smaller
we require an instrument having its aperture larger in the ratio of
$\sqrt{2.512}$ (or $\root 10\of{100}$) to 1: \textit{i.e.} as 1.59:1. Every \textit{tenfold} increase
in the diameter of the object-glass will therefore carry the power of
vision just \textit{five magnitudes lower.}
%% -----File: 481.png---Folio 470-------

\begin{fineprint}
Assuming what seems to be very nearly true for normal eyes and good
telescopes, that the \textit{minimum visible} for a one-inch aperture is a star of the
ninth magnitude, we obtain the following little table of \textit{apertures required to
show stars of a given magnitude}.
\medskip
\begin{center}
\footnotesize
\begin{tabular}{@{}l|c|c|c|c|c|c}
\hline \hline
Star Magnitude \ldots\dotfill
&  7  &  8  &  9  &  10  &  11  &  12\rule{0pt}{3ex}  \\
Aperture \dotfill
& $0^\text{in}.40$ & $0^\text{in}.63$ & $1^\text{in}.00$
& $1^\text{in}.59$ & $2^\text{in}.51$ & $3^\text{in}.98$  \\[1ex]
\hline
Star Magnitude \ldots\dotfill
&  13  &  14  &  15  &  16  &  17  &  18\rule{0pt}{3ex} \\
Aperture \dotfill
&  $6^\text{in}.31$ & $10^\text{in}.00$ & $15^\text{in}.90$
& $25^\text{in}.10$ & $39^\text{in}.80$ & $63^\text{in}.10$  \\[1ex]
\hline \hline
\end{tabular}
\end{center}

\medskip
But on account of the increased thickness necessary in the lenses of large
telescopes, they never quite equal their theoretical capacity as compared with
smaller ones.

The smallest stars visible by the Lick telescope (thirty-six inches aperture),
after allowing for all the advantage of the site, are not quite a whole magnitude
below the smallest visible with the Washington telescope; but the \textit{number}
visible will be at least double; since the smaller stars are vastly the more
numerous.
\end{fineprint}

\nbarticle{823.} \nbparatext{Measurement of Star Magnitudes and Brightness.}---Until
recently all such measurements were mere eye-estimates, and even
yet all photometric measurements depend \textit{ultimately} on the judgment
of the eye. But it is possible by the help of instruments to aid this
judgment very much by limiting the point to be decided, to the
question whether two lights as seen are, or are not, exactly equal,
or else making the decision depend on the visibility or non-visibility
of some appearance.

\nbarticle{824.} \nbparatext{Method of Sequences.}---For some purposes the unassisted
eye is quite as good as any photometric instrument. It judges
directly with great precision of the \textit{order of brightness} in which a
number of objects stand. In the method of ``sequences,'' as it is
called, the observer merely arranges the stars he is comparing, say
to the number of fifty or so, in the order of their brightness, taking
care that the stars in each sequence list are nearly at the same
altitude, and seen under equally favorable circumstances. Then he
makes a second sequence, taking care to include in it some of the
stars that were in the first: and so on. Finally, from the whole set
of sequences, a list can be formed, including all the stars contained in
any of them, arranged in the order of brightness. This process gives,
however, no determination of the light-ratio, nor of the number of
times by which the light of the brightest exceeds that of the faintest.
%% -----File: 482.png---Folio 471-------

\begin{fineprint}
Variable stars are still often observed in this way, the stars with which
they are compared, being such as have their magnitudes already well determined.
\end{fineprint}

\nbarticle{825.} \DPtypo{2. }{}\nbparatext{Instrumental Methods.}---These are based on two different
principles:---

\textit{a}. The measurement is made by causing the star to \textit{disappear} by
diminishing its light in some measurable way. This is usually referred
to as the ``method of \textit{extinctions}.''

\textit{b}. The measurement is effected by causing the light of the star to
appear just \textit{equal} to some other standard light, by decreasing the
brightness of the star or of the standard in some known ratio until
they are perfectly equalized.

\begin{fineprint}
Under the first head come the photometers which act upon the principle
of ``\textit{limiting apertures}.'' The telescope is fitted with some arrangement,
often a so-called ``cat's-eye,'' by which the available aperture of the object-glass
can be diminished at will, and the observation consists in determining
with what area of object-glass the star is just visible. The method is embarrassed
by constant errors from the fact that the greater thickness of the
glass in the middle of the lens comes into account, and, still worse, from the
fact that the image of the star becomes large and diffuse on account of
diffraction when the aperture is very much reduced.
\end{fineprint}

\nbarticle{826.} \nbparatext{The Wedge Photometer.}---The method of producing the
``extinction'' by a ``\textit{wedge}'' of dark, neutral-tinted glass is much
better. The wedge is usually five or six inches long, by perhaps a
quarter of an inch wide, and at the thick end cuts off light enough to
extinguish the brightest stars that are to be observed. In the
Pritchard form of the instrument this wedge is placed close to the
eye at the eye-hole of the eye-piece; in some other forms it is placed
at the principal focus of the object-glass, where micrometer wires
would be put.

In observation the wedge is pushed along promptly until the star
just disappears, and a graduation on the edge of the slider is read.

\begin{fineprint}
The great simplicity of the instrument commends it, and if the wedge is
a good one of really neutral glass (which is not easy to get), the results are
remarkably accurate. But the observations are very trying to the eyes on
account of the straining to keep in sight an object just as it is becoming
invisible. The constant of the wedge must be carefully determined in the
laboratory, \textit{i.e.}, what length of the wedge corresponds to a diminution
of the light of a star by just one magnitude (cutting off 0.602 of its light).
It is convenient to have the slider graduated into inches or millimeters on
%% -----File: 483.png---Folio 472-------
the one edge and magnitudes on the other. The ``Uranometria Nova Oxoniensis''
is a catalogue of the magnitudes of the naked-eye stars to the number
of 2784, between the pole and $10°$ south declination, observed with an
instrument of this kind by Professor Pritchard, and published in 1885.
\end{fineprint}

\sloppy
\nbarticle{827.} \nbparatext{Polarization Photometers.}---The instruments, however, with
which most of the accurate photometric work upon the stars has been
done, are such as compare the light of the star with some standard
by means of an ``\textit{equalizing apparatus}'' based on the application of
the principles of double refraction and polarization.

\fussy
\begin{fineprint}
The light of either the observed star or the comparison star (real or
artificial) is polarized by transmission through a Nicol prism, or else both
pencils are sent through a double refracting prism. The images are viewed
with a Nicol prism in the eye-piece; and by turning this the polarized image
or images can be varied in brightness at pleasure, and the amount of variation
determined by reading a small circle attached to it. In the photometers
of Seidel and Zöllner, who observed comparatively few objects, but very
accurately, the artificial star with which the real stars were compared was
formed by light from a petroleum lamp, shining through a small aperture,
and reflected to the eye by a plate of glass in the telescope tube. Professor
Pickering, in his extensive work embodied in the ``Harvard Photometry''
(published in 1884, and giving the magnitudes of 4260 stars) used the
pole-star as the standard, bringing it by an ingenious arrangement into the
same field with the star observed.
\end{fineprint}

Photometric observations in many cases require large and somewhat
uncertain corrections, especially for the absorption of light by
the atmosphere at different altitudes, and the final results of different
observers naturally fail of absolute accordance. Still the agreement
between the two catalogues of Pickering and Pritchard is remarkably
close, generally within one or two tenths of a magnitude.

\includegraphicsmid{illo219}{\textsc{Fig.~219.}---Pickering's Meridian Photometer.}

\begin{fineprint}
\nbarticle{828.} \nbparatext{The Meridian Photometer.}---This instrument, contrived and
used by Professor Pickering in the observations of the Harvard Photometry,
consists of a telescope with two object-glasses side by side. The telescope is
pointed nearly east and west, and in front of each object-glass is placed a
silvered glass mirror ($M_{1}$ and $M_{2}$, \figref{illo219}{Fig.~219}) at an angle of $45°$. One of the
mirrors is so set as to bring the rays of the pole-star to one object-glass; the
other mirror is capable of being turned around the optical axis of the telescope,
in such a way as to command a star at any part of the meridian, and bring its
light into the other object-glass. At the eye-end is placed, first (\textit{i.e.}, next the
object-glass), a double-image prism $D$, which separates any pencil of light
falling upon it into two, polarized at right angles to each other. The ``ordinary''
rays come through nearly undeflected, but the ``extraordinary'' are
%% -----File: 484.png---Folio 473-------
bent out of their course, as indicated in the \figref{illo219}{figure}, where the pencil $A$, coming
from object-glass No.~1, is divided into two pencils $a_{o}$ and $a_{e}$, and in the
same way the pencil $B$, from the second objective, is divided into $b_{o}$ and $b_{e}$.
The angle of the double-image prism $D$ is so chosen that $a_{o}$ and $b_{e}$ will be
nearly parallel to each other, and a suitable diaphragm\footnote
  {The diaphragm $E$ may be replaced by an eye-stop at $I$.}
$E$ cuts off the two
other pencils $a_{e}$ and $b_{0}$. A Nicol prism $N$ receives the two pencils that come
through the diaphragm, and the eye views the two images through the eye-piece
at $I$. These pencils, being polarized at right angles to each other, will
vary in their brightness when the Nicol is turned, one of them becoming
brighter and the other fainter; and \textit{four} positions of the Nicol can be found
at which the images will appear equal in brightness, whatever may be the
original ratio of brightness between the pole-star and the object observed.
On looking into the instrument the observer sees two stars, the pole-star at
rest, the other moving along as in a transit instrument. He simply turns
the Nicol until the images are equalized, setting the Nicol at all the four different
positions which will produce the effect, and reading the graduated
circle $C$. The whole operation consumes not more than a minute, with the
help of an assistant to record the numbers as read off. The ``Harvard Photometry''
(usually referred to simply as ``H.~P.'') was made by means of an
instrument with object-glasses only two inches and a half in diameter. An
instrument with four-inch lenses is now at work in Cambridge, measuring
the magnitudes of all the nearly 80,000 stars of Argelander's \textit{Durchmusterung},
which are of the eighth magnitude or brighter.

\nbarticle{829.} \nbparatext{Photometry by means of Photography.}---It has been found that,
excepting a few strongly colored stars, the intensity, or more simply the size, of
the image of a star formed upon a photographic plate may be used as a measure
of its brightness as compared with other stars taken on the same plate,
or on similar plates similarly exposed. The comparison becomes easier and
more accurate if the photographic telescope is not made to follow the stars
exactly, but is allowed to lag a little so that the star forms a ``trail.'' It will,
therefore, be possible to use the plates of the great photographic star campaign
to determine star magnitudes as well as positions. But, as has been intimated,
there are some anomalies; certain stars, for instance, that are hardly visible to
%% -----File: 485.png---Folio 474-------
the naked eye, photograph as bright stars, and there are others---red stars---that
are abnormally faint on the plate. The exceptions are numerous
enough to make it necessary to use photographic magnitudes with caution.
\end{fineprint}

\nbarticle{830.} \nbparatext{Star Colors and their Effects on Photometry.}---The stars
differ considerably in color. The majority are of a very pure white,
like Sirius and the sun, but there are not a few of a yellowish hue,
like Capella, or reddish, like Arcturus and Antares; and there are
some, mostly small stars, which are as red as garnets and rubies.
We also have, associated with larger ones in double-star systems,
numerous small stars which are strongly green or blue; and there
are a few large isolated stars, which, like Vega, are of a decidedly
bluish tinge.

\begin{fineprint}
These differences of color embarrass photometric measurements made by
either of the methods described, because it is impossible to make a red star
look identical with a blue one by any mere increase or diminution of brightness,
and because different observers will differ in setting the wedge of an
extinction photometer according to the color of the star. Some eyes are
abnormally sensitive to blue light, some to red. To the writer, for instance,
Vega is decidedly superior to Arcturus, while the majority of observers see
the difference as decidedly the other way.

\nbarticle{831.} \nbparatext{Spectrum Photometry.}---The only completely satisfactory and
scientific method would be to compare the spectra of the stars with some
standard spectrum, \textit{say that of the pole-star}, dividing the spectrum into a considerable
number of portions, and determining and recording the amount
of light in each portion of the spectrum as compared with homologous parts
of the standard spectrum. This, of course, would immensely increase the
work of comparing the brightness of the stars; but it is quite feasible to do
it for a few hundred of the brighter ones, and it would be well worth accomplishment.
If we ever succeed in getting photographic plates equally sensitive
to rays of all wave length, photography would answer the purpose well.
\end{fineprint}

\sloppy
\nbarticle{832.} \nbparatext{Starlight compared with Sunlight.}---The light \textit{received} from
a first-magnitude star like Vega is about $\frac{1}{40000\,000000}$ (one forty
thousand millionth) of that from the sun, according to the determinations
of Zöllner and others. The measurement is not easy, and
must be taken as having a very considerable margin of error.

\fussy
Sirius is nearly equivalent to six of Vega, its light being about
$\frac{1}{7000\,000000}$ of the sun's.

\sloppy
Since the light of a sixth-magnitude star is only $\frac{1}{100}$ of that of a
standard first-magnitude, it follows that it would require 4,000000000000
of stars of the 6th magnitude to give us sunlight.
%% -----File: 486.png---Folio 475-------

\fussy
\nbarticle{833.} \nbparatext{Total Light of the Stars.}---Assuming what is roughly, though
not exactly, true, that Argelander's magnitudes follow the standard
scale, it appears that the 324,000 stars north of the equator enumerated
in his \textit{Durchmusterung} give a light about equal to that of 240 first-magnitude
stars; but it is noticeable that the aggregate amount of light
given by the stars in each of the fainter magnitudes increases rapidly.

\begin{fineprint}
The following is the estimate, substantially according to Newcomb:---\\[1ex]
\begin{tabular}{@{}r@{ }c@{ }c@{ }l@{ }l@{ }r@{ }c@{ }c@{ }r@{.}l@{ }c@{ }l@{}}
    10 & stars& \multicolumn{4}{r@{ }}{(above the  2d} & magnitude) &=  &  6&0  & first-magnitude & stars.
\\
    37 &  ``& from& 2d & to & 3d  & `` & = &   7&3  & `` & ``  \\
   122 &  ``& `` & 3d  & to & 4th & `` & = &   9&6  & `` & ``  \\
   310 & `` & `` & 4th & to & 5th & `` & = &   9&8  & `` & ``  \\
  1016 & `` & `` & 5th & to & 6th & `` & = &  12&7  & `` & ``  \\
  4322 & `` & `` & 6th & to & 7th & `` & = &  21&6  & `` & ``  \\
 13593 & `` & `` & 7th & to & 8th & `` & = &  27&1  & `` & ``  \\
 57960 & `` & `` & 8th & to & 9th & `` & = &  46&   & `` & ``  \\
247544 & `` & `` & 9th & to &$9 \frac{1}{2}$ & `` & = & 100&   & `` & ``  \\[1ex]
\cline{9-10}
  &\multicolumn{6}{c}{Total \dotfill} & = & 240&&& \rule{0pt}{3ex}
\end{tabular}

How much to add for the still smaller magnitudes is very uncertain. Beyond
the tenth magnitude the number of small stars does not increase proportionately
fast, so that if we could carry on the account of stars to the
twentieth magnitude, it is practically certain that we should not find the total
light of the aggregate stars of each succeeding magnitude increasing at
any such rate as from the seventh to the tenth. Perhaps it would be a not
unreasonable estimate to put the total starlight of the northern hemisphere
as equivalent to about 1500 first-magnitude stars, or that of the whole sphere
at 3000. This would make the total starlight on a clear night about $\frac{1}{60}$ of
the light of the full moon, and about $\frac{1}{27\,000000}$ that of the sun. The light
from the stars which are visible to the naked eye would not be as much
as $\frac{1}{25}$ of the whole. But the above estimate of the light received from the
extremely small stars is hardly more than a mere guess, and may hereafter
receive important corrections.
\end{fineprint}

\nbarticle{834.} \nbparatext{Heat from the Stars.}---Attempts have been made to measure
by a sensitive thermopile the heat received from certain stars.
Both Huggins and Stone (about 1870) thought they had obtained
sensible indications of heat from Arcturus and Vega; but their results
have not since been confirmed; and unless the radiation of invisible
energy by these stars is much greater in comparison with their light
than is the case with the sun, it is almost certain that there must be
some illusion in the matter. $\frac{1}{40000\,000000}$ of the sun's \textit{heat} could
hardly be shown by any instrument known to science, and there is no
%% -----File: 487.png---Folio 476-------
present reason to suppose that the total \textit{heat} received from the stars
bears a larger ratio to that received from the sun than star\textit{light} does
to sun\textit{light}.

\nbarticle{835.} \nbparatext{Amount of Light emitted by Certain Stars.}---This, of course,
is something vastly different from that \textit{received} by us. A star may
emit hundreds of times as much light as the sun, and yet, if the star
is remote enough, the amount that reaches the earth will be only an
excessively minute fraction of sunlight. If $l$ be the amount of light
that we \textit{receive} from a star, expressed in terms of sunlight at the earth,
then the total amount of light \textit{emitted}, $L$, is given by the simple equation,
\[
L = l × D^2,
\]
$D$ being the distance of the star in astronomical units, while $L$ is
expressed in terms of the sun's light emission.

Turning to the table of stellar parallax (Appendix, \hyperref[app:IV]{Table~IV.}),
we find that, according to Gill \& Elkin, $D$ for Sirius equals 542,000;
\begin{flalign*}
&\text{whence, for Sirius, } \quad
  L = \frac{542000^2}{7000\,000000} = 42.0;  &&
\end{flalign*}
that is, the light emitted by Sirius is more than forty times as much
as that emitted by the sun.

\begin{fineprint}
If we use the value of the parallax of this star as determined by Abbe,
namely, $p'' = 0''.273$, we shall get $L = 68$, while Gyldén's value of the parallax
$(0''.193)$ gives $L = 155$. In either case, however, it is clear that Sirius emits
vastly more light than the sun.

Similarly for the pole-star $(p = 0''.06)$, $L = 93$; for Vega $(p = 0''.14)$, $L =
69$; $\alpha$~Centauri $(p = 0''.75)$, $L = 1.9$; 70~Ophiuchi $(p = 0''.16)$, $L = 1$;
61~Cygni $(p = 0''.43)$, $L = \frac{1}{15}$; 24,258 Ll.\ $(p = 0''.26)$, $L = \frac{1}{113}$.\footnote
  {In making this calculation the magnitudes of the Harvard Photometry were
  used.}
\end{fineprint}

The companion of Sirius is a little star of the ninth magnitude,
which forms a double-star system with Sirius itself. The light emitted
by this companion does not exceed $\frac{1}{12\,000}$ that of Sirius.

\nbarticle{836.} \nbparatext{Causes of Differences of Brightness in Stars.}---It used to be
thought that the stars were all very much alike in magnitude and constitution;
%% -----File: 488.png---Folio 477-------
not, indeed, without considerable differences, but us much
resembling each other as do individuals of the same race. It is now
quite certain that this is not the case, as is obvious from the short list
of actual light emissions just given.

If the stars \textit{were} all alike, all the differences of apparent brightness
would be traceable simply to differences of \textit{distance}; but as the facts
are, we have to admit other causes to be equally effective. The differences
of brightness are due, \textit{first}, to difference of \textit{distance}; \textit{second},
to difference of \textit{dimensions}, or of light-giving area; \textit{third}, to
difference in the \textit{brilliance of the light-giving surface}, depending upon
difference of temperature and constitution. There are stars near and
remote, large and small, intensely incandescent and barely glowing
with incipient or failing light.

As Bessel puts it, there is no reason why there may not be ``as
many \textit{dark} stars as bright ones.'' As we shall soon see, the companion
of Sirius, though only giving about $\frac{1}{12\,000}$ part as much light as
Sirius itself, is at least $\frac{1}{10}$ part as heavy; so that, \textit{mass for mass}, it
cannot be $\frac{1}{1000}$ part as luminous.

When we compare stars by the thousand, we can say of the tenth-magnitude
stars, for instance, as compared with the fifth, that \textit{as a
class} they are \textit{more remote}; and also, just as truly, that \textit{their average
diameters are smaller}, and also that \textit{their surfaces are less brilliant};
but we must be careful not to make any assertions of this sort regarding
any one star of the tenth magnitude compared with a particular
individual of the fifth, unless we have some absolute knowledge of
their relative distances. The faint star may be the larger of the two,
or the hotter, or the nearer. We must know something beyond their
relative ``magnitudes'' before it is possible to settle such questions.

\nbarticle{837.} \nbparatext{Real Diameter of Stars.}---We have no knowledge whatever
as to the real diameter of any star. As to the apparent \textit{angular} diameter,
we can only say negatively that it is insensible, in no case being
known to reach $0''.01$. If there be a star of the same diameter as our
sun, at such a distance that its parallax equals one second, its apparent
diameter must be $\dfrac{1924''}{206265} \vphantom{\fbox{$\dfrac{0'}{0}$}}$. (The sun's mean angular diameter is
$1924''$ (\artref{Art.}{276}).) This equals $0''.0093$---a quantity far too small
to be reached by any direct measurement, especially since, even in the
Lick telescope, the ``spurious'' disc of a star has a diameter of nearly
$\frac{1}{7}''$, and in smaller telescopes is much larger (about $0''.4$ in a ten-inch
telescope).
%% -----File: 489.png---Folio 478-------

\begin{fineprint}
There is a theoretical connection between the diameter of the diffraction
rings seen around the image of a star in the telescope, and the real (as
opposed to the spurious) diameter of the image; by comparing, therefore,
the actual size of the rings with the size they should have if the star were an
absolute optical point, we might hope to get a determination of the star's
diameter. But no such attempts have succeeded, and at present no way
seems open for obtaining the desired measurement. In some cases, as we
shall see later, the \textit{mass} of a star as compared with that of the sun can be
found; but not its \textit{volume} or its \textit{density}, since these require a knowledge of its
diameter.
\end{fineprint}

\section*{VARIABLE STARS.}

\nbarticle{838.} A close examination shows that many stars change their
brightness. As a general rule, these which do this, and are called
``\textit{variable},'' are reddish in their color; while comparatively few of
the white stars belong to this class. The variable stars may be
classified\footnote
  {This classification is substantially that of Professor Pickering, slightly
  modified, however, by Houzeau.}
as follows:---

\hangitem{I} Cases of slow continuous change.

\hangitem{II} Irregular fluctuations of light: alternately brightening and
darkening without any apparent law.

\hangitem{III} Temporary stars, which blaze out suddenly and then disappear.

\hangitem{IV} Periodic stars of the type of $o$ Ceti, usually of long period.

\hangitem{V} Periodic stars of short period, of the type of $\beta$~Lyrę.

\hangitem{VI} Periodic stars in which the variation is like that which would
be the result of an eclipse by some intervening body---the
Algol type.

\nbarticle{839.} I\@. \textsc{Gradual Changes}. On the whole, the changes in the
brightness of the stars since the time of Hipparchus and Ptolemy
have been surprisingly small. There has been no general increase
or decrease in the brightness of the stars as a whole; and there are
few cases where any individual star has altered its brightness by a
half or even a quarter of a magnitude. The \textit{general appearance} of
the sky is the same as it was 2000 years ago; so that notwithstanding
all the effect of proper motions in the meantime and the whole
amount of the variation that has taken place in the brightness of the
stars, there is no doubt that if either of these old astronomers were to
come to life he would immediately recognize the familiar constellations.

There are a few instances, however, in which it is almost certain
that change has taken place and is going on. In the time of Eratosthenes
%% -----File: 490.png---Folio 479-------
the star in the ``claw of the Scorpion'' (now $\beta$~Librę) was
reckoned the brightest in the constellation. At present, it is a whole
magnitude below Antares, which is now much superior to any star in
the vicinity. So when the two stars Castor and Pollux in the constellation
Gemini were lettered by Bayer, the former, $\alpha$, was brighter than
Pollux, which was lettered $\beta$; but $\beta$ is now notably the brighter of
the two. Taking the whole heavens, we find a considerable number
of such cases; perhaps a dozen or more.

\nbarticle{840.} \nbparatext{Missing Stars.}---It is a common belief that since accurate
star-catalogues began to be made, many stars have disappeared and
not a few new ones have come into existence. While it would not
do to deny absolutely that anything of the kind has ever happened,
it is certainly unsafe to assert that it has.

\begin{fineprint}
There are a considerable number of cases where stars are now missing
from the older catalogues \textit{as published},---nearly, if not quite, a hundred,---but
in almost every instance examination of the original observations shows
that the place as printed was a mistake of some sort which can now be traced,---sometimes
a mistake of the recorder, sometimes in the reduction of the
observation, and sometimes of the press. In a few cases the star observed
was a planet (Uranus, Neptune, or an asteroid); and in some cases the
missing star may have been a ``temporary star,'' as, for instance, 55~Herculis,
which was observed by the elder Herschel. So many of the missing stars
are now satisfactorily explained that it is natural to suppose that the few
cases remaining are of the same sort.

There, is no known instance of a \textit{new} star appearing and remaining permanently
visible.
\end{fineprint}

\nbarticle{841.} II\@. \textsc{Stars  that exhibit Irregular Fluctuations in
Brightness}. The most conspicuous example of this class is the
strange star $\eta$~Argus (not visible in the United States). This star
ranges all the way from the \textit{zero} magnitude (in 1843, when, according
to Sir John Herschel, it was brighter than every star but Sirius) down
to the seventh magnitude, which is its present brightness and has been
ever since 1865.

\includegraphicsmid{illo220}{\textsc{Fig.~220.}---Light Curve of $\eta$~Argus according to Loomis.}

\begin{fineprint}
Between 1677 (when it was observed by Halley as of the fourth magnitude)
and 1800, it oscillated in brightness, so far as we can judge from
the few observations extant, between the fourth and second magnitudes.
About 1810, it rose rapidly in brightness, and between 1826 and 1850 it was
never below the standard first magnitude. When brightest, in 1843, it was
giving more than 25,000 times as much light as in 1865. A singular fact is
that the star is in the midst of a nebula which apparently sympathizes with
%% -----File: 491.png---Folio 480-------
it to some extent in its fluctuations. (There are other instances of connection
of nebulę with variable or temporary stars, as will appear later on.)
Fig.~220 represents the ``light-curve'' of this object from 1800 to 1870,
according to Loomis; who, however, imagines the star to be \textit{periodic}, with
a period of about seventy years. If so, it ought to be again increasing in
brightness by this time.
\end{fineprint}

$\alpha$~Orionis and $\alpha$~Cassiopeię behave in a somewhat similar manner,
only the whole range of variation in their brightness is less than a
single magnitude. The catalogue of variable stars shows a considerable
number of other similar cases.

\nbarticle{842.} III\@. \textsc{Temporary Stars}. There are eleven well authenticated
cases in which new stars have appeared \textit{temporarily},---that is,
for a few weeks or months,---blazing up suddenly and then gradually
fading away. The list is as follows:---

\begin{fineprint}

\hangitemb{1. \phantom{1}134~\textsc{b.c.}}The star of Hipparchus.

\hangitemb{2. \phantom{1}329 \textsc{a.d.}}A star in Aquilla.

\hangitemb{3. 1572 \textsc{a.d.}}Tycho's star in Cassiopeię.

\hangitemb{4. 1600 \textsc{a.d.}}\textit{P} Cygni, 3d magnitude, observed by Jansen.

\hangitemb{5. 1604 \textsc{a.d.}}Kepler's star in Ophiuchus.

\hangitemb{6. 1670 \textsc{a.d.}}11 Vulpeculę, 3d magnitude, observed by Anthelum.

\hangitemb{7. 1848 \textsc{a.d.}}A star of the 5th magnitude, observed by Hind---also in Ophiuchus.

\hangitemb{8. 1860 \textsc{a.d.}}\textit{T} Scorpii, 7th magnitude, in star cluster M~80, observed by Auwers.

\hangitemb{9. 1866 \textsc{a.d.}}\textit{T} Coronę--Borealis, 2d magnitude.

\hangitemb{10. 1876 \textsc{a.d.}}Nova Cygni, 2d magnitude.

\hangitemb{11. 1885 \textsc{a.d.}}A star in the nebula of Andromeda, 6th magnitude.
%% -----File: 492.png---Folio 481-------

As regards the first of these stars, we know almost nothing. Hipparchus
has left no record of its position or brightness; but the Chinese annals mention
a star as appearing in Scorpio at just that date, and probably the same
object; though the Chinese observations \textit{may} refer to a comet. The appearance
of this new star led Hipparchus to form his catalogue of stars.

As to the second on the list, we know hardly more; the records do not
even make it absolutely certain that the object was not a comet, not being
explicit on the point of its motion.
\end{fineprint}

\nbarticle{843.} The third is justly famous. When it was first seen by Tycho
in November, 1572, it was already brighter than Jupiter, having
probably appeared a few days previously. It very soon became as
bright as Venus herself, being even visible by day. Within a week
or two it began to fail, but continued visible to the naked eye for
fully sixteen months before it finally disappeared. It is not certain
whether it still exists or not as a telescopic star: Tycho determined
its position with as much accuracy as was possible to his instruments;
and there are a number of small stars, any one of which is so near to
Tycho's place that it might be the real object.

\begin{fineprint}
There has been an entirely unfounded notion that this star may have
been identical with the ``Star of Bethlehem,'' it being imagined that the
star is \textit{periodically} variable, with a period of 314 years. If so, it might have
been expected to reappear in 1886, and it was so expected by certain persons
``as a sign of the second coming of the Lord.'' It is difficult to see how the
idea came to be so generally prevalent, as it certainly has been. Probably
every astronomer of any note has received hundreds of letters on the subject.
At Greenwich a printed circular was prepared and sent out as a reply to
such inquiries.
\end{fineprint}

The fifth star, observed by Kepler, was nearly, though not quite,
as bright as that of Tycho, and lasted longer---fully two years. It
also has disappeared so that it cannot now be identified.

\nbarticle{844.} The ninth star excited much interest. It blazed out between
the 10th and 12th of May as a star of the second magnitude,
remained at its maximum for three or four days, and then, in five or
six weeks, faded away to its original faintness, for it now is, and was
before the outburst, a nine and one-half magnitude star on Argelander's
catalogue, with nothing noticeable to distinguish it from its
neighbors. While at the maximum its spectrum was carefully studied
by Huggins, and exhibited brightly and strongly the bright lines of
hydrogen, just as if it were a sun like our own, but entirely covered
with outbursting ``prominences'' of incandescent hydrogen.
%% -----File: 493.png---Folio 482-------

The tenth star had a very similar history. It also rose to its full
brightness (second magnitude) on November~24, within \textit{four} hours
according to Schmidt, remained at a maximum for only a day or two,
and faded away to invisibility within a month. But it still exists
as a very minute telescopic star of the fifteenth magnitude. It
was also spectroscopically studied by several observers (by Vogel
especially) with the strange result that the spectrum, which at the
maximum was nearly continuous, though marked by the bright lines
of hydrogen and by bands of other unknown substances, lost more
and more of this continuous character, until at last it became a simple
spectrum of three bright lines \textit{like that of a nebula}.

\nbarticle{845.} The eleventh and last of these temporary stars was very peculiar
in one respect; not in its brightness, for it never exceeded the six
and one-half magnitude, but because it appeared right in the midst
of the great nebula of Andromeda, only $12''$ or $13''$ distant from the
nucleus. It came out suddenly like all the others, and faded away
gradually in about six months, to absolute extinction so far as any
existing telescope can show. It showed under photometric measurements
many fluctuations in brightness, not losing its light smoothly
and regularly but in a rather paroxysmal manner. Its spectrum, even
when brightest, was simply continuous without anything more than the
merest trace of bright lines in it. The eighth star on the list resembled
it in the fact that it appeared in the midst of a star cluster.

\nbarticle{846.} IV\@. \textsc{Variables of the $o$~Ceti Type}. These objects almost
without exception behave like the temporary stars in remaining generally
faint, suddenly brightening up for a short time, and then fading
back to the original condition; but they do it \textit{periodically}. The
periods are generally of considerable length, from six months to two
years; but they are very apt to be considerably irregular, not
unfrequently to the extent of several weeks.

The star $o$ Ceti (often called \textit{Mira}, that is, ``the Wonderful'') may
be taken as the type of this class. Its variability was discovered by
Fabricius in 1596. During most of the time it remains simply a faint
twelfth-magnitude star, but once in about eleven months it runs up to
the fourth, third, or even the second magnitude, and then back again,
occupying usually about 100 days in the rise and fall. Its brightness
increases more rapidly than it fails, and it remains at its maximum
for a week or ten days. At the maximum its spectrum is very beautiful,
containing a large number of intensely bright lines which, however,
%% -----File: 494.png---Folio 483-------
are not yet certainly identified. Its light-curve is $A$, in \figref{illo221}{Fig.~221}.
A large proportion of the known variables belong to this
class, and many suppose that the temporary stars also belong to it,
differing from their classmates only in the length of their periods.

\nbarticle{847.} V\@. \textsc{Variables of the Type} $\beta$~\textsc{Lyrę}. In these the main
characteristic is that there are two or more maxima and minima in
each period, as if we were dealing with several superposed causes of
variation. The light-curve of $\beta$~Lyrę is given by $B$, \figref{illo221}{Fig.~221}. Its
period is about thirteen days.

\nbarticle{848.} VI\@. \textsc{Variables of the Algol Type}. The sixth and last
class consists of stars which seem to suffer a partial eclipse at short
intervals, their light-curves ($C$, \figref{illo221}{Fig.~221}) being the reverse of the
$o$~Ceti type. Of this type of stars, Algol, or $\beta$~Persei, may be taken
as the most conspicuous representative. Its period is $2^\text{d}\: 20^\text{h}\: 48^\text{m}\: 55^\text{s}.4$,
which is subject to almost no variation, except certain slow changes
that appear to be the result of some unknown disturbance. During
most of the time the star remains of the second magnitude. At the
time of obscuration it loses about five-sixths of its light, falling to
the fourth magnitude in about four and one-half hours, remaining at
the minimum for about twenty minutes, and then in three and one-half
hours recovering its original condition. During the minimum
the spectrum undergoes no considerable change, though there are
suspicions of some slight variations in its lines.

\includegraphicsmid{illo221}{\textsc{Fig.~221.}---Light Curves of Variable Stars.}
%% -----File: 495.png---Folio 484-------

Only nine stars are so far known belonging to this class, and among
them are these of the shortest period. One in Cepheus, discovered by
Ceraski in 1880, has a period of two days and eleven hours; another
in Ophiuchus of only twenty hours.

\nbarticle{849.} \nbparatext{Explanation of Variability.}---Evidently no simple explanation
will hold for all the different classes. For the gradually progressive
changes no explanation need be looked for; on the contrary,
it is surprising that such changes are no greater than they are, for
the stars are all growing older.

As for the irregular changes, no sure account can yet be given of
them. Where the range of variation is small, as it is in most cases,
one thinks of spots on the surface like these of our own sun, (but
much more extensive and numerous) and running through a period
just as our sun spots do. Let a star with such spots upon it revolve
on its own axis, as of course it must do, and in the combination we
have at least a possible explanation of a great proportion of all the
known cases, both the irregular variables and the regularly periodic.

\includegraphicsouter{illo222}{\textsc{Fig.~222.}---The Collision Theory of Variable Stars.}

\nbarticle{850.} \nbparatext{Collision Theory.}---For the temporary stars, and those of
the $o$~Ceti type, Mr.~Lockyer has recently (in connection with a
much more extended subject) suggested a collision theory, illustrated
by \figref{illo222}{Fig.~222}. The fundamental
idea that the
phenomena of the temporary
stars may be
due to collisions is not
new. Newton long ago
brought it out, and to
some extent discussed
it; but considering the
probable diameters of
the stars as compared
with the distances between
them, it seems
impossible that collisions
could have been
frequent enough to account
for the number of
temporary stars actually observed. Mr.~Lockyer, however, imagines
that the temporary stars, and also variable stars of the $o$~Ceti class,
%% -----File: 496.png---Folio 485-------
are, in their present stage of development, not compact bodies, but
only pretty dense swarm of meteorites of considerable extent, each
such swarm being accompanied by another smaller one revolving
around it in an eccentric orbit, just as comets revolve around the
sun, or as the components of double stars revolve around each other.
He supposes that the perihelion distance is so small that the swarms
interpenetrate and pass through each other at the perihelion, which
could happen without disturbing the \textit{general} motion of either of the
two meteoric flocks; but while they are thus passing, the collisions
are immensely increased in number and violence, with a corresponding
increase in the evolution of light. There are many good points about
this ingenious theory, but also serious objections to it---as, for instance,
the great irregularity of the periods of stars of this class, an irregularity
which seems hardly consistent with such an orbital revolution.

\nbarticle{851.} \nbparatext{Stellar Eclipses.}---As to the Algol type, the natural explanation
is by means of an eclipse of some sort. The interposition of a
more or less opaque object between the observer and the star,---a
dark companion revolving around it,---would produce just the effect
observed. If, however, this is really the case, the mass of Algol must
be absolutely enormous compared with that of our sun in order to
produce so swift a revolution in the eclipsing body. But Ceraski's
variable of this type in Cepheus is very refractory and exhibits
changes of period and other phenomena that are extremely difficult
to reconcile with the idea that \textit{its} obscuration is due to an eclipse.

\nbarticle{852.} \nbparatext{Number and Designation of Variables.}---The ``Catalogue
of Variable Stars,'' of Mr.\ S.~C.\ Chandler, published in 1888, contains
225 objects. One hundred and sixty are distinctly \textit{periodic}
stars: in 12 cases the periodicity is perhaps uncertain, while 14 are
certainly \textit{irregular}.

\begin{fineprint}
The remainder includes the temporary stars and some twenty-five or
thirty stars in respect to the variations of which very little is yet known.
\hyperref[app:VI]{Table~VI.}\ in the Appendix presents the principal data for the naked-eye
variables which are visible in the United States.

When a star is discovered to be variable which previously had no special
appellation of its own, it is customary to designate it by one of the last
letters in the alphabet, beginning with R. Thus R~Sagittarii is the first
discovered variable in Sagittarius; S~Sagittarii is the second; T~Sagittarii,
the third, and so on.
\end{fineprint}

\nbarticle{853.} \nbparatext{Range of Variation.}---In many cases the whole range is
only a fraction of a magnitude (especially among the more newly
%% -----File: 497.png---Folio 486-------
discovered variables), but in a great number it extends from four
to eight magnitudes, the maximum brightness exceeding the minimum
by from fifty to a thousand times; and in a few cases the
range is greater yet. Not unfrequently considerable changes of
color accompany the changes of brightness; the star as a rule being
whiter at its maximum, and frequently showing bright lines in its
spectrum.

\nbarticle{854.} \nbparatext{Method of Observation.}---There is no better way than that
of comparing the star by the eye, or with the help of an opera-glass,
with surrounding stars of about the same brightness at the time when
its light is near the maximum or minimum; noting to which of them
it is just equal at the moment, and also these which are a shade
brighter or fainter.

\begin{fineprint}
It is possible for an amateur to do really valuable work in this way, by
putting himself in relation with some observatory which is interested in the
subject. The observations themselves require so much time that it is difficult
for the working force in a regular observatory to attend to the matter
properly, and outside assistance is heartily welcomed in gathering the needed
facts. The observations themselves are not specially difficult, require no
very great labor or mathematical skill in their reduction, and, as has been
said, can be made without instruments; but they require patience, assiduity,
and a keen eye.
\end{fineprint}

\section*{STAR SPECTRA.}

\nbarticle{855.} In 1824 Fraunhofer, in connection with his study of the
lines of the solar spectrum, investigated also the spectra of certain
stars, using an apparatus essentially similar to that which is now
employed at Cambridge. He placed a prism in front of the object-glass
of a small telescope and looked at the stars through this, using
a cylindrical lens in the eye-piece to widen the spectrum, which otherwise
would be a mere line.

\begin{fineprint}
He found that Sirius, Castor, and many other stars show very few dark
lines in their spectrum, but strong ones; while, on the other hand, the spectra
of Pollux ($\beta$~Geminorum) and Capella resemble closely the spectrum of the
sun. In all the spectra he recognized the $D$~line, although it was not then
known that it had anything to do with sodium.
\end{fineprint}

\sloppy
\nbarticle{856.} \nbparatext{Observations of Huggins and Secchi.}---Almost as soon as
the spectroscope had taken its place as a recognized instrument of
science it was applied by Huggins to the study of the stars, and
%% -----File: 498.png---Folio 487-------
Secchi followed hard in his footsteps. The former studied the spectra
of comparatively few stars, but with all the dispersive power he
could obtain, and in detail; while Secchi, using a much less powerful
instrument, examined several thousand star spectra, in a more
general way, for purposes of classification.

\fussy
Huggins identified with considerable certainty in the spectra of
$\alpha$~Orionis (Betelgueze) and $\alpha$~Tauri (Aldebaran) a number of elements
that are familiar on the earth, and are most of them prominent in the
solar spectrum. He found in the former sodium, magnesium, calcium,
iron, bismuth, and hydrogen; and in $\alpha$~Tauri he reported in
addition, tellurium, antimony, and mercury; but these latter metals
have not yet been verified.

\nbarticle{857.} \nbparatext{Classification of Stellar Spectra.}---Secchi, in his spectroscopic
survey, found that the 4000 stars which he observed could all
be reduced to four classes.

\includegraphicsmid{illo223}{\textsc{Fig.~223.}---Secchi's Types of Stellar Spectra.}

\textit{The first class comprises the white or blue stars.} To it belong Sirius
and Vega, and, in fact, considerably more than half of all the stars
examined. The spectrum is characterized by the great strength of
the hydrogen lines, which are wide, hazy bands, much like the $H$ and
$K$~lines in the solar spectrum. Other lines are extremely faint or
entirely absent; the $K$~line especially, which in the solar spectrum is
especially prominent, in the spectra of most of these stars is hardly
visible.

\textit{The second class} is also numerous, and \textit{is composed of stars with a
spectrum substantially like that of our sun.} The $H$ and $K$~lines are
both strong. Capella and Pollux ($\beta$~Geminorum) are prominent
examples of this class. There are certain stars which form a connecting
link between these two first classes, stars like Procyon and $\alpha$~Aquilę,
which, while they show the hydrogen lines very strongly, also
exhibit a great number of other lines between them. The first and
second classes together embrace fully seven-eighths of all the stars
he observed.

\textit{The third class} includes most \textit{of the red and variable stars}, some 500
in number, and the spectrum is characterized by dark \textit{bands} instead
of lines (though lines are generally present also). These bands,
which are probably due to carbon, shade from the blue towards the
red; that is, they are sharply defined and darkest at the more refrangible
edge. Occasionally in spectra of this type some of the hydrogen
lines are bright. $\alpha$~\DPtypo{Hercules}{Herculis},
$\alpha$~Orionis, and \textit{Mira} ($o$~Ceti) are fine
examples of this third class.
%% -----File: 499.png---Folio 488-------

\textit{The fourth class} is composed of a very small number of stars, less
than sixty so far as now known, mostly small red stars. This spectrum
is also a banded one; but compared with the third class the
bands are \textit{reversed}, that is, faced the other way, and shaded towards
the blue. These generally show also a number of bright lines.
None of the conspicuous stars belong to this class---none above the
fifth magnitude. The sixth magnitude star, 152~Schjellerup, may
be taken as its finest example ($\alpha$, $12^\text{h}\: 39^\text{m}$; $\delta$, $+40°\: 9'$, in the constellation
of Canes Venatici). \figref{illo223}{Fig.~223} exhibits the light-curves of
these four types of spectrum.\footnote
  {It is difficult to represent spectra accurately by any process of engraving that
  can be readily reproduced in a book like the present. The curve, on the other
  hand, is easily managed, and, though it does not please the eye like the spectrum
  itself, it is capable of conveying all the information that could be obtained from
  the most finished engraving. \textit{Dark lines} are represented by lines running \textit{downward}
  from the upper boundary line of the curve, and \textit{bright lines} by lines running
  \textit{upward}, while the bands and their shading are represented by variations in the
  contour of the curve.}

\nbarticle{858.} Vogel has revised Secchi's classification of spectra as follows,
making only three main classes, but with subdivisions:
%% -----File: 500.png---Folio 489-------

\hangitemc{I. (\textit{a})} Same as Secchi's I. The white stars.

\hangitemc{(\textit{b})} Nearly continuous; all lines wanting or very faint.
$\beta$~Orionis is the type.

\hangitemc{(\textit{c})} Showing the lines of hydrogen bright, and also the
helium line $D_3$ (\artref{Art.}{323}).

\hangitemc{II. (\textit{a})} Same as Secchi's II.

\hangitemc{(\textit{b})} Like II. (\textit{a}), but showing in addition bright lines which
are \textit{not} the lines of hydrogen or helium. (This is rare.)

\hangitemc{III. (\textit{a})} Same as Secchi's III.

\hangitemc{(\textit{b})} Same as Secchi's IV.

\begin{fineprint}
Vogel's classification is based in part on the very doubtful assumption
that stars of Class~I. are hottest and also youngest, while the other classes
belong to stars which are either beginning to fail or are already far gone in
decrepitude. But it is very far from certain that a red star is not just as
likely to be younger than a white one, as to be older. It probably is now at
a \textit{lower temperature}, and possesses a more extensive envelope of gases; but
it may be increasing in temperature as well as decreasing. At any rate we
have no certain knowledge about its age.

\sloppy
\nbarticle{859.} \nblabel{pg:few}\nbparatext{Photography of Stellar Spectra.}---As early as 1863 Huggins
attempted to photograph the spectrum of Vega, and succeeded in getting an
impression of the spectrum, but without any of the lines. In 1872 Dr.~Henry
Draper of New York, working with the reflector which he had himself
constructed, succeeded in getting an impression of the spectrum of the
same star, showing for the first time four of its hydrogen lines. The introduction
of the more sensitive dry plates in 1876, induced Mr.~Huggins to
resume the subject (as did Dr.~Draper soon after), and they soon succeeded in
getting pictures showing many lines. The spectra were about half an inch
long by $\frac{1}{12}$ or $\frac{1}{16}$ of an inch wide. After the lamented death of Dr.~Draper
in 1882, Professor Pickering took up the work at Cambridge (U.\,S.); and with
such success that Mrs.~Draper, who had intended to establish and to endow
her husband's observatory as an establishment for astro-physical research,
and a most fitting monument to his memory, concluded to transfer the
instruments to Cambridge, and there establish the ``Draper Memorial.''
\end{fineprint}

\nbarticle{860.} \nbparatext{The Slitless Spectroscope.}---Professor Pickering has attained
his remarkable success by reverting to the ``slitless spectroscope,''
arranged in the manner first used by Fraunhofer, and later revived
by Secchi. The instrument consists of a telescope with the \textit{objective
corrected not for the visual, but for the photographic rays}, equatorially
mounted and carrying \textit{in front of the object-glass} one or more prisms,
with a refracting angle of nearly $10°$, and large enough to cover the
whole lens.
%% -----File: 501.png---Folio 490-------

\begin{fineprint}
The refracting edge of the prism is placed east and west, so that the
linear spectrum of a star formed on a plate at the focus of the object-glass
runs north and south. If, now, the clock-work of the instrument is adjusted
to follow the star exactly, the image (\textit{i.e.}, the spectrum) will be a mere line,
broken here and there where the dark lines of the spectrum should appear.
By merely retarding or accelerating the clock a trifle, the linear spectrum
will drift a little sidewise upon the plate, and so will form a spectrum having
a width depending on the amount of this drift during the time of exposure.
If the air is calm the lines of the spectrum thus formed are as clean and
sharp as if a slit were used; otherwise not.
\end{fineprint}

\includegraphicsouter{illo224}{\textsc{Fig.~224.}\\
Arrangement of the Prisms in the Slitless Spectroscope.}

\nbarticle{861.} The most powerful instrument used in this work at Cambridge
is Dr.~Draper's eleven-inch photographic refractor, with four huge
glass prisms in a box in front of the object-glass, arranged as indicated
in \figref{illo224}{Fig.~224}. With
this apparatus, photographic
spectra of the brighter stars
are now obtained having,
before enlargement, a length
of fully three inches from
$F$ in the blue of the spectrum
to the extremity of
the ultra violet. It is a
pity, of course, that the
lower portions of the spectrum
below $F$ cannot be
reached in the same way;
but no plates sufficiently
sensitive to green, yellow, and red rays have yet been found. The
exposure necessary to obtain the impression of even the most powerful
photographic rays is from half an hour to an hour. \figref{illo225}{Fig.~225} is
enlarged about one-third from one of these photographs of the
spectrum of Vega, which extends far into the ultra violet.

\includegraphicsmid{illo225}{\textsc{Fig.~225.}---Photographic Spectrum of Vega. Cambridge, 1887.}

These spectra bear tenfold enlargement perfectly, making them
more than two feet long by two inches in width, and then in the spectrum
%% -----File: 502.png---Folio 491-------
of such a star as Capella they show hundreds of lines. It is
simply amazing that the feeble, twinkling light of a star can be made
to produce such an autographic record of the substance and condition
of the inconceivably distant luminary.

\nbarticle{862.} \nbparatext{Peculiar Advantages of the Slitless Spectroscope.}---The
slitless spectroscope has three great advantages. First, that it
utilizes all the light that comes from the star to the object-glass,
much of which in the usual form of the instrument is lost in the
jaws of the slit: Secondly, that by taking advantage of the length
of a large telescope, it produces a very high dispersion with even a
single prism: Thirdly, and most important, it gives on the same
plate and with a single exposure the spectra of all the many stars
whose images fall upon it. With the smaller eight-inch instrument
made at Cambridge, and one prism, as many as 100 or 150 spectra
are sometimes taken together; as, for instance, in a spectrum photograph
of the Pleiades.

\nbarticle{863.} \nbparatext{Disadvantages of the Slitless Spectroscope.}---\textit{Per contra},
the giving up of the slit precludes all the usual methods of identifying
the lines by actually confronting them with comparison spectra;
the comparison prism (\artref{Art.}{315}) cannot be used. This makes it
extremely difficult to utilize these magnificent pictures for purposes
of scientific measurement.

\begin{fineprint}
If it turns out that any of the lines photographed in the spectrum are of
\textit{atmospheric} origin, the difficulty will be largely removed, as these atmospheric
lines could be used as reference points. Professor Pickering has tried, by
the interposition of various vapors in the path of the light within the
telescope tube, to get lines that will answer the purpose, but thus far
without any really satisfactory results. Until this difficulty is overcome it
will be impossible to make these spectra yield us all of the information which
they undoubtedly contain with respect to the motions of the stars in the line
of sight.

Vogel in his photographic work mentioned in connection with this subject
(\artref{Art.}{802}) used an ordinary spectroscope with a slit. This arrangement
required a very long exposure, and limited the dispersion it was possible
to employ; but it permitted him to use a hydrogen Geissler tube placed
within the telescope itself, to furnish a comparison spectrum.
\end{fineprint}

\nbarticle{864.} \nbparatext{Twinkling or Scintillation of the Stars.}---This is a purely
atmospheric effect, usually violent near the horizon and almost null
at the zenith. It differs greatly on different nights according to the
steadiness of the air.
%% -----File: 503.png---Folio 492-------

If the spectrum of a star near the eastern horizon be examined
with a spectroscope so held as to make the spectrum vertical, it will
appear to be continually traversed by dark bands running through
the spectrum from the blue end towards the red. At the western
horizon the bands move in the opposite direction, from red to blue;
on the meridian they merely oscillate back and forth.

\begin{fineprint}
\textbf{Cause of Scintillation.}---Authorities differ as to the exact explanation
of scintillation, but probably it is mainly due to \textit{two} causes (optically speaking),
both depending on the fact that the air is full of streaks of unequal
density that are carried by the wind.

(1) In the first place, light transmitted through such a medium is concentrated
in some places and turned away from others \textit{by simple refraction}:
so that, if the light of a star were strong enough, a white surface illuminated
by it would look like the sandy bottom of a shallow, rippling pool
of water illuminated by sunlight, with light and dark mottlings which
move with the ripples on the surface. So, as we look towards the star,
and the mottlings due to the irregularities of the air move by us, we see the
star alternately bright and faint; in other words, it \textit{twinkles}; and if we
look at it in a telescope we shall see that it not only twinkles, but \textit{dances,
i.e.}, it is slightly displaced back and forth by the refraction.

(2) The other cause of twinkling is ``\textit{interference}.'' Pencils of light
coming from the star (which optically is a mere point), and feebly
refracted by the air in the way above explained, reach the observer by
slightly different routes, and are just in a condition to interfere. The
result of the interference is the temporary destruction of rays of certain
wave-lengths, and the reinforcement of others. At a given moment the
\textit{green} rays, for instance, will be destroyed, while the red and blue will be
abnormally intense; hence the quivering dark bands in the spectrum. If
the star is very near the horizon, the effects are often sufficient to produce
marked changes of color.
\end{fineprint}

\sloppy
\nbarticle{865.} \nbparatext{Why Planets Twinkle Less than Stars.}---This is mainly
because they have \textit{discs of sensible diameter}, so that there is a general
unchanging \textit{average} of brightness for the sum total of all the luminous
points of which the disc is composed. When, for instance,
point $A$ of the disc becomes dark for a moment, point $B$, very near
it, is just as likely to become bright; the interference conditions
being different for the two points. The different points of the disc
\textit{do not keep step}, so to speak, in their twinkling.
\chelabel{CHAPTERXX}
%% -----File: 504.png---Folio 493-------

\fussy
\Chapter{XXI}{Aggregations of Stars}
\nbchapterhang{\stretchyspace
DOUBLE AND MULTIPLE STARS.---ORBITS AND MASSES OF
DOUBLE STARS.---CLUSTERS.---NEBULĘ.---THE MILKY WAY.---DISTRIBUTION
OF STARS.---CONSTITUTION OF THE STELLAR
UNIVERSE.---COSMOGONY AND THE NEBULAR HYPOTHESIS.}

\includegraphicsmid{illo226}{\textsc{Fig.~226.}---Double and Multiple Stars.}

\nbarticle{866.} \nbparatext{Double and Multiple Stars.}---The telescope shows numerous
instances in which two stars lie very near each other, in many
cases so near that they can be seen separate only under a
high magnifying power. These are called ``\textit{double stars}.'' At present
something over 10,000 such couples are known, and the number is
continually increasing. In not a few instances we have \textit{three} stars
%% -----File: 505.png---Folio 494-------
together, two of which are usually very close and the third farther
away; and there are several cases of \textit{quadruple stars}, where there
are two pairs of stars lying close together (as in $\epsilon$~Lyrę), or a pair
of stars with two single stars close by; and there are some cases
where more than four form a ``\textit{multiple star}.'' \figref{illo226}{Fig.~226} represents
a number of such double and multiple stars.

\nbarticle{867.} \nbparatext{Distance, Magnitudes, and Colors.}---The apparent distances
usually range from $30''$ to $\frac{1}{4}''$, few telescopes being able to separate
double stars closer than $\frac{1}{4}''$.

In a very large proportion of cases (perhaps about one-third of all)
the two stars are nearly equal; in many others they are extremely
unequal, a minute star near a large one being usually known as its
``companion.''

Not infrequently the components of a double star present a fine
contrast of color; \textit{never, however, in cases where they are nearly equal
in magnitude.} It is a remarkable fact, as yet wholly unexplained,
that when we have such a contrast of color the tint of the smaller star
always \textit{lies higher in the spectrum} than that of the larger one. The
larger one is \textit{reddish or yellowish}, and the smaller one \textit{green or blue,}
without a single exception
among the many hundreds
of such tinted couples now
known, $\gamma$~Andromedę and
$\beta$~Cygni are fine examples
for a small telescope.

\includegraphicsouter{illo227}{\textsc{Fig.~227.}---Measurement of Distance and Position-Angle of a Double Star.}

\nbarticle{868.}\hspace{0em} \nbparatext{Measurement of
Double Stars.}---Such measures
are generally made
with a filar position-micrometer,
essentially such
as shown in Fig.~28 and
29 (\artref{Art.}{73}). The quantities
to be determined are
the distance and position-angle
of the couple. By
``distance'' we mean simply
the apparent distance in
seconds of arc between the centres of the two star discs. \textit{The position-angle}
of a double star is the \textit{angle made with the hour-circle}
%% -----File: 506.png---Folio 495-------
\textit{by the line drawn from the larger star to the smaller}, reckoning around
from the north through the east, as shown in \figref{illo227}{Fig.~227}.

\begin{fineprint}
Photography may also be used, and promises to become a favorite method.
The first photograph of such an object was by Bond in 1851.
\end{fineprint}

\nbarticle{869.} \nbparatext{Stars Optically and Physically Double.}---Stars may be double
in two different ways. They may be merely \textit{optically} double,---that
is, simply in line with each other, but one far beyond the other; or
they may be really very near together, in which case they are said to
be ``\textit{physically connected},'' because they are then under the influence
of their mutual attraction, and move accordingly.

\sloppy
\nbarticle{870.} \nbparatext{Criterion for distinguishing between Physically and Optically
Double Stars.}---This cannot be done off-hand. It requires a series of
measurements long enough continued to determine whether the relative
movement of the stars is in a curve or a straight line. If the stars
are really close together their attraction will force them to describe
curves around each other. If they are really at a great distance and
only accidentally in line, then their proper motions, being sensibly
uniform and rectilinear, will produce a \textit{relative} motion of the same
kind. Taking either star as fixed, the other star will appear to pass
it in a straight line, and with a steady, uniform drift.

\fussy
\nbarticle{871.} \nbparatext{Relative Number of Stars Optically Double, and Physically
connected}.---Double-star observations practically began with Sir
William Herschel only a little more than a hundred years ago.
When he took up the subject less than 100 such pairs had been recognized,
such as had been accidentally encountered in making observations
of various kinds. The great majority of double stars have been
discovered so recently that sufficient time has not yet elapsed to make
the criterion above given effective with more than a small proportion
of them. But it is already perfectly clear that the optically double
stars are, as the theory of probability shows they ought to be, very
few in number, while several hundred pairs have shown themselves
to be physically connected, \textit{i.e.}, to be what are known as ``\textit{binary}''
stars, or couples which revolve around their common centre of gravity.

\nbarticle{872.} \nbparatext{Binary Stars.}---Sir W.~Herschel began his observations of
double stars in the hope of ascertaining stellar parallax. He had
supposed in the case of couples where one was large and the other
small that the smaller one was usually a long way beyond the other
%% -----File: 507.png---Folio 496-------
(as \textit{sometimes} is really the fact). In this case there should be perceptible
variations in the distance and position of the two stars during
the course of the year; precisely such variations as those by
which, fifty years later, Bessel succeeded in getting the parallax of
61~Cygni (\artref{Art.}{811}). But Herschel, instead of finding the yearly
oscillation of distance and position which he expected, found quite
a different, and, at the time, a surprising thing,---a regular, progressive
change, which showed that one of the stars was slowly describing
a regular orbit around the other. To use his own expression, he
``went out like Saul to seek his father's asses, and found a kingdom,''---the
dominion of gravitation\footnote
  {It is not yet fully \textit{demonstrated} that the motions of binary stars are due to
  gravitation, though it is extremely \textit{probable}, and the burden of proof seems to
  be shifted upon those who are disposed to doubt it. See, however, the footnote
  to \artref{Article}{901}.}
extended to the stars, unlimited
by the bounds of the solar system. $\gamma$~Virginis, $\xi$~Ursę Majoris,
and $\zeta$~Herculis were among the most prominent of the systems
which he pointed out.

At present the number of pairs \textit{known} to be binary is at least
200, and as many more begin to show signs of movement. (Up
to the present time of course only the quicker moving ones are
obvious.) About fifty have progressed so far,---having made at
least one entire revolution or a great part of one,---that their orbits
have been computed more or less satisfactorily.

\nbarticle{873.} \nbparatext{Orbits of Binary Stars.}---The real orbit described by \textit{each}
of such a pair of stars is always found to be an ellipse, and assuming
the applicability of the law of gravitation, the common centre of
gravity must be at the focus. The two ellipses are precisely similar,
the one described by the smaller star being larger than the other in
inverse proportion to the star's mass.

So far as the \textit{relative} motion of the two bodies goes, we may regard
either of them (usually the larger is preferred) as being at rest, and
the other as moving around it in a \textit{relative orbit} of precisely the same
\textit{shape} as either of the two actual orbits which are described around
the centre of gravity. But the relative orbit is larger, having for its
semi-major axis the \textit{sum} of the two semi-axes of the real orbits (\artref{Art.}{427}).

Usually the \textit{relative} orbit is all that we can ascertain, as this alone
can be deduced from the micrometer measures when they consist
only of position-angles and distances measured between the two stars.
%% -----File: 508.png---Folio 497-------

\begin{fineprint}
In a few cases where such measures have been made from small stars in
the same field of view with the couple, but not belonging to the system, or
when the couple has long been observed with the meridian circle, it becomes
possible to work out separately the orbit of each star of the pair with reference
to their common centre of gravity; then we can deduce their relative
masses, as for instance, in the case of Sirius and its companion.
\end{fineprint}

\nbarticle{874.} \nbparatext{Calculation of the Orbit of a Binary Star.}---If the observer
is so placed as to view the orbit perpendicularly, he will see it in its
true form and having the larger star in its focus, while the smaller
moves around it, describing ``equal areas in equal times.'' But if the
observer is anywhere else, the orbit will be apparently more or less
distorted. It will still be an ellipse (since every projection of a conic
is also a conic), but the large star will no longer occupy its focus,
nor will the major and minor axes be apparently at right angles to
each other; nor will they even coincide with the longest and shortest
diameters of the ellipse. In this distorted ellipse the smaller star will,
however, still describe equal areas in equal times around the larger one.

Theoretically five absolutely accurate observations of the position
and distance are sufficient to determine the elements of the relative
orbit, if we assume that the orbital motion is described under the law
of gravitation. Practically a greater number are needed in most cases,
because the motions are so slow and the stars so near each other that
observation-errors of $0''.1$ (which in most calculations are of small
account) here become important. The work requires not only labor,
but judgment and skill, and unless the pair has completed or nearly
completed an entire revolution the result is apt to be seriously uncertain.
So far, as has been said, about fifty such orbits are fairly well
determined. Catalogues, more or less complete, will be found in
Flammarion's book on ``Double Stars,'' also in Gledhill's ``Handbook
of Double Stars,'' and Houzeau's ``Vade Mecum.'' See
\hyperref[app:V]{Table~V.}\ in the Appendix.

\begin{fineprint}
\nbarticle{875.} \nbparatext{Sirius and Procyon.}---The cases of these two stars are remarkable.
In both instances the large stars have been found from meridian-circle
observations to be slowly moving in little ellipses, although when  this
discovery was first made neither of them was known to be double.    In 1862
the minute companion of Sirius was discovered by Clark with the object-glass
of the Chicago telescope, then just finished, and at that time the
largest object-glass in the world. And this little companion was found to be
precisely the object needed to account for the peculiar motion of Sirius itself.

In the case of Procyon, the companion, if it exists, is yet to be discovered;
not, however, because it has not been carefully looked for with the most
powerful instruments.
\end{fineprint}
%% -----File: 509.png---Folio 498-------

\includegraphicsouter{illo228}{\textsc{Fig.~228.}---Orbits of Binary Stars.}

\nbarticle{876.} \nbparatext{Periods.}---The periods of binary stars, so far as at present
known, vary from twelve years (the period of $\delta$~Equulei) to nearly
1600 years, as $\zeta$~Aquarii.

\begin{fineprint}
It is possible that one or two
others may be found with periods
even shorter than fourteen years,
and it is practically certain that
as time goes on, pairs of longer
period than 1500 years will present
themselves. Computed periods
much exceeding 200 years
must, however, be received at
present with much reserve.

Fig.~228 shows the apparent
orbits of several of the most interesting
binaries. The figure
for $\zeta$~Cancri (copied from Gledhill)
is incorrect; 1878 and 1827
should be interchanged, and the
arrow reversed.
\end{fineprint}

\nbarticle{877.} \nbparatext{Size of the Orbits.}---The
angular semi-major axes
of the orbits thus far computed range from about $0''.3$ for $\delta$~Equulei,
to $18''$ for $\alpha$~Centauri. The real dimensions are, of course, only to be
obtained when we know the star's parallax and distance.\footnote
  {The \textit{real} semi-axis of the orbit in astronomical units is simply the \textit{angular}
  semi-axis divided by the parallax.}
Fortunately
several of the stars whose parallaxes have been determined
are also binary stars. Assuming the data as to parallax and orbits
given in the \hyperref[app:V]{tables} in the Appendix (mainly taken from Houzeau's
``Vade Mecum'') we find the following short table of results:---
\begin{center}
\small
\setlength{\tabcolsep}{5pt}
\begin{tabular}{@{}l | c | c | c | c | c@{\,}l@{}}
\hline\hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textsc{Name.}}} & Assumed   & Apparent  &  Real & \multirow{2}{*}{Period.}  & \multicolumn{2}{c}{Mass.}\\
                                                     & Parallax. & Semi-axis.&  Semi-axis.  && \multicolumn{2}{c}{\astrosun = 1.}\\
\hline
$\eta$~Cassiopeię \dotfill        & \phantom{$''$}$0''.15$  &  \phantom{$''$}$8''.64$  &  57.6  &  $195^\text{y}.2$  &  \phantom{0}5.0\\
\multirow{2}{7em}{Sirius \dotfill}&  0.38  & 8.53         &  22.4         &  44.0           & \phantom{0}5.8 & \multirow{2}{*}{$\Big\}$}\\
                                  & (0.19) &  ---         & (44.8)\rlap{\footnotemark}& --- & (46.4)\\
$\alpha$~Geminorum \dotfill       &  0.20  & 7.54         &  37.7         & \llap{9}97.0    &  0.54\\
$\alpha$~Centauri \dotfill        &  0.75  & \llap{1}7.50 &  23.3         &  77.0           &  2.14\\
70 Ophiuchi \dotfill              &  0.16  & 4.79         &  29.9         &  94.5           &  3.0\phantom{0}\\
61 Cygni \dotfill                 &  0.43  & 15.40?       &  35.8\rlap{?} & 450.0?          &  0.23\\
\hline\hline
\end{tabular}
\footnotetext{The distance 44.8 is computed with Gyldén's
parallax, $0''.193$.}
\end{center}
%% -----File: 510.png---Folio 499-------

It is obvious from the table that the double-star orbits are comparable
with the larger orbits of the solar system; that of $\alpha$~Centauri
being just about equal in size to that of Uranus, and that of
$\eta$~Cassiopeię not quite double the size of Neptune's orbit. Of course
the many binary stars whose distance is so great as to make their
parallax insensible while their apparent orbits are as large as those
given in the list must have real orbits of still vaster dimensions.

\nbarticle{878.} \nbparatext{Masses of Binary Stars.}---When we know both the size of
the orbit of a binary and its period, the mass, according to the law
of gravitation, follows at once from the equation of \artref{Article}{536},
\[
M + m = 4\pi^2\frac{a^3}{t^2}.
\]
If $t$ and $a$ are given respectively in \textit{years} and astronomical units
of \textit{distance}, then, by omitting the factor $4\pi^2$, $M + m$ comes out in
\textit{terms of the sun's mass.} The final column of the little table above
gives the \textit{masses} of the six pairs of stars as compared with the mass
of the sun. But the student must bear in mind that the parallaxes
of stars are so uncertain, that these results are to be accepted with a
very large margin of error.

\begin{fineprint}
\nbarticle{879.} \nbparatext{Relation between the ``Mass-Brightness'' of Binary Stars.}---Monck
of Dublin has recently called attention to a curious relation
between the apparent brightness of a binary, its period and (angular) distance
on the one hand, and its ``\textit{mass-brightness},'' or \textit{candle-power per ton}, so
to speak, on the other,---a relation which does not involve a knowledge of
the \textit{parallax} of the stars. He shows that if we let $l$ be the apparent brightness
of any double star, photometrically determined, $a$ the semi-major axis
of its orbit, (in seconds of arc), and $t$ its period in years, while $k$ is its
``\textit{mass-brightness},'' or candle-power per ton,---then we have\footnote
  {This is strictly true only on the assumption that either the two components
  of the double star have the same mass-brightness, or else that the smaller one is
  so much smaller that its motion is practically the same as if it were a mere
  particle. If the stars of the pair are both about alike, or if they differ
  greatly in mass, the equation is practically correct in either case. It breaks
  down when the two stars of a pair \textit{do not differ much in mass, but do differ greatly
  in brightness}---probably an unusual case.}
\[
  k_1 : k_2
= l_1 t_1^{\frac{4}{3}} a_1^2
: l_2 t_2^{\frac{4}{3}} a_2^2 \quad\text{or}\quad
  \frac{k_1}{k_2}
= \frac{l_1}{l_2} \left( \frac{t_1}{t_2} \right)^{\frac{4}{3}}
                  \left( \frac{a_1}{a_2} \right)^2.
\]
(\textit{The Observatory, February,} 1887.)
\medskip
%% -----File: 511.png---Folio 500-------

He takes $\xi$~Ursę Majoris as a standard (\textit{i.e.}, its $k = 1$), and finds for
$\gamma$~Leonis $k = 93$; for $\alpha$~Geminorum, 38; for Sirius, 7; for $\zeta$~Herculis, 4; for
$\eta$~Cor.\ Bor., 1.5; for 70~Ophiuchi, 0.36; for $\eta$~Cassiopeię, 0.22; and for
61~Cygni, 0.08. In other words, $\gamma$~Leonis is enormously brilliant in proportion
to its mass, Sirius is medium, while 61~Cygni is extremely faint. Upon
this scale the sun's \textit{mass-brightness} would be about 2, judging from its mass
and brightness compared with that of Sirius (Arts.~\arnref{832} and \arnref{877}). (We
have given above only a few of Mr.~Monck's numbers.)
\end{fineprint}

\nbarticle{880.} \nbparatext{Have the Stars Planets attending them?}---It is a very
natural supposition that the minute companions which attend some of
the larger stars may be really planetary in their nature, shining more
or less by reflected light. As to this we can only say, that while it is
quite possible that other stars besides our sun may have their retinues
of planets, it is quite certain that such planets could not be seen
by us with any existing telescope. If our sun were viewed from
$\alpha$~Centauri, Jupiter would be a star of less than the twenty-first magnitude,
at a distance of only $5''$ from the sun which itself would be a
smallish first-magnitude star.

\begin{fineprint}
\nbarticle{881.} The statement can be verified as follows: Jupiter at opposition is
certainly not equivalent in brightness to twenty stars like Vega (most
photometric measurements make it from eight to fourteen). Assuming,
however, that it is equal to twenty Vegas, its light received by the earth
would be about $\frac{1}{2000\,000000}$ of the sun's. At opposition our distance
from Jupiter is about four astronomical units, so that seen from the same
distance as the sun, its light would be sixteen times that quantity, or
(nearly) $\frac{1}{125\,000000}$ of the sun's.

Now a ratio of 125,000000 between the light of two stars corresponds to a
difference of $20 +$ magnitudes
\[
\left( \log{\text{125,000000}}\!=\!8.0969; \text{ but } \frac{8.0969}{0.4000}\!=\!20.24 \text{ magnitudes. (\artref{Art.}{820}.)}\right)
\]
Accordingly, if the observer were removed to such a distance that the sun
would appear like a first-magnitude star (as would be the case from
$\alpha$~Centauri), Jupiter would be a star of the twenty-first magnitude. According
to \artref{Art.}{822}, it would require a 25-inch telescope to show a star of the
sixteenth magnitude; it would therefore require an instrument with an
aperture of 250 inches, or nearly 21 feet, to show a star five magnitudes
fainter, even if there were no large star near to add to the difficulty.

\nbarticle{882.} \nbparatext{Triple and Multiple Stars.}---There are a considerable number
of objects of this kind, and some of them constitute physical systems. In
the case of $\zeta$~Cancri the two larger stars revolve around their common
%% -----File: 512.png---Folio 501-------
centre in a nearly circular orbit less than $2''$ in diameter, and with a period
of about sixty years; while the third star, smaller and more distant, moves
around the closed pair in an orbit not yet well determined, but with a period
that must be several hundred years; and in its motion there is evidence of a
peculiar perturbation, as yet unexplained. In $\epsilon$~Lyrę we have two pairs,
each making a very slow revolution, of periods not yet determined, but
probably ranging from 300 to 500 years. And since the pairs have also a
common proper motion it is practically certain that they also are physically
connected, and revolve around their common centre of gravity in a period to
be reckoned by millenniums---the motion during the last hundred years
being barely perceptible. In other cases, as for instance, in the multiple
star $\theta$~Orionis, we have a number of stars not organized in pairs, but at more
or less equal distances from each other: we are confronted by the problem
of $n$ bodies in its most general and unmanageable form. Nature challenges
the mathematicians.
\end{fineprint}

\nbarticle{883.} \nbparatext{Clusters.}---There are in the sky numerous groups of stars
containing from one hundred to many thousand members. Some of
them are made up of stars visible separately to the naked eye, as the
Pleiades; some of them require a small telescope to resolve them, as,
for instance, the Pręsepe in Cancer, and the group of stars in the
sword-handle of Perseus; while others yet, even in telescopes of
some size, look simply like wisps or balls of shining cloud, and break
up into stars only in the most powerful instruments.

\begin{fineprint}
In a large instrument some of the telescopic clusters are magnificent objects,
composed of thousands of stellar sparks compressed into a ball which
is dazzlingly bright at the centre and thinning out towards the edge. In
some of them vividly colored stars add to the beauty of the group. In the
northern hemisphere the finest cluster is that known as 13~M Hercules
($\alpha$, $16^\text{h}\: 37^\text{m}$ and $\delta$,  $36°\: 40'$) not very far from the ``apex of the sun's way.''
\end{fineprint}

\includegraphicsmid{illo229}{\textsc{Fig.~229.}---The Pleiades.}

\nbarticle{884.} \nbparatext{The Pleiades.}---Of the naked-eye clusters the Pleiades is
the most interesting and important. To an ordinary eye six stars are
easily visible in it, the six largest ones indicated in the figure (\figref{illo229}{Fig.~229}).
Eyes a little better see easily five more---these next in size
in the \figref{illo229}{figure} (the two stars of Asterope being seen as one). A
very small telescope (a mere opera-glass) increases the number to
nearly a hundred; and with large instruments more than 400 are
catalogued in the group. A few of the stars, apparently in the cluster,
are really only accidentally on the same line of vision, and are distinguished
by proper motions different from those of the rest of the
group; but the great majority have proper motions nearly the same
%% -----File: 513.png---Folio 502-------
in amount and direction; they have also identical spectra, and therefore
undoubtedly constitute a single system.

\begin{fineprint}
The distances and positions of the principal stars with respect to the
central star Alcyone have been carefully measured three or four times
during the last fifty years. The relative motions during the period have
not proved large enough to admit of satisfactory determination, but it is
clear that such motions exist. A curious and interesting fact is the presence
of \textit{nebulous matter} in considerable quantity. A portion of this nebulosity
hanging around Merope (the northeast star of the dipper-bowl in the \figref{illo229}{figure})
was discovered many years ago; but it was reserved to photography to
detect very recently other wisps of nebulosity attached to other stars, especially
to Maia, and to show that the whole space is covered with streaks and
streamers of it, emitting light of such a character as to impress the photographic
%% -----File: 514.png---Folio 503-------
plate much more strongly than the eye. The eye cannot see the
nebula with the same telescope which is able to photograph it
strongly. The \figref{illo229}{figure} shows roughly the outlines of some of the
principal nebulous filaments.

\nbarticle{885.} \nbparatext{Distance of Star-Clusters and Size of the Component Stars.}---The
question at once arises whether clusters, such as the one mentioned
in Hercules, are composed of stars each comparable with our sun, and
separated by distances corresponding to the distance between the sun
and its neighboring stars, or whether the bodies which compose the
swarm are really very small,---mere sparks of stellar matter:
whether the distance of the mass from us is about the same as that
of the stars among which it seems to be set, or whether it is far
beyond them. Forty years ago the accepted view was that the stars
composing the clusters are no smaller than ordinary stars, and that
the distance of the star-clusters is immensely greater than that of
the isolated stars. There are many eloquent passages in the writings
of that period based upon the belief that these star-clusters are
\textit{stellar universes},---``galaxies,'' like the group of stars
to which the writers supposed the sun to belong, but so
inconceivably remote that in appearance they shrank to these mere
balls of shining dust.

It is now, however, quite certain that the other view is
correct,---that star-clusters are among \textit{our} stars and form
part of \textit{our} universe. Large and small stars are so
associated in the same group in many cases, as to leave us no choice
of belief in the matter. It is true that as yet no parallax has been
detected in any star-cluster; but that is not strange, since a
cluster is not a convenient object for observations of the kind
necessary to the detection of parallax.

\nbarticle{886.} \nbparatext{The Nebulę.}---There are also in the sky a multitude of
faintly shining bodies,---shreds and balls of cloudy stuff that are
known as ``\textit{nebulę}'' (the word meaning strictly a ``little
cloud''). Some 10,000 or 11,000 of these objects are already
catalogued.

Two or three of them are visible to the naked eye. The nebula in the
girdle of Andromeda is the brightest of them, in which, it will be
remembered, the temporary star of 1885 appeared.

The next brightest is the wonderful nebula of Orion, which, in the
beauty and variety of its details, in the interesting relations of
the included stars, the delicate tint of the filmy light, and in its
spectroscopic interest, far exceeds the other,---indeed, all others.
%% -----File: 515.png---Folio 504-------

It is so difficult to represent these delicate objects by any
process available to text-books, that we limit ourselves to cuts of
two only, the great nebula of Andromeda and the ``ring-nebula'' in
Lyra (both from original drawings), referring the reader to the
elaborate engravings that can be found in the ``Philosophical
Transactions,'' the ``Memoirs of the Royal Astronomical Society,''
and other similar works, for adequate representations of other
interesting objects of this kind.
\end{fineprint}

With a small telescope a nebula cannot be distinguished from a close
star-cluster, and it is quite likely that the clusters and nebulę
shade into each other by insensible gradations. Forty years ago it
was supposed that there was no distinction between them except that
of mere remoteness,---that all nebulę could be resolved into stars
by sufficient increase of telescopic power. When Lord Rosse's great
telescope was first erected, it was for a time reported (and the
statement is still often met with) that it had ``resolved'' the
Orion nebula. This was a mistake however. No telescope ever has
resolved that nebula into stars nor ever will, for we now
\textit{know} that it is not composed of stars.

\nbarticle{887.} \nbparatext{Forms and Magnitudes of Nebulę.}---The larger
and brighter nebulę are, many of them, very irregular in form,
stretching out sprays and streamers in all directions, and
containing dark openings or ``lanes.'' The so-called ``fish-mouth''
in the nebula of Orion, and the dark streaks in the nebula of
Andromeda, are striking examples. Some of these bodies are of
enormous volume. The nebula of Orion, with its outlying streamers,
extends over several square degrees, and the nebula of Andromeda
covers more than one. Now, as seen from even the nearest star, the
apparent distance of Neptune from the sun is only $30''$, and the
diameter of its orbit $1'$. It is perfectly certain that neither of
these nebulę is as near as $\alpha$~Centauri, and therefore the
cross-section of the Orion nebula, as seen from the
%% -----File: 516.png---Folio 505-------
earth, must be \textit{at least} many thousand times the area of
Neptune's orbit.

\includegraphicsmid{illo230}{\textsc{Fig.~230.}---The Great Nebula in Andromeda.}

We do not know what is the real shape of either of these nebulę,
whether it is a thin, flat sheet, or a voluminous bulk; but some
things about these two nebulę and several others favor strongly the
idea that their thickness does not correspond to their apparent
area.

\includegraphicsouter{illo231}{\textsc{Fig.~231.}---The Annular Nebula in Lyra.}

\nbarticle{888.} \nbparatext{The Smaller Nebulę.}---The smaller nebulę are for
the most part elliptical in outline, some nearly circular, others
more elongated, and some narrow, slender streaks of light. Generally
they are brighter at the centre, and in many cases the centre is
occupied by a star. Indeed, there is a considerable number of
so-called ``\textit{nebulous stars},'' that is, stars with a hazy
envelope
around them.

There are some nebulę which present nearly a uniform disc of light,
and are known as ``\textit{planetary}'' nebulę, and there are some
which are dark in the centre and are known as ``\textit{annular}''
or ring nebulę. The finest of these annular nebulę is the one in the
constellation of Lyra, about half-way between the stars $\beta$ and
$\gamma$; it is shown in \figref{illo231}{Fig.~231}.

There are also a number of double nebulę, and some of these exhibit
a remarkable spiral structure when examined by telescopes of the
largest aperture like the great reflector of Lord Rosse. The
so-called ``\textit{whirlpool}'' nebula in the constellation ``Canes
Venatici'' is the most striking specimen. This spiral structure,
however, is to be made out \textit{only} in large telescopes; in
fact, very little of the real beauty of most of these objects is
accessible to instruments of less than 12 inches
aperture.

\sloppy
\nbarticle{889.} \nbparatext{Variable Nebulę.}---There are several nebulę which
\textit{vary in their brightness} from time to time; one especially,
near $\epsilon$~Tauri, at times has been visible with a small
telescope, while at other times it
%% -----File: 517.png---Folio 506-------
is entirely invisible even with large ones. So far no regular
periodicity has been ascertained in such cases.

\fussy
\nbarticle{890.} \nbparatext{Their Spectra.}---One of the earliest and most remarkable
achievements of the spectroscope was its demonstration of the fact
that the light of many of the nebulę proceeds mainly from
\textit{luminous gas}. They give a spectrum of six or seven bright
lines\footnote{ The wave-lengths of the lines are the following, in
the order of brightness: (1) 500.4 ? (2) 495.8 ? (3) 486.1 Hydrogen
(\textit{F}); (4) 434.0 Hydrogen$_\gamma$; (5) 410.1 Hydrogen
(\textit{h}); (6) 587.4 Helium (?); (7) 448?} (\figref{illo232}{Fig.~232}), three of
which are fairly conspicuous; while the rest are very faint.

\includegraphicsmid{illo232}{\textsc{Fig.~232.}---Spectrum of the Gaseous Nebulę.}

\begin{fineprint}
Three of the lines, $F$, $H_{\gamma}$, and $h$, are due to
\textit{Hydrogen}. One, the faintest of all, recently discovered by
Copeland, seems to be identical with the $D_3$ line of the solar
chromosphere; but the origin of the rest remains unknown. The
brightest line of the whole number is in the green, $\lambda$,
500.4, and was for a while referred to nitrogen; but under closer
examination the identification breaks down, though the student will
find it still called ``the nitrogen line'' in many astronomical
works.

Mr. Lockyer identifies it with a ``fluting'' in the low-temperature
spectrum of magnesium, which he has found in the spectrum of
meteorites; but this seems rather doubtful, as the nebula line is
fine and sharp, and does not look at all like the relic of a
\textit{band}; nor has the coincidence been satisfactorily
established by actually confronting the nebula spectrum with that of
magnesium.

All the nebulę which give a gaseous spectrum at all, present this
same spectrum entire or in part. If the nebula is faint only the
brightest line appears, while the $H_{\gamma}$ line and the other
fainter lines are seen only in the brightest nebulę and under
favorable circumstances.
\end{fineprint}

\nbarticle{891.} But not all the nebulę by any means give a gaseous spectrum:
those which do so---about half the whole number---are of a more or
less distinct greenish tint, which is at once recognizable in the
telescope. The \textit{white} nebulę, the nebula of Andromeda at
their head, give only a continuous and perfectly expressionless
spectrum,
%% -----File: 518.png---Folio 507-------
unmarked by any lines or bands, either bright or dark. This must
not be interpreted as showing that these nebulę cannot be gaseous;
for a gas \textit{under pressure} gives just such a spectrum; but so also do
masses of solid or liquid when heated to incandescence. The spectroscope
simply declines to testify in this case. The \textit{telescopic} evidence
as to the nature of the white nebulę is the same as for the green.
They withstand all attempts at resolution, none more firmly than the
Andromeda nebula itself, the brightest of them all.


\nbarticle{892.} \nbparatext{Changes in the Nebulę.}---The question has been raised
whether some of the nebulę have not sensibly changed, even within
the few years since it has become possible to observe them in detail.
It is quite certain that in important respects the early \textit{drawings} differ
seriously from these of recent observers; but the appearance of a
nebula depends so much upon the telescope and the circumstances
under which it is used, the features are so delicate and indefinite, and
the difficulty of representing them on paper is such, that very little
reliance can be placed on discrepancies between drawings, unless supported
by the evidence of \textit{measures} of some kind.

\begin{fineprint}
Thus far, the best authenticated instance of such a change, according
to Professor Holden, is in the so-called ``trifid'' nebula, in Sagittarius.
In this object there is a peculiar three-legged area of darkness which
divides the nebula into three lobes. A bright triple star, which in the
early part of the century was described and figured by Herschel and
other observers as in the \textit{middle} of one of these dark lanes, is now certainly
in the edge of the nebula itself. The star does not seem to have moved with
reference to the neighboring stars, and it seems therefore necessary to suppose
that the nebula itself has drifted and changed its form.

As to the nebula of Orion, Professor Holden's conclusion is, that while
the \textit{outlines} of the different features have probably undergone but little
change, their \textit{relative brightness} and \textit{prominence} have been continually fluctuating.
This, however, can hardly be considered certain; to settle the
question will probably require another fifty years or so, and the comparison,
not of drawings but of \textit{photographs}---for it is now possible to photograph
the brighter nebulę.
\end{fineprint}

\nbarticle{893.} \nbparatext{Photographs of Nebulę.}---The first success in photographing
an object of this sort was obtained in 1880 by Dr.~Henry Draper
in experiments upon the nebula of Orion; in 1881 and 1882 he greatly
improved upon his first essays; and not long before his death in 1882
he obtained a really fine photograph of the beautiful object. Mr.~Common,
of England, with his three-foot silver-on-glass reflector, and
%% -----File: 519.png---Folio 508-------
exposures ranging from half an hour upward, has since then obtained
pictures of the same object, still finer, and that seem to leave little to
be desired. They fail of perfection only because the stars which
jewel the nebula so brilliantly as seen by the eye, become in a photograph
mere blotches, encroaching upon and cutting out large patches
of the most interesting portions of the nebula.

\begin{fineprint}
A number of nebulę have also been photographed by others as well
as by Common; but he still maintains the lead, and it is hoped that
when his new five-foot reflector is finished, he will procure for us an
authentic record of the present appearance of all the most important
objects of this class.
\end{fineprint}

\nbarticle{894.} \nbparatext{Nature of the Nebulę.}---As to the constitution of these
clouds we can only speculate. In the green nebulę we can say with
confidence that hydrogen and some other gas or gases are certainly
present, and that the gases emit most of the light that reaches us
from such objects. But how much solid or liquid matter in the form
of grains and drops may be included within the gaseous cloud we
have no means of knowing.

\begin{fineprint}
The idea of Mr.~Lockyer (a part of his wide induction as to what we
may call the ``meteoritic constitution of the universe'') is that they are
clouds of ``sparse meteorites, the collisions of which bring about a rise
of temperature sufficient to render luminous one of their chief constituents,---magnesium.''

How far this theory will stand the test of time and future investigations
remains to be seen. At first view it seems very doubtful whether the \textit{collisions}
in such a body could be frequent or violent enough to account for
its luminosity, and one is tempted to look to other causes for the source
of light.
\end{fineprint}

\nbarticle{895.} \nbparatext{Number and Distribution of Nebulę.}---Sir William Herschel
was the first extensive investigator of these interesting objects,
and left his unfinished work as a legacy to his son, Sir John Herschel,
who completed the survey of the heavens by a residence of
several years at the Cape of Good Hope. His ``General Catalogue''
is the standard of reference for objects of this kind, and contains
about 5000 of them. Between 1000 and 2000 more have been
added since its publication, most of them extremely faint. It is
hardly possible that any important ones remain to be found.

As to their distribution, it is a curious and important fact that it
is \textit{in contrast} to the distribution of the stars. The stars, as we shall
%% -----File: 520.png---Folio 509-------
soon see, gather especially in and about the Milky Way, as do also
the star-clusters; but the nebulę specially crowd together in regions
as far from the Milky Way as it is possible to get. As has been
pointed out by more than one, this shows, however, not a want of
relation between the stars and the nebulę, but some ``relation of
contrariety.'' Precisely what this is, and why the nebulę avoid the
regions thickly starred, is not yet clear. Possibly the stars \textit{devour
them}, that is, gather in and appropriate surrounding nebulosity so
that it disappears from their neighborhood.

\nbenlargepage
\nbarticle{896.} \nbparatext{Distance of the Nebulę.}---On this point we have very little
absolute knowledge. Attempts have been made to measure the parallax
of one or two, but so far unsuccessfully. Still it is probable,
indeed almost certain, that they are at the same \textit{order of distance} as
the stars. The wisps of nebulosity which photography shows attached
to the stars in the Pleiades (and a number of similar cases
appear elsewhere), the nebulous stars of Herschel, and numerous
nebulę which have a star exactly in the centre,---these compel us to
believe that in such cases the nebulosity is really \textit{at the star}. Then
in the southern hemisphere there are two remarkable luminous clouds
which look like detached portions of the Milky Way (though they
are not near it), and are known as the Nubeculę (major and minor).
These are made up of stars and star-clusters, and of nebulę also,
all swarming together, and so associated that it is not possible to
question their real proximity to each other.

\begin{fineprint}
\nbarticle{897.} Fifty years ago a very different view prevailed. As has been said
already, astronomers at that time very generally believed that there was no
distinction between nebulę and star-clusters except in regard to distance,
the nebulę being only clusters too remote to show the separate stars. They
considered a nebula, therefore, as a ``universe of stars,'' like our own ``galactic
cluster'' to which the sun belongs, but as far beyond the ``star-clusters''
as these were believed to be beyond the isolated stars. In some respects this
old belief strikes one as grander than the truth even. It made our vision
penetrate more deeply into space than we now dare think it can.
\end{fineprint}

\nbarticle{898.} \nbparatext{The Galaxy, or Milky Way.}---This is a luminous belt which
surrounds the heavens nearly in a great circle. It varies much in
width and brightness, and for about a third of its extent, from Cygnus
to Scorpio, is divided into two nearly parallel streams. In
several constellations, as in Cygnus, Sagittarius, and Argo Navis,
it is crossed by dark straight-edged bars that look as if some
%% -----File: 521.png---Folio 510-------
light cloud lay athwart it, and in the constellation of Centaurus there
is a dark pear-shaped orifice,---the ``coal sack,'' as it is called.

The galaxy intersects the ecliptic at two opposite points near the
solstices, making with it an angle of about $60°$. The northern
``galactic pole,'' as it is called, lies, according to Sir John Herschel,
in declination $+37°$, and right ascension $12^\text{h}\: 47^\text{m}$; the southern
``galactic pole'' is of course at the opposite point in the southern
hemisphere. As Herschel remarks, the ``galactic plane'' ``is to
sidereal what the ecliptic is to planetary astronomy, a plane of
ultimate reference, the ground plan of the sidereal system.''

The Milky Way is made up almost wholly of small stars from the
eighth magnitude down. It contains also a large number of star-clusters,
but (as has been already mentioned) very few true nebulę.
In some places the stars are too thickly packed for counting, especially
in the bright knots which abound here and there.

\begin{fineprint}
(An excellent detailed description of its appearance and course may be
found in Herschel's ``Outlines of Astronomy.'')
\end{fineprint}

\nbarticle{899.} \nbparatext{Distribution of Stars in the Sky: Star-Gauges.}---It is
obvious that the stars are not uniformly scattered over the heavens.
They show a decided tendency to collect in groups here and there,
and to form connected streams; but besides this, an enumeration of
the stars in the great star-catalogues shows that the number increases
with considerable regularity from the galactic poles, where they are
most sparse, towards the galactic circle, where they are most crowded.
The ``star-gauges'' of the Herschels make this fact still more
obvious.

\begin{fineprint}
These gauges consisted merely in the counting of the number of stars
visible in the field of view ($15'$ in diameter) of the twenty-foot reflector.
Sir William Herschel made 3400 of these gauges, directing the telescope to
different parts of the sky; and his son followed up the work at the Cape of
Good Hope. Struve's discussion of these gauges in their relation to the
galactic circle gives the following result:---
\begin{center}
\begin{tabular}{r@{}p{0.5\textwidth}@{}l}
  \scriptsize Distance from \rlap{Galaxy.}& & \scriptsize\llap{Number} of Stars in Field.\\
  $90°$ & \dotfill &  \phantom{00}4.15  \\
  $75°$ & \dotfill &  \phantom{00}4.68  \\
  $60°$ & \dotfill &  \phantom{00}6.52  \\
  $45°$ & \dotfill &  \phantom{0}10.36  \\
  $30°$ & \dotfill &  \phantom{0}17.68  \\
  $15°$ & \dotfill &  \phantom{0}30.30  \\
   $0°$ & \dotfill &            122.00  \\
\end{tabular}
\end{center}
\end{fineprint}
%% -----File: 522.png---Folio 511-------

\nbarticle{900.} \nbparatext{Structure of the Heavens.}---Our space does not permit a
discussion of the untenable conclusions reached by Herschel and
others by combining the unquestionable data derived from observation,
with the unfounded and untrue assumptions that the stars are
substantially of a size and spaced at approximately equal distances.
Many of these conclusions relating to the form and dimensions of
the Milky Way, and of the stellar universe to which our sun belongs,
have become almost classical; but they are none the less incorrect.

It is certain, however, that the faint stars as a class are smaller
and darker and more remote than are the bright ones \textit{as a class:} and
accepting this, we can safely draw from the star-gauges a few general
conclusions, as follows:---

\begin{fineprint}
We present them substantially as given by Newcomb in his ``Popular
Astronomy,'' p.~491.
\end{fineprint}

1. ``The great mass of the stars which compose this (stellar)
system are spread out on all sides in or near a widely extended
plane, passing through the Milky Way. In other words, the large
majority of the stars which we can see with the telescope are contained
in a space having the form of a round, flat disc, the diameter
of which is eight or ten times its thickness.\DPtypo{}{''}

2. ``Within this space the stars are not scattered uniformly, but
are for the most part collected into irregular clusters or masses, with
comparatively vacant spaces between them.'' They are ``gregarious,''
to use Miss Clerke's expression.

3. Our sun is near the centre of this disc-like space.

4. The \textit{naked-eye stars} ``are scattered in this space with a near
approach to uniformity,'' the exceptions being a few star-clusters and
star-groups like the Pleiades and Coma Berenices.

5. ``The disc described above does not represent the form of the
stellar system, but only the limits within which it is mostly contained.''
The circumstances are such as to ``prevent our assigning
any more definite form to the system than we could assign to a cloud
of dust.''

6. ``On each side of the galactic region the stars are more evenly
and thinly scattered, but probably do not extend out to a distance at
all approaching the extent of the galactic region,'' or if they do they
are very few in number; but it is impossible to set any definite
boundaries.

7. On each side of the galactic and stellar region we have a nebular
region, comparatively starless, but occupied by great numbers of
nebulę.
%% -----File: 523.png---Folio 512-------

As to the Milky Way itself, it is not yet certain whether the stars
which compose it are distributed pretty equally near the galactic
circle, or whether they form something like a ring with a comparatively
vacant space in the middle. The ring theory seems at present
rather the more probable of the two.

As to the distance of the remotest stars in the stellar system, it is
impossible to say anything very definite, but it seems quite certain
that it must be at least so great that light would occupy from 10,000
to 20,000 years in traversing it. If one asks what is beyond the
stellar system, whether the star-filled space extends indefinitely or
not, no certain answer can be given.

\nbarticle{901.} \nbparatext{Do the Stars Form a System?}---That is, do they form an
\textit{organized unit}, in which, as in the solar system, each of the different
members has its own function and permanently maintains its relation
to the rest? Gravitation certainly operates, as the binary stars
demonstrate,\footnote
  {This is perhaps rather too strong an expression. It would be truer to say
  that none of the observed phenomena of the binary stars \textit{contravene} the universality
  of gravitation, and for the most part they are just what gravitation easily
  accounts for. But they do not \textit{demonstrate} that the central force varies inversely
  with the square of the distance, because we do not know the angle made by the
  orbit-plane with the line of sight, except from calculations based on the assumption
  that the central force really varies in that way. The orbits, as directly
  observed, are consistent with several other laws of central force than the law
  of inverse squares. (See ``Astron.\ Journal,'' Vol.~VIII., article by Prof.\ A.~Hall.)}
and the stars are moving swiftly in various directions
with enormous velocities, as shown by their proper motions, and by
the spectroscope. The question is whether these motions are controlled
by gravitation, and whether they carry the stars in \textit{orbits} that
can be known and predicted.

That the stars are organized into a system or systems \textit{of some sort}
can hardly be doubted, for this seems to be a necessary consequence
of their mutual attraction. But that the system is one at all after
the pattern of the solar system, in which the different members move
in \textit{closed orbits},---orbits that are permanent except for the slow
changes produced by perturbation,---this is almost certainly impossible,
as was said a few pages back.

\begin{fineprint}
\nbarticle{902.} \nbparatext{Is there a Revolution of the Whole Mass of Stars?}---A
favorite idea has been that the mass of stars which constitutes our system
%% -----File: 524.png---Folio 513-------
has a slow rotation like that of a body on its axis, the plane of this general
revolution coinciding with the plane of the galaxy. Such a general motion
is not in any way inconsistent with the independent motions of the individual
stars, and there is perhaps a slight inherent probability in favor of
such a movement; but thus far we have no evidence that it really exists---indeed,
there hardly \textit{could} be any such evidence at present, because exact
Astronomy is not yet old enough to have gathered the necessary data.

\sloppy
\nbarticle{903.} \nbparatext{Central Suns.}---A number of speculative astronomers, Mädler
perhaps most prominently, have held the belief that there is a ``\textit{central sun},''
standing in some such relation to the stellar system as our sun does to the
solar system. It is hardly necessary to say that the notion has not the
slightest foundation, or even probability.

\fussy
Lambert supposed \textit{many} such suns as the centres of subordinate stellar
systems, and because we cannot see them, he imagined them to be dark.

If we conceive of boundaries drawn around our stellar system, and count
all the stars within the limits as members of it, leaving out of the account all
that fall outside, then, of course, our system so limited has at any moment
a perfectly definite \textit{centre of gravity}. There is no reason why some particular
star may not be very near that centre, and in that sense a ``central sun'' is
possible; but its central position would not give it any preėminence or rule
over its neighbors, or put it in any such relation to the rest of the stars as
the sun bears to the planets.
\end{fineprint}

\nbarticle{904.} \nbparatext{Orbits of Sun and Stars.}---It is practically certain that the
motions of the stars are not \textit{orbital} in any strict sense. Excepting
stars which are in clusters, all other stars are simultaneously acted
upon by many forces drawing in various and opposite directions;
and these forces must in most cases be so nearly balanced that
the resultant cannot be very large. The motions of the stars must
consequently, as a rule, be \textit{nearly} rectilinear.

Still the balancing of the forces will seldom be exact, and accordingly
the path of a star will almost always be \textit{slightly} curved; and
since the amount and direction of the resultant force which acts on
the star is continually changing, the curvature of its motion will
alter correspondingly, and the result will be a path which does not
lie in any one plane, but is bent about in all ways like a piece of
crooked wire. It is hardly likely, however, that the curvature of a
star's path would, in any ordinary case, be such as could be detected
by the observations of a single century, or even of a thousand
years.

As has been said before, in connection with the proper motions of
the stars, the probability is that the separate stars move nearly independently,
%% -----File: 525.png---Folio 514-------
``like bees in a swarm.'' In the solar system the central
power is supreme, and perturbations or deviations from the path
which the central power prescribes are small and transient. In the
stellar system, on the other hand, the central force, if it exists at all
(as an attraction towards the centre of gravity of the whole mass of
stars) is trifling. Perturbation prevails over regularity, and ``\textit{individualism}''
is the method of the greater system of the stars, as solar
despotism is that of the smaller system of the planets.

\nbarticle{905.} \nbparatext{Cosmogony.}---Unquestionably one of the most interesting,
and also most baffling, topics of speculation is the problem of the
way in which the present condition of the universe came about. By
what processes have moons and earths and Jupiters and Saturns,
come to their present state and into their relation to the sun? What
has been their past history, and what has the future in store for
them? How has the sun come to his present glory and dominion?
and in the stellar universe, what is the meaning and mutual relation
of the various orders of bodies we see,---of the nebulę, the star-clusters,
and the stars themselves?

In a forest, to use a comparison long ago employed by the elder
Herschel, we see around us trees in all stages of their life history.
There are the seedlings just sprouting from the acorn, the slender
saplings, the sturdy oaks in their full vigor, those also that are old
and near decay, and the prostrate trunks of the dead. Can we
apply the analogy to the heavens, and if we can, which of the objects
before us are to be regarded as in their infancy, and which of them
as old and near dissolution?

\nbarticle{906.} \nbparatext{Fundamental Principles of a Rational Cosmogony.}---In
the present state of science many of the questions thus suggested
seem to be hopelessly beyond the reach of investigation, while
others appear like problems which time and patient work will
solve, and others yet have already received clear and decided
answers. In a general way it may be said that the \textit{condensation
and aggregation of rarefied masses of matter under the force of
gravitation; the conversion into heat of the $($potential$)$ ``energy of
position'' destroyed by the process of condensation; the effect of this
heat upon the contracting mass itself, and the radiation of energy into
space and to surrounding bodies as waves of light and heat,}---these
principles contain nearly all the explanations that can thus far be
given of the present state of the heavenly bodies.
%% -----File: 526.png---Folio 515-------

\nbarticle{907.} \nbparatext{The Planetary System.}---We see that our planetary system
is not a mere accidental aggregation of bodies. Masses of matter
coming hap-hazard towards the sun would move, as comets do, in
orbits, always conic sections to be sure, but of every degree of eccentricity
and inclination. There are a multitude of relations actually
observed in the planetary system which are wholly independent of
gravitation and demand an explanation.

1. The orbits are all \textit{nearly circular}.

2. They are all nearly \textit{in one plane} (excepting the cases of some
of the little asteroids).

3. The revolution of all is \textit{in the same direction.}

4. There is a curiously \textit{regular progression of distances} (expressed
by Bode's law, which, however, breaks down at Neptune).

5. There is a roughly regular \textit{progression of density}, increasing
both ways from Saturn, the least dense of all the planets in the
system.

As regards the planets themselves, we have

6. The \textit{plane} of the planets' rotation \textit{nearly coinciding with that of
the orbit} (probably excepting Uranus).

7. The \textit{direction} of the rotation \textit{the same as that of the orbital
revolution} (excepting probably Uranus and Neptune).

8. The \textit{plane} of \textit{orbital revolution of the satellites} coinciding nearly
with that of the planet's rotation.

9. The \textit{direction} of the satellites' revolution also coinciding \textit{with
that of the planet's rotation}.

10. The largest planets rotate most swiftly.

\begin{fineprint}
\nbarticle{908.} \nbparatext{Origin of the Nebular Hypothesis.}---Now this is evidently a
good arrangement for a planetary system, and therefore some have inferred
that the Deity \textit{made} it so, perfect from the first. But to one who considers
the way in which other perfect works of nature usually come to their perfection---their
processes of growth and development---this explanation seems
improbable. It appears far more likely that the planetary system \textit{grew} than
that it was \textit{built} outright.

Three different philosophers in the last, century, Swedenborg, Kant, and
La~Place (only one of them an astronomer), independently proposed essentially
the same hypothesis to account for the system as we now know it.
La~Place's theory, as might have been expected from his mathematical and
scientific attainments, was the most carefully and reasonably worked out in
detail. It was formulated before the discovery of the great principle of the
``conservation of energy,'' and before the mechanical equivalence of heat
with other forms of energy was known, so that in some respects it is defective,
%% -----File: 527.png---Folio 516-------
and even certainly wrong. In its main idea, however, that the solar
system once existed as a nebulous mass and has reached its present state as
the result of a series of purely physical processes, it seems certain to prove
correct, and it forms the foundation of all the current speculations upon the
subject.
\end{fineprint}

\nbarticle{909.} \nbparatext{La Place's Theory.}---(\textit{a}) He supposed that at some past time,
which may be taken as the starting-point of our system's history
(though it is not to be considered as \textit{the beginning of the existence of
the substance} of which our system is composed), the matter now collected
in the sun and planets was in the form of a \textit{nebula}.

(\textit{b}) This nebula was a \textit{cloud of intensely heated gas}, perhaps hotter,
as he supposed, than the sun is now.

(\textit{c}) This nebula under the action of its own gravitation, assumed
an approximately globular form with a rotation around an axis. As
to this movement of rotation, it appears to be necessary to account
for it by supposing that the different portions of the nebula, before
the time which has been taken as the starting-point, had motions of
their own. Then, unless these motions happened to be balanced in the
most perfect and improbable manner, a motion of rotation would set
in of itself as the nebula contracted, just as water whirls in a basin
when drawn off by an orifice in the bottom. The velocity of this rotation
would become continually swifter as the volume of the nebula
diminished, the so-called ``moment of momentum'' remaining necessarily
unchanged.

\nbarticle{910.} (\textit{d}) In consequence of this rotation, the mass, instead of
remaining spherical, would become much flattened at the poles, and as
the rotation went on and the motion became accelerated, the time
would come when the centrifugal force at the equator of the nebula
would become equal to gravity, and ``rings of nebulous matter''
would be \textit{abandoned} (not thrown off), resembling the rings of Saturn,
which, indeed, suggested this feature of the theory.

(\textit{e}) A ring would revolve for a while as a whole, but in time would
\textit{break}, and the material would \textit{collect into a single globe}. La~Place supposed
that the ring would revolve as if it were solid, the outer edge,
therefore, moving more swiftly than the inner. If this were so, the
mass formed from the collection of the matter of the ruptured ring
\textit{would necessarily rotate in the same direction} as the ring had revolved.

(\textit{f}) The planet thus formed would continue to revolve around the
central mass, and might itself in turn abandon rings which might
break, and so furnish it with a retinue of satellites.
%% -----File: 528.png---Folio 517-------

\nbarticle{911.} It is obvious that this theory meets completely most of the
conditions of the problem. It explains every one of the facts just
mentioned as demanding explanation in the solar system. Indeed, it
explains them almost \textit{too} well; for as the theory stands it meets a
most serious difficulty in the \textit{exceptional} cases of the planetary system,
such as the anomalous and retrograde revolutions of the satellites of
Uranus and Neptune. Another difficulty lies in the swift revolution
of Phobos (\artref{Art.}{589}), the inner satellite of Mars. According to the
unmodified nebular hypothesis, no planet or satellite could have a
time of revolution less than the time of rotation which the central
body would have, if expanded until its radius becomes equal to the
radius of the satellite's orbit; still less could it have a period shorter
than the central body \textit{now} has.

\sloppy
\nbarticle{912.} \nbparatext{Necessary Modifications.}---The principal modifications which
seem essential to the theory in the light of our present knowledge, are
the following. (The small letters indicate the articles of the original
theory to which reference is made.)

\fussy
(\textit{b}) It is not probable that the original nebula could have been \textit{at a
temperature} even nearly as high as the present temperature of the
sun. The process of condensation of a gaseous cloud from loss of
heat by radiation, would cause the temperature to \textit{rise}, according to
the remarkable and almost paradoxical law of Lane. (\artref{Art.}{357}),
until the mass had begun to liquefy or solidify. And it appears probable
that the original nebula, instead of being \textit{purely gaseous}, was
rather a \textit{cloud of dust} than a ``\textit{fire-mist}''; \textit{i.e.}, that it was made up of
finely divided particles of solid or liquid matter, each particle enveloped
in a mantle of permanent gas. Such a nebula in condensing
would \textit{rise} in temperature at first as if purely gaseous, so that its central
mass after a time would reach the solar stage of temperature, the
solid and liquid particles melting and vaporizing as the mass grew
hotter. At a subsequent stage, when yet more of the original energy
of the mass had been dissipated by radiation, the temperature of the
bodies which were formed from and within the nebula would fall again.

\begin{fineprint}
And yet La Place \textit{may} have been right in ascribing a high temperature to
the original nebula. If that were really the case, the only difference would
be that the nebula would be longer in reaching the condition of a solar
system; but it is not \textit{necessary}, as he supposed, to assume that the original
temperature was high, and that the matter was originally in a purely gaseous
condition, in order to account for the present existence of such a group of
bodies as the incandescent sun and its cool attendant planets.
\end{fineprint}
%% -----File: 529.png---Folio 518-------

\nbarticle{913.} (\textit{d}) As regards the manner in which the planetary bodies
were probably liberated from the parent mass, it seem to be very
doubtful whether the matter accumulated at the equator of the rotating
mass would usually separate itself as a ring. If a plastic mass
in swift rotation is not absolutely homogeneous and symmetrical, it
is more likely to become distorted by a lump formed somewhere on
its equator, which lump may be finally detached and circulate around
its primary. The formation of a \textit{ring}, though possible, would seem
likely to be only a rare occurrence.

La Place seems to have believed also that the outer rings must
necessarily have been abandoned first, and the others in regular succession,
so that the \textit{outer} planets are much the older. It seems, however,
quite possible, and even probable, that several of the planets
may be of about the same age, more than one ring having been
liberated at the same time; or several planets having been formed
from different zones of the same ring.

(\textit{e}) In the case where a ring was formed, it is practically certain
that it could not have revolved as a solid sheet; \textit{i.e.}, with the same
angular velocity for all the particles, and with the outer portions,
therefore, moving more swiftly than the inner. If, for instance, the
matter which now constitutes the earth were ever distributed to form
a ring occupying anything like half the distance from Venus to Mars,
it must have been of a tenuity comparable only to that of a comet.
The separate particles of such a ring could have had very little control
over each other, and must have moved substantially as independent
bodies; the outer ones, like remoter planets, making their circuits
in longer periods and moving \textit{more slowly} than those near the
inner edge of the ring.

\nbarticle{914.} \nbparatext{Explanation of the Anomalous Rotation of Uranus and Neptune.}---When
the matter of such a ring concentrates into a single
mass, the direction of the rotation of the resultant planet depends upon
the manner in which the matter was originally distributed in the ring.
If the ring be nearly of the same density throughout, the resulting
planet (which would be formed at about the middle of the ring's
width) must have a \textit{retrograde} rotation like Uranus and Neptune.
But if the particles of the ring are more closely packed near its
inner edge, so that the resultant planet would be formed much within
the middle of its width, its axial rotation must be \textit{direct.} In the
first case, illustrated in \figref{illo233}{Fig.~233} (\textit{a}), the particles near the \textit{inner edge
of the ring} would control the rotation, having a greater moment of
%% -----File: 530.png---Folio 519-------
rotation with respect to $M$, where the planet is supposed to be formed,
than those at the outer edge. The rotation, therefore, will be
\textit{retrograde}, on account of their greater velocity.

In the other case, \figref{illo233}{Fig.~233} (\textit{b}), where the inner edge of the ring
is densest, and the planet is formed as at $N$, much nearer the inner
than the outer edge of the ring, the aggregate moment of rotation
with respect to $N$ is greater for the \textit{particles beyond $N$} (because of
their greater distance from it) than that of the swifter moving particles
within, and this determines a \textit{direct} rotation.

\includegraphicsmid{illo233}{\textsc{Fig.~233.}---Rotation of Planets formed from Rings according to the Nebular Hypothesis.}

The fact that the satellites of Uranus and Neptune revolve backwards
is not, therefore, at all a bar to the acceptance of the nebular
hypothesis, as sometimes represented. If a new planet should ever
be discovered outside of Neptune it is altogether probable that its
satellites would be found to retrograde.

\begin{fineprint}
This is not the only way in which the retrograde rotation of the outer
planets may be accounted for. There are a number of other possible
assumptions as to the constitution of the mass thrown off from the parent
nebula, and the manner in which its particles ultimately coalesce to form a
planet, which lead to a similar result.

\nbarticle{915.} Faye has recently propounded a modification of the nebular hypothesis
which makes the planets of the ``terrestrial group'' (Mercury, Venus, the
%% -----File: 531.png---Folio 520-------
Earth, and Mars) older than the outer ones. He supposes that the planets
were formed by local condensations (not by the formation of rings) \textit{within}
the revolving nebula. At first, before the nebula was much condensed at the
centre, the inward attraction would be at any point \textit{directly proportional to
the distance of that point from the centre of gravity} of the nebula; \textit{i.e.}, the
force could be expressed by the equation $F = ar$. After the condensation
has gone so far that practically almost the whole of the matter is collected
at the centre of the nebula, the force is \textit{inversely proportional to the square of
the distance},---the ordinary law of gravitation,
\begin{flalign*}
&\textit{i.e., } && F = \frac{b}{r^2}.  &&\phantom{i.e. }
\end{flalign*}
At any intermediate time, during the gradual condensation of the nebula,
the intensity of the central force will, therefore, be given by an expression
having the form
\[
F = ar + \frac{b}{r^2},
\]
$r$ being the distance of the body acted upon from the centre of gravity of
the nebula, while $a$ and $b$ are coefficients which depend upon its age; $a$ continually
decreasing as the nebula grows older, while $b$ increases. The planets
formed within the nebula when it was young, \textit{i.e.}, when $a$ was large and $b$
was small, would have \textit{direct} rotation upon their axes, while those formed
after $a$ had sensibly vanished would have a retrograde rotation; and this he
supposes to be the case with Uranus and Neptune, which he considers
\textit{younger} than the inner planets. Faye's work ``L'Origine du Monde,'' 1885,
contains an excellent summary of the views and theories of the different
astronomers who have speculated upon the cosmogony.
\end{fineprint}

\nbarticle{916.} \nbparatext{Tidal Evolution.}---Within a few years Prof.\ G.~H. Darwin
(son of the great naturalist) has made some important investigations
upon the effect of \textit{tidal reaction} between a central mass and a
body revolving about it, both of them being supposed to be of such
a nature (\textit{i.e.}, not absolutely \textit{rigid}), that tides can be raised upon
them by their mutual attraction. We have already alluded to the
subject in connection with the tides (\artref{Art.}{484}). He finds in this
reaction an explanation of many puzzling facts. It appears, for
instance, that if a planet and its satellite have ever had their times of
rotation of the same length as the time of their orbital revolution
around their common centre of gravity, then, starting from that
time, either of two things might happen,---the satellite might begin
to recede from the planet, or it might fall back to the central mass.
The condition is one of unstable equilibrium, and the slightest cause
might determine the subsequent course of things in either of the
%% -----File: 532.png---Folio 521-------
two opposite directions. Whenever the time of rotation of the
planet is \textit{shorter} than the orbital period of the satellite (as it would
naturally become by condensation continuing after the separation of the
satellite), the tendency would be, as explained in \artref{Article}{484}, slightly
to accelerate the satellite, and so to cause it continually to recede
by an action the reverse of that produced by the hypothetical resisting
medium which is supposed to disturb Encke's comet. This, it
will be remembered, is thought to be the case with our moon.

\nbarticle{917.} But if by any means the rotation of the planet were \textit{retarded},
so that its day should become \textit{longer} than the period of the satellite,
the tides produced by the satellite upon the planet will then retard
the motion of the satellite like a resisting medium, and so will cause
a continual shortening of its period, precisely as in the case of
Encke's comet. If nothing intervenes, this action will in time bring
down the satellite upon the planet's surface. Now in the case of
Mars there is a known cause operating to retard its rotation (namely,
the tides which are raised by the \textit{sun} upon the planet), and those
who accept the theory of tidal evolution suggest that this was the
cause which first made the length of the planet's day to exceed the
period of the satellite, and so enabled the planet to establish upon
the satellite that retardation which has shortened its little month,
and must ultimately bring it down upon the planet.

Processes such as these of tidal evolution must necessarily be
extremely slow. How long are the periods involved, no one can yet
estimate with any precision, but it is certain that the years are to be
counted by the million.

\begin{fineprint}
(We have already referred the reader (\DPtypo{\artref{Art.}{484*}}{\artref{Art.}{484}}) to the last chapter of
Ball's ``Story of the Heavens'' as containing an excellent and easily understood
explanation of this subject.)
\end{fineprint}

\nbarticle{918.} \nbparatext{Conclusions derived from the Theory of Heat.}---As Professor
Newcomb has said, ``Kant and La~Place seem to have arrived at
the nebular hypothesis by reasoning \textit{forwards}. Modern science obtains
a similar result by reasoning \textit{backwards} from actions which we
now see going on before our eyes.''

We have abundant evidence that \textit{the earth} was once at a much
higher temperature than now. As we penetrate below the surface
we find the temperature continually rising at a rate of about 1°\:F.
for every fifty or sixty feet, thus indicating that at the depth of a
few miles the temperature must be far above incandescence. Now,
%% -----File: 533.png---Folio 522-------
since the surface temperature is so much lower, this implies one (or
\textit{both}) of two things,---either that heat-making processes are going
on within the earth (which may be true to some extent), or else that
the earth has been much hotter than it now is, and is cooling off,---and
this seems to be a most probable supposition. It is just as
reasonable, as Sir W.~Thomson puts it, to suppose that the earth
has lately been intensely heated as to suppose that a warm stone that
one picks up in the field has been lately somewhere in the fire.

\nbarticle{919.} \nbparatext{Evidence derived from the Condition of the Moon and
Planets.}---In the case of the \textit{moon} we find a body bearing upon its
surface all the marks of past igneous action, but now in appearance
intensely cold. The \textit{planets}, so far as we can judge from what we
can see through the telescope, corroborate the same conclusion.
Their testimony is not very strong, but it is at least true that nothing
in the aspect of any of them militates against the view that they also
are bodies cooling like the earth; and in the cases of Jupiter and
Saturn many phenomena go to show that they are still (or at least
\textit{now}) at a high temperature,---as might be expected of bodies of
such an enormous mass, which, necessarily, other things being equal,
would cool much more slowly than smaller globes like the earth.

\begin{fineprint}
The ratio of surface to mass is smaller as the diameter of a globe grows
larger, and upon this ratio the rate of cooling of a body largely depends. In
short, everything we can ascertain from the observation of the planets agrees
completely with the idea that they have come to their present condition by
\textit{cooling down from a molten or even gaseous state.}
\end{fineprint}

\nbarticle{920.} \nbparatext{The Sun's Testimony.}---In the sun we have a body steadily
pouring forth an absolutely inconceivable amount of heat, without
any visible source of supply. Thus far the only reasonable hypothesis
to account for this, and for a multitude of other phenomena which
it shows us, is the one which makes it a great cloud-mantled ball
of incandescent gases, slowly shrinking under its own central gravity,
converting continually a portion of its ``potential energy of position''\footnote
  {By ``\textit{potential} energy of position'' is meant the energy due to the separated
  condition of its particles from each other. As they fall together and towards
  the centre in the shrinkage of the sun, they ``do work'' in precisely the same
  way as any falling weight.}
into the \textit{kinetic-energy} of heat, which at present is mainly radiated off
into space.
%% -----File: 534.png---Folio 523-------

\begin{fineprint}
We say \textit{mainly}, because it is not impossible that the sun's temperature is
even yet slowly rising, and that the maximum has not yet been reached.
We are not sure whether \textit{all} the heat produced by the sun's annual shrinkage
is radiated into space, or whether a portion is retained within its mass, thus
raising its temperature; or whether, again, it radiates \textit{more} than the amount
thus generated, so that its temperature is slowly diminishing.

\nbarticle{921.}---That the sun is really shrinking is admittedly only an inference,
for the shrinkage must be far too slow for direct observation. Our case is
like that of a man who, to use one of Professor Newcomb's illustrations,
when he comes into a room and finds a clock in motion, concludes that the
clock-weight is descending, even though its motion is too slow to be observed.
Knowing the construction of the clock and the arrangement of its gearing,
and the number of teeth in each of its different wheels, he states confidently
just how many thousandths of an inch the weight sinks at each vibration of
the pendulum; and looking into the clock-case and measuring the length of
the space in which the weight can move, and noting its present place, he
proceeds to state how long ago the clock was wound up, and how long it has
yet to run. We must not push the analogy too far, but it is in some such
way that we conclude from our measurements of the sun's annual output of
energy in the form of heat, how fast it is shrinking, and we find that its
diameter must diminish not far from 250 feet in a year; at least, the loss of
potential energy corresponding to that amount of shrinkage would account
for one year's running of the solar mechanism.
\end{fineprint}

\nbarticle{922.} \nbparatext{Age of the Solar System.}---Looking backward, then, in
imagination we see the sun growing continually larger through the
reversed course of time, expanding and becoming ever less and less
dense, until at some epoch in the past it filled all the space now
included within the largest orbit of the solar system.

How long ago that was no one can say with certainty. If we could
assume that the amount of potential energy lost by contraction, converted
into the actual energy of heat and radiated into space, has been
the same each year through all the intervening ages, and moreover,
that \textit{all} the heat radiated has come from this source \textit{only}, without subsidy
from any original store of heat contained in an original ``fire
mist,'' or from energy derived from outside sources, then it is not difficult
to conclude that the sun's past history must cover some 15,000000
or 20,000000 years.

But the assumption that the loss of heat has been even nearly uniform
is extremely improbable, considering how high the present temperature
of the sun must be as compared with that of the original
nebula, and how the ratio of surface to solid content has increased
with the lessening diameter.
%% -----File: 535.png---Folio 524-------

Nor is it unlikely that the sun may have received energy from other
sources than its own contraction. Altogether it would seem that we
must consider the 15,000000 years to be the least possible value of a
duration which may have been many times more extended. If the
nebular hypothesis and the theory of the solar contraction be true,
the sun must be as old as that,---how much older no one can tell.

\nbarticle{923.} \nbparatext{Future Prospects.}---Looking forward toward the future, it
is easy to conclude also that at its present rate of radiation and contraction
the sun must, within 5,000000 or 10,000000 years, become so
dense that the conditions of its constitution will be radically changed,
and to such an extent that life on the earth, as we now know life, would
probably be impossible. If nothing intervenes to reverse the course
of things, the sun must at last solidify and become a dark, rigid
globe, frozen and lifeless among its lifeless family of planets. At
least, this is the necessary consequence of what now seems to science
to be the true account of its present activity and the story of its life.

\nbarticle{924.} \nbparatext{Stars, Star-Clusters, and the Nebulę.}---It is obvious that the
same nebular hypothesis applies satisfactorily to the explanation of
the relation of these different classes of bodies to each other. In fact,
Herschel, appealing only to the law of continuity, had concluded before
La~Place formulated his theory, that nebulę develop sometimes into
clusters, sometimes into double or multiple stars, and sometimes into
single ones. He showed the existence in the sky of all the intermediate
forms between the nebula and the finished star. For a time,
about the middle of our century, while it was generally supposed that
all nebulę were nothing but star-clusters, too remote to be resolved by
existing telescopes, his views fell rather into abeyance; but when the
spectroscope demonstrated the substantial differences between the
gaseous nebulę and the star-clusters, they regained acceptance in
their essential features; with perhaps the reservation, that many are
disposed to believe that the rarest even of nebulous matter, instead
of being purely gaseous, is full of solid and liquid particles like a
cloud of fog or smoke.

\nbarticle{925.} \nbparatext{The Present System not Eternal.}---One lesson seems to
stand out clearly,---that the present system of stars and worlds is
not an eternal one. We have before us irrefragable evidence of
continuous, uncompensated progress, inexorable in one direction.
The hot bodies are losing their heat, and distributing it to the cold
ones, so that there is a steady, unremitting tendency towards a
%% -----File: 536.png---Folio 525-------
uniform (and therefore useless) temperature throughout the universe:
for heat does \textit{work}, and is \textit{available} as energy \textit{only when it
can pass from hotter to cooler bodies}, so that this warming up of
cooler bodies at the expense of hotter ones always involves a loss,
not of energy (for that is indestructible), but of \textit{available} energy.
To use the technical language now usually employed, energy is
unceasingly ``\textit{dissipated}'' by the processes which maintain the
present life of the universe; and this dissipation of energy can
have but one ultimate result,---that of absolute stagnation when a
uniform temperature has been everywhere attained. If we carry our
imagination backwards we reach at last a ``beginning of things,''
which has no intelligible antecedent: if forwards, an end of things in
stagnation. That by some process or other this end of things will
result in ``new heavens and a new earth'' we can hardly doubt,
but science has as yet no word of explanation.

\nbarticle{926.} \nbparatext{Mr.~Lockyer's Meteoric Hypothesis.}---The idea that the
heavenly bodies in their present state may have been formed by the
aggregation of \textit{meteoric} matter, rather than by the condensation of a
\textit{gaseous} mass, is not new, and not original with Mr.~Lockyer, as he himself
points out. But his adoption and advocacy of the theory, and the
support he brings to it from spectroscopic experiments on the light
emitted by fragments of meteoric stones under different conditions,
has given it such currency within the last two years that his name
will always be justly associated with it. We have already referred
to it in several places (Arts.~\arnref{850} and \arnref{894} especially).

He believes that he finds in the spectra of meteorites, under various
conditions, an explanation\footnote
  {In some cases the explanation appears to be at least doubtful, especially in
  instances where the presence of a sharply defined \textit{line} in the spectrum of a heavenly
  body is referred to the degradation of a \textit{band} observed in the meteoric
  spectrum.}
of the spectra of comets, nebulę, and
all the different types of stars, as well as the spectra of the Aurora
Borealis and the Zodiacal Light.

Assuming this, he considers that \textit{nebulę} are meteoric swarms in
the initial stages of condensation, the separate individuals being still
widely separated, and collisions comparatively infrequent.

As aggregation goes on, the nebulę become \textit{stars}, which run through
a long life-history, the temperature first increasing slowly to a maximum,
and then falling to non-luminosity. During this life-history
the stars pass through successive stages, each stage characterized by
%% -----File: 537.png---Folio 526-------
its own typical spectrum; and according to these views he divides the
stars into six spectroscopic classes, as follows:---

I\@. Those which, like the nebulę, show \textit{bright lines or flutings} in the
spectrum, without \textit{dark} lines or bands. Vogel's II.~(\textit{b}) and I.~(\textit{c})
(\artref{Art.}{858}) are included in this class. $\gamma$~Cassiopeię and 152~Schjellerup
are types.

II\@. Those which show both \textit{dark lines and dark flutings} in the spectrum.
Vogel's III. (\textit{a}): ($\alpha$~Orionis).

III\@. Those which show the \textit{fine dark lines} of metals. Vogel's
II.~(\textit{a}) is included here \textit{in part}. $\alpha$~Aurigę and the sun are typical.

IV\@. Those whose spectra are characterized by \textit{the conspicuous hydrogen
lines}, all other lines and markings being faint. These stars are at
the summit of the temperature curve. Vogel's I.~(\textit{a}): Sirius and Vega.

V\@. This class, on the \textit{descending} branch of the temperature curve
(stars past middle life), should have sensibly the same spectrum as
these of Class~III\@. We cannot be sure in which of the two classes
our sun, for instance, should be counted.

VI\@. The \textit{red stars}, their spectrum characterized by heavy absorption
bands. These are stars verging to extinction according to Mr.~Lockyer's
view. Vogel's III.~(\textit{b}).

It is impossible to go into detail and give here Mr.~Lockyer's ingenious
applications of the theory to explain the phenomena of the
different classes of variable stars. (See \artref{Art.}{850}, however.)

\sloppy
This hypothesis has recently been much strengthened by a most
interesting mathematical investigation of Prof.\ G.~Darwin, who
shows that, if we assume a meteoric swarm comparable in dimensions
with our solar system, composed of individual masses such
as fall on the earth, and endowed with such velocities as meteors are
known to have, such a swarm, seen from the distance of the stars,
\textit{would behave like a mass composed of a continuous gas.} This is not
strange since, according to the kinetic theory of gases, a gas is
simply a swarm of \textit{molecules}, behaving in just the way the meteorites
are supposed to act. But it follows that the \textit{meteoric} theory of a
nebula does not in the least invalidate, or even to any great extent
modify, the reasoning of La~Place in respect to the development of
suns and systems from a \textit{gaseous} nebula. The old hypothesis has no
quarrel with the new.

While it would be premature to indorse this speculation of Mr.~Lockyer's
as an established discovery (since there remain in it many
obscure and doubtful points), there can be little doubt that it marks
an epoch in the history of the science.
\chelabel{CHAPTERXXI}

%% -----File: 538.png---Folio 527-------
\fussy
\clearpage
\pdfbookmark[1]{Appendix. Tables of Astronomical Data}{Appendix. Tables of Astronomical Data}
\chapter*{APPENDIX.}
\chslabel{APPENDIX}
\markboth{APPENDIX.}{}
\thispagestyle{empty}
\nbrule\bigskip

\small
\begin{center}

\nblabel{app:greek}
\nbappheading{THE GREEK ALPHABET.}
\medskip

\begin{tabular}{@{}ll @{\qquad} ll @{\qquad} ll@{}}
\scriptsize Letters. & \scriptsize Name. & \scriptsize Letters.  & \scriptsize Name.& \scriptsize Letters.    & \scriptsize Name.\\
A, $\alpha$,         & Alpha.            & I, $\iota$,           & Iota.            & P, $\rho$, $\varrho$,   & Rho.\\
B, $\beta$,          & Beta.             & K, $\kappa$,          & Kappa.           & $\Sigma$, $\sigma$, $\varsigma$, & Sigma.\\
$\Gamma$, $\gamma$,  & Gamma.            & $\Lambda$, $\lambda$, & Lambda.          & T, $\tau$,              & Tau.\\
$\Delta$, $\delta$,  & Delta.            & M, $\mu$,             & Mu.              & $\Upsilon$, $\upsilon$, & Upsilon.\\
E, $\epsilon$,       & Epsilon.          & N, $\nu$,             & Nu.              & $\Phi$, $\phi$,         & Phi.\\
Z, $\zeta$,          & Zeta.             & $\Xi$, $\xi$,         & Xi.              & X, $\chi$,              & Chi.\\
H, $\eta$,           & Eta.              & O, $o$,               & Omicron.         & $\Psi$, $\psi$,         & Psi.\\
$\Theta$, $\theta$, $\vartheta$, &Theta. & $\Pi$, $\pi$, $\varpi$, & Pi.            & $\Omega$, $\omega$,     & Omega.\\
\end{tabular}

\medskip
\nbapprule
\medskip

\nblabel{app:misc}
\nbappheading{MISCELLANEOUS SYMBOLS.}
\medskip

\begin{tabular}{l@{\hspace{4em}}l}
  \conjunction, Conjunction.   & A.R., or $\alpha$, Right Ascension.\\
  $\Box$, Quadrature.              & Decl., or $\delta$, Declination.\\
  \opposition, Opposition.     & $\lambda$, Longitude (Celestial).\\
  $\nbAscnode$, Ascending Node.    & $\beta$, Latitude (Celestial).\\
  $\nbDescnode$, Descending Node.  & $\phi$, Latitude (Terrestrial).\\
  \multicolumn{2}{p{.8\textwidth}}{\hangindent=2.5em $\omega$, Angle between line of nodes and line of apsides. Also
obliquity of the ecliptic.}
\end{tabular}
\end{center}

\DPtypo{}{\nbapprule
\medskip}

{\centering%\medskip
\nblabel{app:spheroid}
\nbappheading{DIMENSIONS OF THE TERRESTRIAL SPHEROID.}

\footnotesize(According to Clarke's Spheroid of 1878.\\For the spheroid of 1866, see \artref{Art.}{145}.)\par}

Equatorial \DPtypo{semidiameter}{semi-diameter},---

{\centering $20\,926\,202 \text{ feet} = 3963.296 \text{ miles} = 6\,378\,190 \text{ metres.}$\par}

Polar \DPtypo{semidiameter}{semi-diameter},---

{\centering $20\,854\,895 \text{ feet} = 3949.790 \text{ miles} = 6\,356\,456 \text{ metres.}$

\smallskip
Ellipticity, or Polar Compression, $\frac{1}{293.46}$.\par}

\medskip
\nbapprule
\medskip

Length (in metres) of $1°$ of meridian in lat.\ $\phi =$\\
  \hspace*{6em}$111\,132.09 - 556.05 \cos{2\phi} + 1.20\cos{4\phi}$.

Length (in metres) of $1°$ of parallel, in lat.\ $\phi =$\\
  \hspace*{6em}$111\,415.10 \cos{\phi} - 94.54\cos{3\phi}$.
%% -----File: 539.png---Folio 528-------

\medskip
\begin{tabular}{@{} l@{ }r@{ }l}
  $1°$ of lat.\ at &    pole &$= 111\,699.3$ metres $= 69.407$ miles. \\
  $1°$ of lat.\ at & equator &$= 110\,567.2$ metres $= 68.704$ miles.
\end{tabular}

\medskip
These formulę correspond to the Clarke Spheroid of 1866, used by
the U.S. Coast and Geodetic Survey.

\medskip
\nbapprule

{\centering\medskip
\nblabel{app:time}
\nbappheading{TIME CONSTANTS.}\par}

\medskip
\begin{tabular}{@{} l@{ }l}
  The sidereal   day &$= 23^\text{h}\: 56^\text{m}\:  4^\text{s}.090$ of mean solar time.\\
  The mean solar day &$= 24^\text{h}\:  3^\text{m}\: 56^\text{s}.556$ of sidereal time.
\end{tabular}

\medskip
To reduce a time-interval expressed in units of \textit{mean solar time} to
\textit{units of sidereal time}, multiply by 1.00273791; Log.\ of $0.00273791
= [7.4374191]$.

\medskip
To reduce a time-interval expressed in units of \textit{sidereal time} to
units of \textit{mean solar time}, multiply by $0.99726957 = (1 - 0.00273043)$;
Log.\ $0.00273043 = [7.4362316]$.

\medskip
\begin{tabular}{@{} l@{}l@{}l@{}l@{ } l@{\:}l@{\:}l@{\:} r@{.}l}
Tropical year (Lev&errier, red&uced to 19&00),
& $365^\text{d}$ & $5^\text{h}$ & $48^\text{m}$ & $45^\text{s}$&51.
\\
Sidereal year    &  ``  &  ``  &  ``
& 365 & 6 &\phantom{0}9 & 8&97.
\\
Anomalistic \rlap{year} &  ``  &  ``  &  ``
& 365 & 6 &13 & 48&09.
\end{tabular}

\medskip
\begin{tabular}{@{} l l@{\:}l@{\:}l@{\:} r}
Mean synodical month (new moon to new),
& $29^\text{d}$ & $12^\text{h}$ & $44^\text{m}$ & $2^\text{s}.684$.
\\
Sidereal month, \dotfill
&   27 & \phantom{0}7 & 43 &  11.545.
\\
Tropical month (equinox to equinox),\dotfill
&    27 & \phantom{0}7  &  43 &  4.68.
\\
Anomalistic month (perigee to perigee),\dotfill
&  27 &  13 &  18 &  37.44.
\\
Nodical month (node to node),  \dotfill
& 27 & \phantom{0}5 & \phantom{0}5 &  35.81.
\end{tabular}

\bigskip
\nbapprule
\medskip

Obliquity of the ecliptic (Leverrier),\\
  \hspace*{\fill}$23°\: 27'\: 08''.0 - 0''.4757 (t-1900)$.

Constant of precession (Struve), \hfill $50''.264 + 0''.000227\, (t-1900)$.

Constant of notation (Peters), \hfill \PadTo[l]{$50''.264 + 0''.000227\, (t-1900)$.}{$9''.223$.}

Constant of aberration (Nyrén), \hfill \PadTo[l]{$50''.264 + 0''.000227\, (t-1900)$.}{$20''.492$.}

\clearpage
%% -----File: 540.png---Folio 529-------

\begin{sidewaystable}
\centering
\tiny
\noindent
\renewcommand{\arraystretch}{1.1}
\nblabel{app:I}
\begin{tabular}{l|l|c| D{,}{}{-1}|D{,}{}{-1}|D{.}{.}{-1}|D{,}{}{-1}| c|c| *{3}{D{,}{}{-1}|} D{,}{}{-1}}
\multicolumn{13}{c}{\footnotesize TABLE I.---PRINCIPAL ELEMENTS OF THE SOLAR SYSTEM.}\\[2ex]
\hline \hline
  \multicolumn{2}{c|}{\textsc{Name.}}
& \multicolumn{1}{m{1em}|}{\rotatebox{90}{\textsc{Symbol.}}}
& \multicolumn{1}{m{5.5em}|}{\centering Semi-Major Axis of Orbit.}
& \multicolumn{1}{m{4em}|}{\centering Mean Dist.\\ Millions of Miles.}
& \multicolumn{1}{m{6em}|}{\centering \rule{0pt}{3ex}Sidereal Period\\ (mean solar days).\rule[-2ex]{0pt}{3ex}}
& \multicolumn{1}{m{3em}|}{\centering Period in Years.}
& \multicolumn{1}{m{5em}|}{\centering Orbit-Velocity\\ (miles per second).}
& \multicolumn{1}{m{4em}|}{\centering Eccen\-tricity.}
& \multicolumn{1}{m{5em}|}{\centering Inclination to Ecliptic.}
& \multicolumn{1}{m{5em}|}{\centering Longitude of Ascending Node.}
& \multicolumn{1}{m{5em}|}{\centering Longitude of Perihelion.}
& \multicolumn{1}{m{5em} }{\centering Longitude at Epoch, Jan.~1, 1850.}
\\
\hline
\multirow{4}{*}{\rotatebox{90}{\parbox{4.55em}{\centering Terrestrial Planets.}}}
& Mercury\rule{0pt}{3ex} & \Mercury
& 0.38,7099 & 3,6.0 & 87.96926 & 0,.24
& 23 to 35 & .20560
&   7\rlap{°}\ 0,0\rlap{$'$}\ \phantom{0}8''
&  46\rlap{°}\ 3,3\rlap{$'$}\ \phantom{0}9''
&  75\rlap{°}\ \phantom{0},7\rlap{$'$}\ 14''
& 327\rlap{°}\ 1,5\rlap{$'$}\ 20''
\\
& Venus & \Venus
& 0.72,3332 & 6,7.2 & 224.7008 & 0,.62
& 21.9 & .00684
& 3\ 2,3\ 35 & 75\ 1,9\ 52 & 129\ 2,7\ 15 & 245\ 3,3\ 15
\\
& The Earth & $\oplus$
& 1.00,0000 & 9,2.9 & 365.2564 & 1,.00
& 18.5 & .01677
& 0\ 0,0\ 00 & 0\ 0,0\ 00 & 100\ 2,1\ 22 & 100\ 4,6\ 44
\\
& Mars & \Mars
& 1.52,3691 & 14,1.5 & 686.9505 & 1,.88
& 15.0 & .09326
& 1\ 5,1\ \phantom{0}2 & 48\ 2,3\ 53 & 333\ 1,7\ 54 & 103\ 2,5\ \phantom{0}3  \\[1ex]
\hline

& Ceres\rule{0pt}{3ex} & \textcircled{\raisebox{-0.25ex}{1}}
& 2.76,7265 & 25,7.1 & 1681.414 & 4,.60
& 11.1 & .07631
& 10\ 3,7\ 10 & 30\ 4,6\ 39 & 149\ 3,7\ 49 & 83\ 4,0\ 31 \\[1ex]
\hline
\multirow{4}{*}{\rotatebox{90}{\parbox{5em}{\centering Major Planets.}}}
& Jupiter\rule{0pt}{3ex} & \Jupiter
& 5.20,2800 & 48,3.3 & 4332.580 & 11,.86
& \phantom{0}8.1 & .04825
& 1\ 1,8\ 41 & 98\ 5,6\ 17 & 11\ 5,4\ 58 & 160\ \phantom{0},1\ 19
\\
& Saturn & \Saturn
& 9.53,8861 & 88,6.0 & 10759.22 & 29,.46
& \phantom{0}6.0 & .05607
& 2\ 2,9\ 40 & 112\ 2,0\ 53 & 90\ \phantom{0},6\ 38 & 14\ 5,2\ 28
\\
& Uranus & \Uranus
& 19.18,329 & 178,1.9 & 30686.82 & 84,.02
& \phantom{0}4.2 & .04634
& 0\ 4,6\ 20 & 73\ 1,3\ 54 & 170\ 5,0\ \phantom{0}7 & 29\ 1,7\ 51
\\
& Neptune & \Neptune
& 30.05,508 & 279,1.6 & 60181.11 & 164,.78
& \phantom{0}3.4 & .00896
& 1\ 4,7\ \phantom{0}2 & 130\ \phantom{0},6\ 25 & 45\ 5,9\ 43 & 334\ 3,3\ \ 29 \\[1ex]
\hline \hline
\end{tabular}
\bigskip

\setlength{\tabcolsep}{3pt}
\noindent
\begin{tabular}{l|l|c|l| D{,}{,}{-1}|D{.}{.}{-1}|c|c| c|c|c|l| D{,}{}{-1}|c|c|c}
\hline \hline
  \multicolumn{2}{c|}{\multirow{2}{3em}{\centering \textsc{Name.}}}
& \multirow{2}{*}{\rotatebox{90}{\textsc{Symbol.}}}
& \multicolumn{1}{c|}{\multirow{2}{5em}{\centering Apparent Angular Diameter.}}
& \multicolumn{2}{c|}{\textsc{Mean Diameter}\rule{0pt}{4ex}}
& \multicolumn{2}{c|}{\textsc{Mass.}}
& \multicolumn{1}{c|}{\textsc{Volume.}}
& \multicolumn{2}{c|}{\textsc{Density.}}
& \multicolumn{1}{c|}{\multirow{2}{5em}{\centering Time of Axial Rotation.}}
& \multicolumn{1}{c|}{\multirow{2}{4.8em}{\centering Inclination Equator to Orbit.}}
& \multicolumn{1}{c|}{\multirow{2}{4em}{\centering Oblate\-ness, or Elliptic\-ity.}}
& \multicolumn{1}{c|}{\multirow{2}{4em}{\centering Gravity at Surface.}}
& \multicolumn{1}{c }{\multirow{2}{4em}{\centering Albedo.}}
\\[2ex]
\cline{5-11}
  \multicolumn{2}{c|}{} &&
& \multicolumn{1}{c|}{in Miles.\rule{0pt}{4ex}}
& \multicolumn{1}{c|}{$\oplus=1$.}
& \multicolumn{1}{c|}{$\astrosun=1$.}
& \multicolumn{1}{c|}{$\oplus=1$.}
& \multicolumn{1}{c|}{$\oplus=1$.}
& \multicolumn{1}{c|}{$\oplus=1$.}
& \multicolumn{1}{c|}{Water $=1$.}
&&&&&
\\[2ex]
\hline
& Sun       & \astrosun\rule{0pt}{3ex}
& $32'\: 04''$ mean & 866,400 & 109.4 & 1.00000 & 331100 & 1\:310\:000
& 0.25 & 1.39 & $25^\text{d}\: 7^\text{h}\: 48^\text{m}$
& 7\text{°}\: 1,5' & ? & \llap{2}7.65
\\
& Moon      & \leftmoon
& $31'\: 07''$ \quad ``   & 2,163 & 0.273 & $\frac{1}{26821000}$ & $\frac{1}{81}$ & 0.020
& 0.61 & 3.40 & $27\phantom{^d}\: 7\phantom{^d}\: 43$
& 6\phantom{\text{°}}\: 3,3 & ? & $\frac{1}{6}$ & 0.17 \\[1ex]
\hline
\multirow{4}{*}{\rotatebox{90}{\parbox{5em}{\centering Terrestrial Planets.}}}
& Mercury   & \Mercury\rule{0pt}{3ex}
& $5''$ to $13''$  & 3,030 & 0.382 & $\frac{1}{2668700}$? & $\frac{1}{8}$? & 0.056
& 2.23\rlap{?} & 12.44? & \qquad ?
& ,? & ?  & 0.85 & 0.13
\\
& Venus     & \Venus
& $11''$ to $67''$ &  7,700 & 0.972 & $\frac{1}{425000}$ &  0.78 & 0.920
& 0.86 & 4.85 & $23^\text{h}\: 21^\text{m}\: 23^\text{s}$
& ,? & ? & 0.83 & 0.50
\\
& The Earth & $\oplus$
& \qquad $\ldots$   & 7,918 & 1.000 & $\frac{1}{331100}$ & 1.000
& 1.000 & 1.00 & 5.58 & $23\phantom{^d}\: 56\: \phantom{0}4.09$
& 23\text{°}\: 2,7'\: 12'' & $\frac{1}{203}$ & 1.00 & 0.20\rlap{?}
\\
& Mars      & \Mars
& $3''.6$ to $24''.5$ &  4,230 & 0.534 & $\frac{1}{3093500}$ & $\frac{1}{9.31}$ & 0.152
& 0.72 & 4.01 & $24\phantom{^d}\: 37\: 22.67$
& 24\phantom{\text{°}}\: 5,0 & $\frac{1}{220}$ & 0.38 & 0.26 \\[1ex]
\hline
& Ceres     &  \textcircled{\raisebox{-0.25ex}{1}}\rule{0pt}{3ex}
& \qquad \ldots & \multicolumn{1}{c|}{100?} & \multicolumn{1}{c|}{\ldots}
& \ldots?  & $\frac{1}{70000}$?
& $\frac{1}{50000}$? & ? & ? & \qquad ? & ,? & ? & ? \\[1ex]
\hline
\multirow{4}{*}{\rotatebox{90}{\parbox{5em}{\centering Major Planets.}}}
& Jupiter   & \Jupiter\rule{0pt}{3ex}
& $32''$ to $50''$  & 86,500 & 10.92 & $\frac{1}{1048}$ & 316.0 & 1309
& 0.24 & 1.33 & $\phantom{0}9^\text{h}\: 55^\text{m}\ ±$
& 3\text{°}\: 0,5' & $\frac{1}{1710}$ & 2.65 & 0.62
\\
& Saturn    & \Saturn
& $14''$ to $20''$ & 73,000 & 9.17 & $\frac{1}{3490}$ & \phantom{0}94.9 & \phantom{0}760
& 0.13 & 0.72 & $10^\text{h}\: 14^\text{m}\: 24^\text{s}$
& 26\phantom{\text{°}}\: 4,9 & $\frac{10}{92}$ & 1.18 & 0.52
\\
  & Uranus    & \Uranus
& $3''.8$ to $4''.1$ & 31,900 & 4.03 & $\frac{1}{22600}$ & \phantom{0}14.7 & \phantom{00}65
& 0.22 & 1.22 & \qquad ? & ,? & $\frac{1}{11}$ & 0.91 & 0.64
\\
& Neptune   & \Neptune
& $2''.7$ to $2''.9$ & 34,800 & 4.39 & $\frac{1}{19380}$ & \phantom{0}17.1 & \phantom{00}85
& 0.20 & 1.11 & \qquad ? & ,? & ? & 0.88 & 0.46 \\[1ex]
\hline \hline
\end{tabular}
\renewcommand{\arraystretch}{1}
\end{sidewaystable}

%% -----File: 541.png---Folio 530-------
%% -----File: 542.png---Folio 531-------
\clearpage

\begin{sidewaystable}
\centering
\tiny
\noindent
\setlength{\tabcolsep}{1pt}
\nblabel{app:II}
\begin{tabular}{c| l@{}| l@{~}c| D{.}{.}{-1}| D{,}{}{-1}| *{3}{r@{}r@{\:}}r@{}l |*{3}{r@{}r@{\:}}r@{}l| *{2}{r@{}l@{\ }}r@{}l@{\,} | c| D{,}{}{-1}| r@{}c@{\,} | c | l@{}}
\multicolumn{34}{c}{\footnotesize TABLE II.---THE SATELLITES OF THE SOLAR SYSTEM.}\\[2ex]
\hline \hline

& & \rule{0pt}{3ex}
&& \multicolumn{1}{c|}{\multirow{3}{7em}{\centering Dist.\ in Equa\-torial Radii of Planet}}
& \multicolumn{1}{c|}{\multirow{3}{4em}{\centering Mean Distance in Miles.}}
&&&&&&&&
&&&&&&&&
& \multicolumn{6}{c|}{\multirow{3}{5em}{\centering Inc.\ of Orbit to Ecliptic.}}
& \multicolumn{1}{c|}{\multirow{3}{6em}{\centering Inc.\ to Plane of Planet's Orbit.}}
& \multicolumn{1}{c|}{\multirow{3}{3em}{\centering Eccen\-tricity.}}
& \multicolumn{2}{c|}{\multirow{3}{2em}{\scalebox{0.9}{\rotatebox{90}{\parbox{3.5em}{\centering Diam'r in Miles}}}}}
%& \multicolumn{2}{c|}{\multirow{3}{2em}{\scalebox{0.9}{\rotatebox{90}{\parbox{3.5em}{\centering Diam'r in Miles}}}}}
& \multicolumn{1}{c|}{\multirow{3}{4em}{\centering Mass in Terms of Primary.}}
&
\\
& \multicolumn{1}{c|}{\textsc{Name}.}
& \multicolumn{2}{c|}{Discovery.} &&
& \multicolumn{8}{c|}{Sidereal Period.}
& \multicolumn{8}{c|}{Synodic Period.}
&&&&&&&&&&&
& \multicolumn{1}{c }{Remarks.}
\\
&&&& &&&&&&&&&
 &&&&&&&&&&&&&&&&&&~&&~% ~ needed to help LaTeX realise it needs to draw column separator lines in the third row of the header
\\[1ex]
\hline

  & Moon \dotfill \rule{0pt}{3ex}
& \multicolumn{2}{@{\ }c@{\ }|}{\dotfill} & 60.27035 & 23,8\,840
& 27&$^\text{d}$ &  7&$^\text{h}$ & 43&$^\text{m}$ & 11&$^\text{s}.5$
& 29&$^\text{d}$ & 12&$^\text{h}$ & 44&$^\text{m}$ & 2&$^\text{s}.7$
&  5&$°$ & 08&$'$ & 40&$''$
& \multicolumn{1}{c|}{ --- \qquad --- }
& 0.05,491 &  2162&& $\frac{1}{81}$
& \parbox{7.5em}{\hspace{0.5em}Specific gravity\\ 3.44.}
\\[1ex]
\hline
\multicolumn{34}{c}{\rule[3.5ex]{0pt}{0pt}SATELLITES OF MARS}
\\[1ex]
\hline
1 & Phobos \dotfill \rule{0pt}{3ex}
& Hall, & 1877 &   2.771 &  ,5\,850
& &&  7&$^\text{h}$ & 39&$^\text{m}$ & 15&$^\text{s}.1$
& \multicolumn{8}{c|}{ --- \qquad --- }
& 26&$°$ & 17&$'$\rlap{.2} &&
& $28°$ & ,0 & 7&? &  ?
& \raisebox{0.8ex}{\multirow{2}{8em}{\hspace{0.5em}Orbits sensibly coincident with planet's equator.}}
\\
2 & Deimos \dotfill \rule{0pt}{3ex}
& \quad `` & ``  &   6.921 &  1,4\,650
& 1&$^\text{d}$ & 6&  & 17&  & 54&.0
& \multicolumn{8}{c|}{ --- \qquad --- }
& 25&           & 47&\rlap{.2}           &&
&  $28°$\rlap{$ \pm$}
& ,0 & 5&? & ?  &
\\[1ex]
\hline
\multicolumn{34}{c}{\rule[3.5ex]{0pt}{0pt}SATELLITES OF JUPITER} \\[1ex]
\hline
1 & Io \dotfill \rule{0pt}{3ex}
&  Galileo, & 1610 &   5.933  &  26,1\,000
&  1&$^\text{d}$ & 18&$^\text{h}$ & 27&$^\text{m}$ & 33&$^\text{s}.5$
& 1&$^\text{d}$ & 18&$^\text{h}$ & 28&$^\text{m}$ & 35&$^\text{s}.9$
& 2&$°$ & 08&$'$  & 3&$''$
& --- \qquad ---   &  ,0 & 2500&   & .00001688
& \multirow{4}{8em}{\hspace{0.5em}The diameters are Englemann's. The rest of the data are from Damoiseau.}
\\
2 & Europa \dotfill \rule{0pt}{3ex}
& \quad `` & `` &   9.430 & 41,5\,000
&  3&  & 13&  & 13&  &  42&.1
&  3&  & 13&  & 17&  &  53&.7
&  1&  & 38&  &  57&
& --- \qquad ---  &  ,0 & 2100&   & .00002323   &
\\
3 & Ganymede \dotfill \rule{0pt}{3ex}
& \quad `` & `` &  15.057 &   66,4\,000
&  7&  & 3&  & 42&  & 33&.4
&  7&  & 3&  & 59&  & 35&.9
&  1&  & 59&  & 53&
& --- \qquad ---  & .0,013 & 3550&   & .00008844   &
\\
4 & Callisto \dotfill \rule{0pt}{3ex}
& \quad `` & `` &  26.486 & 1\,16,7\,000
& 16&   & 16&   & 32&   & 11&.2
& 16&   & 18&   &  5&   &  6&.9
&  1&   & 57&   & 00&
& --- \qquad ---  & .0,072 & 2960&  & .00004248  &
\\[1ex]
\hline
\multicolumn{34}{c}{\rule[3.5ex]{0pt}{0pt}SATELLITES OF SATURN} \\[1ex]
\hline
1 & Mimas \dotfill \rule{0pt}{3ex}
& W. Herschel, & 1789 &  3.11 &  11,7\,000
&  &  & 22&$^\text{h}$ & 37&$^\text{m}$ & 5&$^\text{s}.7$
& \multicolumn{8}{l|}{Long.\ of Ascend.}
& 28&$°$   & 10&$'$   &10&$''$
& \multicolumn{1}{l|}{About $27°$.} &  ,0 &  600&? & ?
& \multirow{5}{7em}{\hspace{0.5em}The planes of the 5 inner orbits sensibly coincide with the plane of the ring.}
\\
2 & Enceladus \dotfill \rule{0pt}{3ex}
& \quad `` \qquad `` & `` &  3.98 & 15,7\,000
& 1&$^\text{d}$ & 8&  &  53&  &  6&.9
& \multicolumn{8}{l|}{Node of orbits on}   &&& ``  &&&
& \multicolumn{1}{l|}{Inclination of the} &  ,0 &  800&? & ?  &
\\
3 & Tethys \dotfill \rule{0pt}{3ex}
& J. D. Cassini, & 1684 &  4.95 &  18,6\,000
& 1&  &21&  &18&  & 25&.6
& \multicolumn{8}{l|}{ecliptic for 1900*,} &&& ``  &&&
& \multicolumn{1}{l|}{5 inner satellites}  &  ,0  & 1100&? &  ?  &
\\
4 & Dione \dotfill \rule{0pt}{3ex}
& \quad `` \qquad `` & ``  &  6.34 &  23,8\,000
& 2&  &17&  & 41&  & 9&.3
& \multicolumn{8}{l|}{$168°\: 10'\: 35''$.}
&&& ``  &&&
& \multicolumn{1}{l|}{to plane of celes-}   &   ,0 & 1200&? & ?  &
\\
5 & Rhea \dotfill \rule{0pt}{3ex}
& \quad `` \qquad ``  & 1672  &  8.86 &  33,2\,000
& 4&  & 12&  & 25&  & 11&.6
& \multicolumn{8}{l|}{(5 inner satellites}  &&& `` &&&
& \multicolumn{1}{l|}{tial equator =}  &   ,0 &  1500&?  & ?  &
\\
6 & Titan \dotfill \rule{0pt}{3ex}
& Huyghens, &  1655  &  20.48 &  77,1\,000
&  13&  & 22&  & 41&   & 23&.2
& \multicolumn{8}{l|}{and ring.)}
& 27&  & 38&   & 49&
&\multicolumn{1}{l|}{$6°\: 57'\: 43''$ (1900)}
&  0.0,299 & 3500&?  & $\frac{1}{4600}$
& \multirow{3}{7.5em}{$\left\{ \parbox{6.5em}{\hspace{0.5em}Discovered independently by Lassell.} \right.$}
\\
7 & Hyperion \dotfill \rule{0pt}{3ex}
& G. P. Bond, &  1848  &  25.07 &  93,4\,000
&  21&  &   6&   & 39&   & 27&.0
&&&&&&&&
& 27&  & 4&.8  &&
& --- \qquad ---  &  0.1,189 &  500&?  &  ?  &
\\
8 & Iapetus \dotfill \rule{0pt}{3ex}
& J. D. Cassini,  &  1671  &  59.58  & 2\,22,5\,000
&  79&   &  7&   & 54&   &17&.1
&&&&&&&&
& 18&  & 31&.5  &&
& --- \qquad ---  &  0.0,296 & 2000&? & ?  &
\\[1ex]
\hline
\multicolumn{34}{c}{\rule[3.5ex]{0pt}{0pt}SATELLITES OF URANUS} \\[1ex]
\hline
1 & Ariel  \dotfill \rule{0pt}{3ex}
& Lassell,  &  1851 &   7.52 &  12,0\,000
&  2&$^\text{d}$ & 12&$^\text{h}$ & 20&$^\text{m}$ & 21&$^\text{s}.1$
& \multicolumn{8}{l|}{Long.\ of Ascend.}
&  97&$°$ & 51&$'$ &  &
\phantom{$'$}\llap{\raisebox{-2ex}[0ex][0ex]{$=-82°\: 09'$}}
&  & ,0 & 500&?  &  ? &
\\
2 & Umbriel  \dotfill \rule{0pt}{3ex}
& \quad `` & `` & 10.46 &  16,7\,000
&  4&   &  3&   &  27&  & 37&.2
& \multicolumn{8}{l|}{Node of orbits on}
& \multicolumn{2}{r}{``} &\multicolumn{2}{c }{ ``}  &&
& \multicolumn{1}{l|}{Inc.\ to celestial} &  ,0 & 400&? & ?
& \hspace{0.5em}Retrograde.
\\
3 & Titania \dotfill \rule{0pt}{3ex}
& W. Herschel, & 1787 &  17.12  &  27,3\,000
&  8&   &  16&   & 56&   & 29&.5
& \multicolumn{8}{l|}{plane of ecliptic}
& \multicolumn{2}{r}{``} &\multicolumn{2}{c }{ ``}  &&
&\multicolumn{1}{l|}{equator $75°\: 18'$} & ,0 & 1000&? & ? &
\\
4 & Oberon \dotfill \rule{0pt}{3ex}
& \quad `` \qquad ``  & ``  &  22.90 &  36,5\,000
&  13&   &  11&   &  7&   & 6&.4
& \multicolumn{8}{l|}{$= 165°\: 32'$ (1900).}
& \multicolumn{2}{r}{``} &\multicolumn{2}{c }{ ``}  &&
& \multicolumn{1}{l|}{(1900).} & ,0 & 800&? & ? &
\\[1ex]
\hline
\multicolumn{34}{c}{\rule[3.5ex]{0pt}{0pt}SATELLITE OF NEPTUNE} \\[1ex]
\hline
1 & Nameless \dotfill \rule{0pt}{3ex}
& Lassell,  &  1846 &  12.93  & 22,5\,000
&  5&$^\text{d}$ &  21&$^\text{h}$ &  2&$^\text{m}$ & 44&$^\text{s}.2$
& \multicolumn{8}{m{8em}|}{Long.\ Asc.\ Node, $184°\: 25'$ (1900).}
& \raisebox{1.2ex}{145}&\raisebox{1.2ex}{$°$} & \raisebox{1.2ex}{12}& \raisebox{1.2ex}{$'$}
& &\phantom{$'$}\llap{\raisebox{-1.2ex}{$=-34°\: 48'$}}
& \multicolumn{1}{l|}{$120°\: 05'$ (1900)}
&  ,0 & 2000&? & ?  & \hspace{0.5em}Retrograde.
\\[1ex]

\hline
\hline
\end{tabular}

\end{sidewaystable}
\clearpage

%% -----File: 543.png---Folio 532-------

\begin{sidewaystable}
\footnotesize
\noindent
\setlength\tabcolsep{3pt}
\nblabel{app:III}
\begin{tabular}{@{}r | l@{}| r@{, }l@{ }r | r@{.}l | r@{.}l | r@{.}l | c | r@{}l@{ }r@{}l@{ }r@{}l | r@{}l@{ }r@{}l@{ }r@{}l | r@{}l@{ }r@{}l@{ }r@{}l@{} }
\multicolumn{30}{c}{TABLE~III.---PERIODIC COMETS WHICH HAVE BEEN OBSERVED AT MORE THAN ONE PERIHELION PASSAGE.}
\\
\multicolumn{30}{c}{FROM THE ANNUAIRE DU BUREAU DES LONGITUDES, 1888.}
\\[1ex]
\hline\hline
\multirow{2}{*}{\begin{sideways}\textsc{\footnotesize No}.\end{sideways}}
& \multicolumn{1}{c|}{\multirow{2}{*}{\textsc{\footnotesize Name}.}}
& \multicolumn{3}{c|}{\footnotesize Perihelion\rule{0pt}{3.5ex}}
& \multicolumn{2}{c|}{\footnotesize Period.}
& \multicolumn{2}{c|}{\footnotesize Perihelion}
& \multicolumn{2}{c|}{\footnotesize Aphelion}
& \multirow{2}{*}{\footnotesize Eccentricity.}
& \multicolumn{6}{c|}{\footnotesize Longitude of}
& \multicolumn{6}{c|}{\footnotesize Longitude of}
& \multicolumn{6}{c}{\multirow{2}{*}{\footnotesize Inclination.}}
\\[-.5ex]
 &
 & \multicolumn{3}{c|}{\footnotesize Passage.}
 & \multicolumn{2}{c|}{\footnotesize (Years.)}
 & \multicolumn{2}{c|}{\footnotesize Distance.}
 & \multicolumn{2}{c|}{\footnotesize Distance.} &
 & \multicolumn{6}{c|}{\footnotesize Perihelion.}
 & \multicolumn{6}{c|}{\footnotesize Node.}  &
\\[1.5ex]
\hline
1 & Encke \dotfill\rule{0pt}{3ex}
& 1885 & Mar.\ &  7 & 3&307 & 0&342309 & 4&096935 & 0.845781
& 158&$°$ & 32&$'$ & 45&$''$
& 334&$°$ & 36&$'$ & 55&$''$
&  12&$°$ & 54&$'$ & 00&$''$
\\[.5ex]
2 & Tempel \dotfill
& 1883 & Nov.\ & 20 & 5&209 & 1&344665 & 4&665563 & 0.552541
& 306 &&  7 &&  4 &    & 121 &&  2 &&  8 &    & 12 && 45 && 17 &
\\[.5ex]
3 & Tempel-Swift \dotfill
& 1886 & May &  9 & 5&505 & 1&072638 & 5&162744 & 0.655951
& 43 && 9 && 54 &    & 297 && 0 && 39 &    & 5 && 23 && 37 &
\\[.5ex]
4 & Brorsen \dotfill
& 1879 & Mar.\ & 30 & 5 & 462 & 0 & 589892 & 5 & 612808 & 0.809797
& 116 && 15 && 3 &    & 101 && 19 && 16 &    & 29 && 23 && 10 &
\\[.5ex]
5 & Winnecke \dotfill
& 1886 & Sept.\ & 4 & 5 & 812 & 0 & 883240 & 5 & 582030 & 0.726775
& 276 && 4 &&  &    & 101 && 56 &&  &    & 14 && 27 &&  &
\\[.5ex]
6 & Tempel \dotfill
& 1885 & Sept.\ & 25 & 6 & 507 & 2 & 073322 & 4 & 897332 & 0.405128
& 241 && 21 && 50 &    &  72 && 24 &&  9 &    & 10 && 50 && 27 &
\\[.5ex]
7 & Biela, 2d nucleus, \dotfill
& 1852 & Sept.\ & 23 & 6 & 629 & 0 & 860592 & 6 & 196874 & 0.755119
& 108 && 58 && 17 &    & 245 && 58 && 29 &    &  12 && 33 && 50 &
\\[.5ex]
8 & D'Arrest \dotfill
& 1884 & Jan.\ & 13 & 6 & 686 & 1 & 326420 & 5 & 771986 & 0.626277
& 319 && 11 && 11 &    & 146 &&  7 && 21 &    & 15 && 41 && 47 &
\\[.5ex]
9 & Faye \dotfill
& 1881 & Jan.\ & 22 & 7 & 566 & 1 & 738140 & 5 & 970090 & 0.549017
&  50 && 48 && 47 &    & 209 && 35 && 25 &    &  11 && 19 && 40 &
\\[.5ex]
10 & Tuttle \dotfill
& 1885 & Sept.\ & 11 & 13 & 760 & 1 & 024728 & 10 & 459624 & 0.821544
& 116 && 28 && 59 &    & 269 && 42 &&  1 &    &  55 && 14 && 23 &
\\[.5ex]
11 & Pons-Brooks\dotfill
& 1884 & Jan.\ & 25 & 71 & 48 & 0 & 77511 & 33 & 67129 & 0.954996
&  93 && 20 && 48 &    & 254 &&  6 && 15 &    & 74 &&  3 && 20 &
\\[.5ex]
12 & Olbers \dotfill
& 1887 & Oct.\ &  8 & 72 & 63 & 1 & 19961 & 33 & 61592 & 0.931088
& 149 && 45 && 47 &    &  84 && 29 && 41 &    &  44 && 33 && 53 &
\\[.5ex]
13 & Halley \dotfill
& 1835 & Nov.\ & 15 & 76 & 37 & 0 & 58895 & 35 & 41121 & 0.967281
& 165 && 48 && 48 &    &  55 && 10 && 15 &    & 162 && 15 &&  7 &
\\[1ex]
\hline\hline
\end{tabular}
\end{sidewaystable}
%% -----File: 544.png---Folio 533-------

\clearpage

\begin{center}
\nblabel{app:IV}
TABLE~IV.---STELLAR PARALLAX.
\end{center}

{\footnotesize
The star-places are only approximate,---sufficient merely for identification. The parallaxes
are mostly from Houzeau's ``Vade Mecum,'' but a few are included from later authorities.

\bigskip
\setlength\tabcolsep{3pt}
\tiny
\begin{tabular}{@{}r@{ }l@{}|l| r@{\ \ }r@{\quad}|@{ }r@{}r@{\ \ }r| l|l@{\,}|l|r|rl}
\hline\hline

  \multicolumn{2}{@{}c|}{\strut}
& \multicolumn{1}{c|} {\multirow{4}{*}{\begin{sideways}Magnitude.\mbox{\ \ }\end{sideways}} }
& \multicolumn{5}{c|}{\strut}
& \multicolumn{1}{c|}{\strut}
& \multicolumn{1}{c|}{\strut}
& \multicolumn{1}{c|}{\strut}
& \multicolumn{1}{c|}{\multirow{4}{*} {\begin{sideways}Parallax.\mbox{\quad\ }\end{sideways}}}
& \multicolumn{2}{c} {\multirow{4}{*}{\begin{sideways}\begin{tabular}{c}Distance.\\ Light-Years.\end{tabular} \end{sideways}}}\\[5ex]

\multicolumn{2}{@{}c|}{\textsc{Name of Star.}}
& & \multicolumn{5}{c|}{Approx.~Place.}
& \multicolumn{1}{c|}{Authority. }
& \multicolumn{1}{c|}{Date. }
& \multicolumn{1}{c|}{Method. }
&& \\

&& & \multicolumn{5}{c|}{(1900)} & &&&&\\[1ex]

&& & \multicolumn{2}{c }{$\alpha$}
   & \multicolumn{3}{c|}{$\delta$} & &&&&\\[1ex]

\hline &&&&&&&&&&&&\\[-1.5ex]

1. & $\beta$~Cassiopeię \dotfill & 2
& 0\rlap{$^\text{h}$} & 4\rlap{$^\text{m}$} & $+$&58\rlap{$°$} & 36\rlap{$'$}
& Pritchard & 1887 & Photography & $0''.187$ & 17.4\\

2. & Groombridge 34\DPtypo{,}{}
\dotfill & 8 & 0 & 12 &$+$&43 & 26
& Auwers & 1867 & Mer.\ Circle & 0.307 & 10.6\\

3. & $\eta$~Cassiopeię \dotfill & 4 & 0 & 43 & $+$&57 & 17
& O. Struve & 1856 & Micrometer & 0.154 & 21.3\\

4. & $\mu$~Cassiopeię \dotfill & 5.5 & 1 & 1 & $+$&54 & 26
& O. Struve &1856 & Micrometer & 0.342 & 9.5\rlap{\rdelim\}{2}{10pt}}\\
&& & && &&&  Pritchard & 1887 & Photography & 0.036 & 90.6\\

5. & Polaris \dotfill & 2 & 1 & 22 & \DPtypo{}{$+$} &88 & 46
& Peters & 1846--47 & Mer.\ Circle & 0.073 & 44.7\rlap{\rdelim\}{2}{10pt}}\\
&& & && &&&  Pritchard & 1887 & Photography & 0.052 & 62.7\\

6. & $\epsilon$~Eridani \dotfill & 4.4 & 3 & 16 &  $-$&43 & 27
& Elkin & 1882 & Heliometer & 0.143 & 22.8\\

7. & $o_{2}$ Eridani \dotfill & 4.4 & 4 & 11 & $-$& 7 & 46
& Gill & 1882 & Heliometer & 0.166 & 19.6\rlap{\rdelim\}{2}{10pt}}\\
&& & && &&&  Hall & 1884 & Micrometer & 0.223 & 14.6\\

8. & $\alpha$~Aurigę \dotfill & 1  & 5 & 9 & $+$&45 & 54
& Peters & 1846 & Mer.\ Circle & 0.046 &70.9\rlap{\rdelim\}{2}{10pt}}\\

\multicolumn{2}{c|}{(Capella) }
 & & && &&&  O. Struve & 1856 & Micrometer & 0.305 & 10.7\\

9. & $\alpha$~Canis Maj. \dotfill & 1 & 6 & 41 & $-$&16 & 35
& Gyldén & 1864 & Mer.\ Circle & 0.193 & 16.9\rlap{\rdelim\}{3}{10pt}}\\
\multicolumn{2}{c|}{(Sirius) }
 & & && &&&  Abbe & 1868 & Mer.\ Circle & 0.273 & 11.9\\
&& & && &&&  Gill \& Elkin & 1882 & Heliometer & 0.380 & 8.6\\

10. & $\alpha$~Geminorum \dotfill & 1.5& 7 & 28 & $+$&32 & 6
& Johnson & 1856 & Heliometer& 0.198 & 16.5\\
\multicolumn{2}{c|}{(Castor) } & & && &&& &&&&\\

11. & $\alpha$~Canis Min. \dotfill & 1 & 7 & 34 & $+$& 5 & 29
& Auwers & 1873 & Micrometer & 0.123 & 26.5\\
\multicolumn{2}{c|}{(Procyon) } & & && &&& &&&&\\

12. & $\iota$~Ursę Maj. \dotfill & 3.5 & 8 & 52 & $+$&48 & 26
& Peters & 1846 & Mer.\ Circle & 0.133 & 24.5\\

13. & Lalande 21\,185 \dotfill & 7 & 10 & 56 & $+$&36 & 42
& Winnecke
& \begin{tabular}{@{}l}1858\rlap{\rdelim\}{2}{10pt}} \\
                       1872
  \end{tabular}
& Micrometer & 0.506 & 6.6\\

14. & Lalande 21\,258 \dotfill & 8.5 & 11 & 1 & $+$&44 & 02
& Auwers & 1863 & Micrometer & 0.262 & 12.4\rlap{\rdelim\}{2}{10pt}}\\
&& & && &&&  Krüger & 1864 & Micrometer & 0.260 & 12.6\\

15. & Groombr'ge 1830 \dotfill & 6.5 & 11 & 7 & $+$&38 & 32
& Peters & 1846 & Micrometer &0.226 & 14.4\rlap{\rdelim\}{5}{10pt}}\\
&& & && &&&  Wichmann  & 1848 & Micrometer & 0.182 & 17.9\\
&& & && &&&  O. Struve & 1850 & Micrometer & 0.034 & 95.9\\
&& & && &&&  Johnson   & 1854 & Heliometer & 0.033 & 98.8\\
&& & && &&&  Brünnow   & 1873 & Micrometer & 0.095 & 34.3\\

16. & Oeltzen 11\,677 \dotfill & 9.5 & 11 & 15 & $+$&66 & 23
& Geelmuyden & 1880 & Mer.\ Circle & 0.265 & 12.3\\

17. & $\alpha$~Bootis \dotfill & 1 & 14 & 11 & $+$&19 & 42
& Peters & 1846 & Mer.\ Circle& 0.127 & 25.7\rlap{\rdelim\}{2}{10pt}}\\
\multicolumn{2}{c|}{(Arcturus) }
& & && &&&  Johnson & 1856 & Heliometer & 0.138 & 23.6\\

18. & $\alpha$~Centauri \dotfill & 1 & 14 & 33 & $-$&60 & 25
& Henderson & 1842 & Mer.\ Circle & 0.913 & 3.6\rlap{\rdelim\}{3}{10pt}}\\
&& & \multicolumn{5}{c|}{(Probably best.)}
& Gill \& Elkin & 1882 & Heliometer & 0.750 & 4.35\\

19. & Oeltzen 17\,415 \dotfill & 9 & 17 & 37 & $+$&68 & 28
& Krüger & 1863 & Micrometer & 0.243 & 13.4\\

20. & $\gamma$~Draconis \dotfill & 2 & 17 & 54 & $+$&51 & 30
& \begin{tabular}{@{}l@{}l}
    \hspace*{2pt}\llap{\ldelim\{{2}{6pt}} & Pond \\ & Gyldén
  \end{tabular}
& \begin{tabular}{@{}l}1817\rlap{\rdelim\}{2}{10pt}} \\ 1877 \end{tabular}
& Mural Circle & 0.127 & 25.7\\

21. & 70, p, Ophiuchi \dotfill & 4.5 & 18 & 00 & $+$& 2 & 33
& Krüger & 1859--63 & Micrometer & 0.162 & 20.1\\

22. & $\alpha$~Lyrę \dotfill & 1 & 18 & 34 & $+$&38 & 41
& W. Struve & 1840 & Micrometer & 0.262 & 12.4\rlap{\rdelim\}{6}{10pt}}\\

\multicolumn{2}{c|}{(Vega) }
 & & && &&&  Peters    & 1846 & Mer.\ Circle & 0.103 & 32.4 \\
&& & && &&&  Johnson   & 1856 & Heliometer   & 0.140 & 23.3 \\
&& & && &&&  O. Struve & 1859 & Micrometer   & 0.140 & 23.3 \\
&& & && &&&  Brünnow   & 1873 & Micrometer   & 0.188 & 17.3 \\
&& & && &&&  Hall      & 1881 & Micrometer   & 0.134 & 24.3 \\

23. & $\alpha$~Draconis \dotfill & 5 & 19 & 33 & $+$&69 & 30
& Brünnow & 1870--73 & Micrometer & 0.234 & 14.0\\

24. & 61 Cygni \dotfill & 5.5 & 21 & 2 & $+$&38 & 15
& Bessel & 1838--40 & Heliometer & 0.348 & 9.4\rlap{\rdelim\}{8}{10pt}}\\
&& & && &&&  Pogson    & 1853     & Heliometer  & 0.384 & 8.5  \\
&& & && &&&  Johnson   & 1854     & Heliometer  & 0.397 & 8.2  \\
&& & && &&&  O. Struve & 1852--53 & Micrometer  & 0.505 & 6.5  \\
&& & && &&&  Auwers    & 1863     & Micrometer  & 0.564 & 5.8  \\
&& & && &&&  Ball      & 1878     & Micrometer  & 0.465 & 7.0  \\
&& & && &&&  Hall      & 1880--86 & Micrometer  & 0.270 & 12.1 \\
&& & && &&&  Pritchard & 1887     & Photography & 0.432 & 7.55 \\

25. & $\epsilon$~Indi \dotfill & 5.5 & 21 & 56 &  \DPtypo{}{$-$}  &57 & 69
& Gill \& Elkin & 1881--82 & Heliometer & 0.222 & 14.6\\

26. & Lacaille 9352 \dotfill & 7.5 & 22 & 59 &  \DPtypo{}{$-$}  &36 & 26
& Gill & 1881 & Heliometer & 0.285 & 11.4\\

27. & Bradley 3077 \dotfill & 6 & 23 &  8 &  \DPtypo{}{$+$}  &56 & 37
& Brünnow & 1873 & Micrometer & 0.069 & 47.3\\

28. & 85 Pegasi \dotfill & 6 & 23 & 57 &  \DPtypo{}{$+$}  &26 & 35
& Brünnow & 1873 & Micrometer & 0.054 & 60.4\\
\hline\hline
\end{tabular}
}%end group for footnotesize, etc
%% -----File: 545.png---Folio 534-------
\clearpage

\begin{sidewaystable}
\footnotesize
\nblabel{app:V}
\begin{tabular} {@{}r | l@{}l@{}| r @{} l @{ } r @{.} l | l @{} r @{} l @{ } r @{} l | r @{.} l | r @{.} l | c | c | l @{} l @{ } r @{} l | p{3.25cm}@{} }
\multicolumn{23}{c}{TABLE V.---ORBITS OF BINARY STARS\@.  FROM HOUZEAU'S ``VADE MECUM.''}\\[1ex]
\hline\hline
& \multicolumn{2}{@{}c@{}|}{\textsc{Name of Star.\rule{0pt}{4ex}}}
& \multicolumn{4}{c|}{$\alpha$ (1880).}
& \multicolumn{5}{c|}{$\delta$ (1880).}
& \multicolumn{2}{c|}{Period.}
& \multicolumn{2}{c|}{a.}
& c.
&Periastron.
& \multicolumn{4}{c|}{Magnitudes.}
&Calculator of Orbit.
\\[2ex]
\hline
  1 & $\delta$~Equulei \dotfill\rule{0pt}{3ex}
&&21&$^{\text{h}}$ & $8^{\text{m}}$&6    &$+$& 9&$°$&32&$'$
& \multicolumn{2}{l}{$10^{\text{y}} \text{ to } 12^{\text{y}}$}\vline&$0''$&30    &?    &  ?    &   &4.5,& 5&
&\parbox[t]{9em}{\parindent=0em\hangindent=1em No satisfactory orbit yet computed.}
\\
 2 & 42 Comę Ber. \dotfill   && 13 &   & 4&2    & $+$&18 &    & 10 &   & 25&71  & 0&657 & 0.480 & 1869.92 &   & 6,  & 6 &    & Dubiago.
\\
 3 & $\zeta$~Herculis \dotfill    && 16 &   & 38&8    & $+$&31&    & 49 &   & 34&411 & 1&284 & 0.463 & 1864.79 &   & 3,  & 5&.5   & Doberck.
\\
 4 & $\eta$~Coronę Bor. \dotfill    && 15 &   & 18&2    & $+$&30&    & 43 &   & 41&576 & 0&827   & 0.263 & 1850.26 &   & 6,  & 6&.5   & Duner.
\\
 5 & Sirius \dotfill    && 6 &   & 39&7    & $-$&16 &    & 32 &   & 44&0       & 8&53    & 0.591 & 1889.44    & $-$&1,  & 9&   & Pritchard.
\\
 6 & $\mu_{2}$ Herculis \dotfill    && 17 &   & 41&8    & $+$&27&    & 48 &   & 54&25      & 1&46    & 0.302 & 1877.13    &   & 9.5, & 10&.5   & Doberck.
\\
 7 & $\zeta$~Cancri \dotfill  && 8 &   & 5&3    & $+$&18&    & 01 &   & 60&327       & 0&853   & 0.391 & 1866.02    &   & 5.5, & 6&.2   & Seeliger.
\\
 8 & $\xi$~Ursę Maj. \dotfill   && 11 &   & 11&8    & $+$&32&    & 13 &   & 60&80      & 2&580   & 0.416 & 1875.26    &   & 4,  & 5 &   & Pritchard.
\\
 9 & $\alpha$~Centauri \dotfill   && 14 &   & 32&0    & $-$&60&    & 22 &   & 77&42      & 17&50    & 0.526 & 1875.97    &   & 1,  & 4 &   & Elkin.
\\
10 & 70, p, Ophiuchi \dotfill   && 17 &   & 59&4    & $+$& 2 &    & 32 &   & 94&44      & 4&790   & 0.467 & 1808.90    &   & 4.5, & 6 &   & Pritchard.
\\
11 & $\gamma$~Coronę Bor. \dotfill    && 15 &   & 37&7    & $+$&26 &    & 40 &   & 95&50      & 0&70    & 0.350 & 1843.70    &   & 4,  & 7 &   & Doberck.
\\
12 & $\omega$~Leonis \dotfill   && 9 &   & 22&0    & $+$& 9 &    & 35 &    & 110&82      & 0&890   & 0.536 & 1841.81    &   & 6,  & 7 &   & Doberck.
\\
13 & $\xi$~Bootis \dotfill    && 14 &   & 45&8    & $+$&19 &    & 36 &    & 127&35      & 4&860   & 0.708 & 1770.69    &   & 4.5, & 6&.5   & Doberck.
\\
14 & 4 Aquarii \dotfill   && 20 &   &45&0    & $-$& 6 &    & 5 &    & 129&84      & 0&717   & 0.461 & 1751.96    &   & 6,  & 7 &   & Doberck.
\\
15 & $\gamma$~Virginis \dotfill   && 12 &   & 35&6    & $-$& 0 &    & 47 &    & 185&01      & 3&97    & 0.896 & 1836.68    &   & 3,  & 3&.2   & Thiele.
\\
16 & $\eta$~Cassiopeię \dotfill   && 0 &   & 41&8    & $+$&57 &    & 11 &    & 195&24      & 8&639   & 0.624 & 1901.95    &   & 4,  & 7&.3   & Gruber.
\\
17 & 36 Andromedę \dotfill    && 0 &   & 48&0    & $+$&23 &    & 53 &    & 349&10      & 1&54    & 0.634 & 1798.80    &   & 6,  & 7 &   & Doberck.
\\
18 & $\gamma$~Leonis \dotfill   && 10 &   & 31&3    & $+$&20 &    & 27 &    & 407&04      & 1&98    & 0.733 & 1741.00    &   & 2,  & 3&.5   & Doberck.
\\
19 & $\sigma$~Coronę Bor. \dotfill    && 16 &   & 10&2    & $+$&34 &    & 10 &    & 845&86      & 5&88    & 0.752 & 1826.93    &   & 5.5, & 6&.5   & Doberck.
\\
20 & $\alpha$~Geminorum \dotfill    && 7 &   & 27&0    & $ +$&32 &    & 2 &    &\ 996&85      & 7&538   & 0.344 & 1750.33    &   & 2.5, & 3 &   & Thiele.
\\[1ex]
\hline\hline
\end{tabular}
\end{sidewaystable}
%% -----File: 546.png---Folio 535-------
\clearpage

\begin{center}
\nblabel{app:VI}
TABLE VI.---THE PRINCIPAL VARIABLE STARS.
\end{center}

{\small
A selection from S. C. Chandler's catalogue of 225 variables (Astronomical Journal, Sept.\
1888), containing such as are visible to the naked eye, have a range of variation exceeding half
a magnitude, and can be seen in the United States.

\bigskip
\tiny
\noindent
\renewcommand{\arraystretch}{1.4}
\setlength\tabcolsep{4pt}
\begin{tabular}{r|l| *{2}{r@{}c@{\,}r@{}l|} l@{ }c@{}r@{}l| D{!}{}{-1}| @{}l }
\hline\hline

& \multicolumn{1}{c|}{\multirow{2}{5em}{\centering \textsc{Name}.\rule{0pt}{4ex}}}
& \multicolumn{8}{c|}{Place, 1900.\rule{0pt}{3ex}}
& \multicolumn{4}{m{5em}|}{\multirow{2}{4.5em}{\centering \rule{0pt}{3ex}Range of Variation.}}
& \multicolumn{1}{c|}{\multirow{2}{7em}{\centering Period (days).\rule{0pt}{4ex}}}
& \multicolumn{1}{c}{\multirow{2}{*}{Remarks.\rule{0pt}{4ex}}}
\\[1ex]
\cline{3-10}

  \rotatebox{90}{\rlap{\textsc{\:No}.}}
&& \multicolumn{4}{c|}{$\alpha$\rule{0pt}{3ex}}
 & \multicolumn{4}{c|}{$\delta$} &&&&&
\\[1ex]
\hline

1 & T Ceti\dotfill \rule{0pt}{3ex}
& 0&${}^\text{h}$& 16${}^\text{m}$.&7  & $-20$&°& 37&$'$
& 5.1 & to & 7 && 65!\ ± &\hspace*{0.6em}Irregular.
\\
2 & R Andromedę\dotfill
& 0 && 18.8 && $+38$ &&  1 && 5.6 && 13 && 411!.2 &
\\
3 & R Sculptoris\dotfill
& 1 && 22.4 && $-33$ &&  4 && 5.8 && 7&.8 & 207!
& \multirow{3}{*}{$\left\{\parbox{7em}{\textit{Mira}. Variations in length of period.}\right.$}
\\
4 & $o$ Ceti\dotfill
& 2 && 14.3 && $-\phantom{0}3$ && 26 && 1.7 && 9&.5 & 331!.3363 &
\\
5 & $\rho$~Persei\dotfill
& 2 && 58.7 && $+38$ && 27 && 3.4 && 4&.2 &  33!  &
\\
6 & $\beta$~Persei\dotfill
&  3 &&  1.6 && $+40$ && 34 && 2.3 && 3&.5
& 2^\text{d}\, 23^\text{h}\, 48!^\text{m}\, 55^\text{s}.43
& \smash{$\Bigl\{$\parbox{7em}{\textit{Algol}. Period now shortening.}}
\\
7 & $\lambda$~Tauri\dotfill
&  3 && 55.1 && $+12$ && 12 && 3.4 && 4&.2
& 3^\text{d}\, 22^\text{h}\, 52!^\text{m}\, 12^\text{s}
&
\\
8 & $\epsilon$~Aurigę\dotfill
&  4 && 54.8 && $+43$ && 41 && 3   && 4&.5
&\multicolumn{1}{c|}{\hspace*{0.6em}Irregular}
&\smash{$\Bigl\{$\parbox{7em}{Algol type, but irregular.}}
\\
9 & $\alpha$~Orionis\dotfill
&  5 && 49.7 && $+\phantom{0}7$ && 23 && 1 && 1&.6 & 196!\ ?
& \hspace*{0.6em}Irregular.
\\
10 & $\eta$~Geminorum\dotfill
&  6 &&  8.8 && $+22$ && 32 && 3.2 && 4&.2 & 229!.1  &
\\
11 & $\zeta$~Geminorum\dotfill
&  6 && 58.2 && $+20$ && 43 && 3.7 && 4&.5
& 10^\text{d}\, \phantom{0}3^\text{h}\, 41!^\text{m}\, 30^\text{s} &
\\
12 & R Canis Maj.\dotfill
&  7 && 14.9 && $-16$ && 12 && 5.9 && 6&.7
& 1^\text{d}\, \phantom{0}3^\text{h}\, 15!^\text{m}\, 55^\text{s}
& \hspace*{0.6em}Algol type.
\\
13 & R Leonis Min.\dotfill
&  9 && 39.6 && $+34$ && 58 && 6   && 13 && 373!.5 & \hspace*{0.6em}Period short'ing.
\\
14 & R Leonis\dotfill
&  9 && 42.2 && $+11$ && 54 && 5.2 && 10 && 312!.87 &
\\
15 & U Hydrę\dotfill
& 10 && 32.6 && $-12$ && 52 && 4.5 &&  6&.3 & 194!.65 &
\\
16 & R Ursę Maj.\dotfill
& 10 && 37.6 && $+69$ && 18 && 6.0 && 13&.2 & 305!.4
& \hspace*{0.6em}Period short'ing.
\\
17 & R Hydrę\dotfill
& 13 && 24.2 && $-22$ && 46 && 3.5 &&  9&.7 & 496!.91
& \hspace*{0.6em}Period short'ing.
\\
18 & S Virginis\dotfill
& 13 && 27.8 && $-\phantom{0}6$ && 41 && 5.7 && 12&.5 & 376!.0 &
\\
19 & R Bootis\dotfill
& 14 && 32.8 && $+27$ && 10 && 5.9 && 12&.2 &  223!.9 &
\\
20 & $\delta$~Librę\dotfill
& 14 && 55.6 && $-\phantom{0}8$ && 7 && 5.0 && 6&.2
& 2^\text{d}\, \phantom{0}7^\text{h}\, 51!^\text{m}\, 22^\text{s}.8 & \hspace*{0.6em}Algol type.
\\
21 & S Coronę\dotfill
& 15 && 17.3 && $+31$ && 44 && 6.0 && 12&.5 & 360!.57 &
\\
22 & R Coronę\dotfill
& 15 && 44.4 && $+28$ && 28 && 5.8 && 13&
& \multicolumn{1}{c|}{\quad Irregular} &
\\
23 & R Serpentis\dotfill
& 15 && 46.1 && $+15$ && 26 && 5.6 && 13&   & 357!.6 &
\\
24 & $\alpha$~Herculis\dotfill
& 17 && 10.1 && $+14$ && 30 && 3.1 &&  3&.9
& \multicolumn{2}{c}{Two or three months, but very irreg.}
\\
25 & U Ophiuchi\dotfill
& 17 && 11.5 && $+\phantom{0}1$ && 19 && 6.0 && 6&.7
& 20^\text{h}\, \phantom{0}7!^\text{m}\, 41^\text{s}.6
& \smash{$\Bigl\{$\parbox{7em}{Shortest period known.}}
\\
26 & X Sagittarii\dotfill
& 17 && 41.3 && $-27$ && 48 && 4   &&  6&   & 7!.01185 &
\\
27 & W Sagittarii\dotfill
& 17 && 58.6 && $-29$ && 35 && 5   &&  6&.5 & 7!.59445 &
\\
28 & Y Sagittarii\dotfill
& 18 && 15.5 && $-18$ && 54 && 5.8 &&  6&.6 & 5!.76900 &
\\
29 & R Scuti\dotfill
& 18 && 42.1 && $-\phantom{0}5$ && 49 && 4.7 && 9&   & 71!.10
& \multirow{3}{8em}{$\Biggl\{$\parbox{7em}{Secondary minimum about midway.}}
\\
30 & $\beta$~Lyrę\dotfill
& 18 && 46.4 && $+33$ && 15 && 3.4 &&  4&.5
& 12^\text{d}\ 21^\text{h}\, 46!^\text{m}\, 58^\text{s}.3 &
\\
31 & R Cygni\dotfill
& 19 && 34.1 && $+49$ && 58 && 5.9 && 13&   & 425!.7 &
\\
32 & $\chi$~Cygni\dotfill
& 19 && 46.7 && $+32$ && 40 && 4.0 && 13&.5 & 406!.045
& \hspace*{0.6em}Period length'ing
\\
33 & $\eta$~Aquilę\dotfill
& 19 && 47.4 && $+\phantom{0}0$ && 45 && 3.5 && 4&.7
& 7^\text{d}\, \phantom{0}4^\text{h}\, 14!^\text{m}\, \phantom{0}0^\text{s}.0 &
\\
34 & S Sagittę\dotfill
& 19 && 51.4 && $+16$ && 22 && 5.6 && 6&.4
& 8^\text{d}\, \phantom{0}9^\text{h}\, 11!^\text{m} &
\\
35 & X Cygni\dotfill
& 20 && 39.5 && $+35$ && 13 && 6.4 && 7&.7
& 15^\text{d}\, 14^\text{h}\, 24!^\text{m}\,
& \smash{$\Bigl\{$\parbox{7em}{Minimum not constant.}}
\\
36 & T Vulpeculę\dotfill
& 20 && 47.2 && $+27$ && 52 && 5.5 && 6&.5
& 4^\text{d}\, 10^\text{h}\, 29!^\text{m}
\\
37 & T Cephei\dotfill
& 21 &&  8.2 && $+68$ && \phantom{0}5  &&  5.6 && 9&.9 & 383!.20 &
\\
38 & $\mu$~Cephei\dotfill
& 21 && 40.4 && $+58$ && 19 && 4   &&  5&   & 432!\ ? &
\\
39 & $\delta$~Cephei\dotfill
& 22 && 25.4 && $+57$ && 54 && 3.7 &&  4&.9
& 5^\text{d}\, \phantom{0}8^\text{h}\, 47!^\text{m}\, 39^\text{s}.97 &
\\
40 & $\beta$~Pegasi\dotfill
& 22 && 58.9 && $+27$ && 32 && 2.2 &&  2&.7
& \multicolumn{1}{c|}{\quad Irregular} &
\\
41 & R Aquarii\dotfill
& 23 && 38.6 && $-15$ && 50 && 5.8 && 11&   & 387!.16 &
\\
42 & R Cassiopeię\dotfill
& 23 && 53.3 && $+50$ && 50 && 4.8 && 12&   & 429!.00
\\[0.5ex]\hline\hline
\end{tabular}
} %end group for font size changes etc
\chelabel{APPENDIX}
%% -----File: 547.png---Folio 536-------
%% -----File: 548.png---Folio 537-------

\clearpage
\pdfbookmark[1]{Index}{Index}
\chapter*{INDEX.}
\chslabel{INDEX}
\markboth{INDEX.}{}
\setlength\headheight{20.5pt}
\renewcommand\headrule{\vspace{-6pt}}
\fancyhead[C]{\begin{tabular}{c}\footnotesize\leftmark\\[-0.5ex]
\tiny[All references, unless expressly stated to the contrary, are to \textit{articles} and not to \textit{pages}.]\end{tabular}}
\fancyhead[R]{\begin{tabular}{r}\thepage\\[-0.5ex]
\tiny~\end{tabular}}

\thispagestyle{empty}
\nbrule\bigskip

{
{\centering\scriptsize
[All references, unless expressly stated to the contrary, are to \textit{articles} and not to \textit{pages}.]
\par}

\setlength\columnseprule{0.5pt}%
\setlength\columnsep{1.5em}%
\begin{multicols}{2}
\sloppy
\footnotesize
\setlength\parindent{0pt}%
\setlength\parskip{0pt plus 0.1pt}%

\idxsection*{A.}

\idxb{Aberration} of light, annual, \idxart{99}, \idxart{224}--\idxart{226};
used to determine the solar parallax, \idxart{692};
diurnal, \idxart{226*};
spherical and chromatic, \idxart{39}.

\idxc{Aboul Wefa}, discoverer of the lunar variation, \idxart{457}.

\idxb{Absolute} scale of stellar magnitude, \idxart{819}.

\idxb{Acceleration} of Encke's comet, \idxart{710};
of Winnecke's comet, \idxart{711};
of the sun's equator, \idxart{283}--\idxart{285};
secular, of moon's mean motion, \idxart{459}--\idxart{461};
secular, of moon's mean motion as affected by meteors, \idxart{778}.

\idxb{Achromatic} object-glasses for telescopes, \idxart{41}.

\idxb{Actinometer} of Violle, \idxart{341}.

\idxc{Adams, J.~C.}, the discovery of Neptune, \idxart{654};
investigation of the orbit of the Leonids, \idxart{785}.

\idxb{Adjustments} of the transit instrument, \idxart{60}.

\idxb{Aerolites}.    See \idxsee{Meteorites}.

\idxb{Age}, relative, of the planets, \idxart{913}, \idxart{915};
of the solar system, \idxart{922};
of the sun, \idxart{359}.

\idxb{Air-currents }at high elevations, \idxart{773}, \textit{note}.

\idxc{Airy, G.~B.}, density of the earth, \idxart{169}.

\idxb{Albedo} defined and determined, \idxart{546};
of Jupiter, \idxart{644};
of Mars, \idxart{583};
of Mercury, \idxart{558};
of the Moon, \idxart{259};
of Neptune, \idxart{660};
of Saturn, \idxart{636};
of Uranus, \idxart{648};
of Venus, \idxart{572}.

\idxb{Algol}, or $\beta$~Persei, \idxart{848}.

\idxb{Almagest} of Ptolemy, \idxart{500}, \idxart{700}, \idxart{795}.

\idxb{Almucantar} defined, \idxart{12}.

\idxb{Altitude} defined, \idxart{21};
parallels of, \idxart{12};
of pole equals latitude, \idxart{30};
of sun, how measured with sextant, \idxart{77}.

\idxb{Altitude} and azimuth instrument, \idxart{71}.

\idxb{Amplitude} defined, \idxart{22}.

\idxb{Andromeda}, the nebula in, \idxart{886};
temporary star in the nebula of, \idxart{845}.

\idxb{Andromedes}, the, \idxart{780}, \idxart{784}, \idxart{786}.

\idxb{Angle}, position, of a double star, \idxart{868};
of the vertical, \idxart{156}.

\idxb{Angular} and linear dimensions, \idxart{5};
velocity under central force, its law, \idxart{408}, \idxart{409}.

\idxb{Annual} equation of the moon's motion, \idxart{458};
motion of the sun, \idxart{172}, \idxart{173}.

\idxb{Annular} eclipse, \idxart{382};
nebula in Lyra, \idxart{888}.

\idxb{Anomalistic} month, the, \idxart{397}, \textit{note};
revolution of the moon, \idxart{250};
year defined, \idxart{216}.

\idxb{Anomaly} defined, mean and true, \idxart{189}.

\idxb{Apertures}, limiting, in photometry, \idxart{825}.

\idxb{Apex} of the sun's way, \idxart{805}.

\idxb{Apparition}, perpetual, circle of, \idxart{33}.

\idxb{Apsides}, line of, defined, \idxart{183};
its revolution in case of the earth's orbit, \idxart{199};
its revolution in case of the moon's orbit, \idxart{454};
its revolution in case of the planets' orbits, \idxart{527}.

\idxb{Arc} of meridian, how measured, \idxart{147}.

\idxb{Areal} or areolar velocity, law of, under
central force, \idxart{402}--\idxart{406}.

\idxb{Areas}, equable description of, in earth's orbit, \idxart{186}, \idxart{187}.

\idxc{Argelander}, his \textit{Durchmusterung} and zones, \idxart{795}, \idxart{833};
his star magnitudes, \idxart{817}, \idxart{833}.

\idxb{Argus}, $\eta$, \idxart{841}.

\idxb{Ariel}, the inner satellite of Uranus, \idxart{650}.

\idxb{Aries}, first of, \idxart{17}.

\idxc{Aristarchus}, method of determining the sun's distance, \idxart{666}, \idxart{670}.

\idxb{Artificial} horizon, the, \idxart{78}.

\idxb{Ashes} of meteors, \idxart{775}.

\idxb{Aspects} of planets defined by diagram, \idxart{494}.

\idxb[Asteroid]{Asteroids}, the, or minor planets, \idxart{592}--\idxart{601}\DPtypo{,}{;}
theories as to their origin, \idxart{600}.

\idxb{Astręa}, the fifth asteroid, discovered by Heneke, \idxart{593}.
%% -----File: 549.png---Folio 538-------

\idxb{Astro-Physics} defined, \idxart{2}.

\idxb{Atlases} of the stars, \idxart{793}.

\idxb{Atmosphere} of the moon, \idxart{255}--\idxart{257};
of Venus, \idxart{573};
height of the earth's, \idxart{98}.

\idxb{Attraction}, intensity of the solar, on the earth, \idxart{436};
within a hollow sphere,\idxart{169};
of universal gravitation, \idxart{161}, \idxart{162}.

\idxb{Axis} of the earth, its direction, \idxart{14};
of the earth, disturbed by precession, \idxart{206};
of the sun, its direction, \idxart{282}.

\idxb{Azimuth} defined, \idxart{22};
determination of, \idxart{127};
method of reckoning, \idxart{22};
of transit instrument, its adjustment, \idxart{60}.

\idxsection* {B.}

\idxc{Baily}, determination of the density of
the earth, \idxart{166}.

\idxb{Balance}, common, used in determining
the density of the earth, \idxart{170};
torsion, used in determining the density of the
earth, \idxart{165}.

\idxb{Barometer}, changes of, affecting atmospheric
refraction, \idxart{91};
effect on height of the tides, \idxart{480}.

\idxb{Barometric} error of a clock and its compensation,
52.

\idxb{Beginning} of the day, \idxart{123};
of the year, \idxart{222}.

\idxc{Benzenberg}, experiments on the deviation
of falling bodies, \idxart{138}.

\idxc{Bessel}, the parallax of 61 Cygni, \idxart{809}, \idxart{811};
formation of comets' tails, \idxart{728};
his ``zones,'' \idxart{795}.

\idxc{Biela's} comet, \idxart{744}--\idxart{746}.

\idxb{Bielids}, the, \idxart{746}, \idxart{780}, \idxart{784}, \idxart{786}.

\idxb{Bielid} meteorite, Mazapil, \idxart{784}.

\idxb[Binary Stars]{Binary stars}, \idxart{872}--\idxart{875};
number known at present, \idxart{872};
their masses, \idxart{877}, \idxart{878};
their ``mass-brightness,'' \idxart{879};
their orbits, \idxart{875}--\idxart{877}.

\idxb{Bissextile} year, explanation of term, \idxart{219}.

\idxb{Black Drop}, the, at a transit of Venus,
681.

\idxc{Bode's} law, \idxart{488}, \idxart{489}.

\idxb{Bolides}, or detonating meteors, \idxart{768}.

\idxb{Bolometer}, the, of Langley, \idxart{343}.

\idxc{Bond, G. P.}, first photograph of a double
star, \idxart{868}.

\idxc{Bond, W. C.}, discovery of Hyperion, \idxart{643};
of Saturn's dusky ring, \idxart{637}.

\idxc{Boyle}, law of, \idxart{360}, \textit{note}.

\idxc{Brahe, Tycho}.  See \idxsee{Tycho}.

\idxc{Bredichin}, his theory of comets' tails,
731, \idxart{732}.

\idxb{Brightness} of comets, \idxart{699};
of planets in various positions, Mercury, \idxart{551};
Venus, \idxart{563}, \idxart{568};
Mars, \idxart{579};
Asteroids, \idxart{596}, \idxart{599};
Jupiter, \idxart{610};
Saturn, \idxart{632};
Uranus, \idxart{647};
Neptune, \idxart{660};
of an object in the telescope, \idxart{38};
of shooting stars, \idxart{773};
of stars, causes of the difference in this respect, \idxart{836};
of stars, its measurement, \idxart{823}--\idxart{831}.

\idxsection* {C.}

\idxb{Calendar}, the, \idxart{217}--\idxart{223}.

\idxb{Callisto}, the outer satellite of Jupiter,
621, \idxart{627}.

\idxb{Calories} of different magnitude, \idxart{338}, \textit{note.}

\idxb{Candle} power, its mechanical equivalent, \idxart{776};
power of sunlight, \idxart{332}, \idxart{333}.

\idxb{Candle} standard, \idxart{333}, \textit{note.}

\idxb{Capture} theory of comets, \idxart{740}.

\idxb{Cardinal} points defined, \idxart{20}.

\idxc{Carlini}, earth's density, \idxart{168}.

\idxc{Carrington}, law of the sun's rotation,
283, \idxart{284}.

\idxc{Cassegrainian} telescope, \idxart{48}.

\idxc{Cassini, J. D.}, discovery of the division
in  Saturn's  ring, \idxart{637};
discovery of four satellites of Saturn, \idxart{643}.

\idxb{Catalogues} of stars, \idxart{795}.

\idxc{Cavendish}, the torsion balance, \idxart{165}.

\idxb{Celestial} latitude and longitude, \idxart{178}, \idxart{179};
sphere, conceptions of it, \idxart{4}.

\idxb{Cenis, Mt.}, determination of the earth's
density, \idxart{168}.

\idxb{Central} force, motion under it, \idxart{400}--\idxart{410};
force, its measure in case of circular motion, \idxart{411}.

\idxb{Central} suns, \idxart{807}, \idxart{903}.

\idxb{Centrifugal} force of the earth's rotation,
154.

\idxb{Ceres}, discovery of, \idxart{592}.

\idxc{Chandler, S. C.}, catalogue of variable
stars, \idxart{852}, Appendix, \hyperref[app:VI]{Table VI.}

\idxb{Changes} on the moon's surface, \idxart{268};
in the nebulę, \idxart{892}.

\idxb{Characteristics} of different meteoric
swarms, \idxart{783}.

\idxb{Charts} of the stars, \idxart{798}.

\idxb{Chemical} elements recognized in comets, \idxart{724}, \idxart{725};
elements recognized in stars, \idxart{856};
elements recognized in the sun, \idxart{315}--\idxart{317}.

\idxb{Chromatic} aberration of a lens, \idxart{39}.

\idxb{Chromosphere}, the, \idxart{291}, \idxart{322}, \idxart{363}.

\idxb{Chronograph}, the, \idxart{56}.

\idxb{Chronometer}, the, \idxart{54};
longitude by, \idxart{121} [A].

\idxb{Circle}, the meridian, \idxart{63}.

\idxb{Circles} of perpetual apparition and occultation, \idxart{33}.

\idxb{Circular} motion, central force in, \idxart{411}.

\idxc{Clairaut's} equation concerning the
ellipticity of the earth, \idxart{155}.
%% -----File: 550.png---Folio 539-------

\idxc{Clarke, Col}., dimensions of the earth,
145 and \hyperref[app:spheroid]{Appendix}.

\idxb{Classification} of stellar spectra, \idxart{857}, \idxart{858}.

\idxc{Clerke, Miss A. M.}, her history of astronomy,
Preface, \idxart{570}, \idxart{626}, \textit{note}, \idxart{723},
746, \idxart{900}.

\idxb{Clocks}, general remarks on, \idxart{50}.

\idxb{Clock-breaks} (electric), \idxart{57}.

\idxb{Clock-error}, or correction, and rate, \idxart{53};
or correction determined by transit instrument, \idxart{59}.

\idxb{Clusters} of stars, \idxart{883}--\idxart{885}.

\idxc{Coggia's} Comet, \idxart{730}.

\idxb{Collimating} eye-piece, \idxart{67}.

\idxb{Collimation} of transit instrument, \idxart{60}.

\idxb{Collimator}, the, used with transit instrument, \idxart{60};
of a spectroscope, \idxart{311}.

\idxb{Collision} theory of variable stars, \idxart{850}.

\idxb{Colors} of stars in photometry, \idxart{830};
of double stars, \idxart{867}.

\idxb{Colures} defined, \idxart{25}.

\idxb{Comet}, Biela's, \idxart{744}; Donati's, \idxart{727}, \idxart{730}, \idxart{747};
Encke's, \idxart{710}, \idxart{743};
great, of 1882, \idxart{748}--\idxart{752};
Halley's, \idxart{742};
Winnecke's, \idxart{711}.

\idxb{Comets}, acceleration of Encke's and Winnecke's,
710, \idxart{711};
brightness of, \idxart{699}, \idxart{723};
capture theory of, \idxart{740};
chemical elements in, \idxart{724}, \idxart{725};
constituent parts, \idxart{713};
contraction of head when near the sun, \idxart{715};
danger from, \idxart{753}, \idxart{754};
density of, \idxart{720};
designation of, \idxart{697};
dimensions of, \idxart{714}, \idxart{717};
ejection theory, \idxart{741};
fall upon earth or sun, probable effect, \idxart{754};
groups of, with similar orbits, \idxart{705};
their light, \idxart{721};
their masses, \idxart{718}, \idxart{719};
and meteors, their connection, \idxart{785}--\idxart{787};
nature of, \idxart{737};
their orbits, \idxart{700}--\idxart{709};
origin of, \idxart{738}--\idxart{741};
perihelia, distribution of, \idxart{706};
physical characteristics, \idxart{712};
planetary families of, \idxart{739};
their spectra, \idxart{724}, \idxart{726};
superstitions regarding them, \idxart{695};
their tails or trains, \idxart{713}, \idxart{717}, \idxart{728}--\idxart{736};
variations in brightness, \idxart{723};
visitors in the solar system, \idxart{709}.

\idxb{Comparison} of starlight with sunlight,
334, \idxart{832}.

\idxb{Compensation} pendulums, \idxart{51}.

\idxb{Compensation} of pendulum for barometric
changes, \idxart{52}.

\idxb{Components} of the disturbing force, \idxart{445}.

\idxb{Co-ordinates}, astronomical, \idxart{20}.

\idxc{Common, A. A.}, photographs of nebulę,
893.

\idxb{Conics}, the. \idxart{422}, \idxart{423}.

\idxb{Connection} between comets and meteors,
785--\idxart{787}.

\idxb{Constant} of aberration, the, \idxart{225};
the solar, \idxart{338}--\idxart{340}.

\idxb{Constancy}, secular, of the mean distances
and periods of the planets, \idxart{526}.

\idxb{Constellations}, list of, \idxart{792};
their origin, \idxart{791}.

\idxb{Contact} observations, transit of Venus,
679--\idxart{682}.

\idxb{Contraction} theory of solar heat, \idxart{356}.

\idxb{Conversion} of R. A. and Decl.\ to latitude
and longitude, \idxart{180}.

\idxc{Copernicus}, his system, \idxart{503};
``Triumphans,'' \idxart{809}.

\idxc{Cornu}, determination of the earth's density, \idxart{166};
photometric observation of
eclipses of Jupiter's satellites, \idxart{630}.

\idxb{Corona}, the solar, \idxart{291}, \idxart{327}--\idxart{331}, \idxart{364}.

\idxb{Cosmogony}, \idxart{905}--\idxart{917}.

\idxb{Cotidal} lines, \idxart{475}.

\idxb{Craters} on the moon, \idxart{265}--\idxart{267}.

\idxc{Crew, H.}, spectroscopic observations of
the sun's rotation, \idxart{285}, \textit{note}.

\idxb{Crust} of meteorites, \idxart{761}.

\idxb{Curvature} of comet's tails, \idxart{729}.

\idxb{Curvilinear} motion the effect of force, \idxart{401}.

\idxb{Cycle}, the metonic, \idxart{218}.

\idxb{Cyclones} as proofs of the earth's rotation,
143.

\idxsection* {D.}

\idxc{Dalton}, his law of gaseous mixtures, \idxart{360},
\textit{note}.

\idxb{Danger} from comets, \idxart{753}, \idxart{754}.

\idxb{Darkening} of the sun's limb, \idxart{337}.

\idxc{Darwin, G. H.}, rigidity of the earth, \idxart{171};
tidal evolution, \idxart{484}, \idxart{916}.

\idxc{Dawes}, diameter  of  the  spurious  discs
of stars, \idxart{43};
nucleoli in sun spots, \idxart{293}.

\idxb{Day}, the civil and the astronomical, \idxart{117};
effect of tidal friction upon its length, \idxart{461};
changes in its length, \idxart{144};
where it begins, \idxart{123}.

\idxb{Declination} defined, \idxart{23};
parallels of, \idxart{23};
determined with the meridian circle, \idxart{128}.

\idxb{Degree} of the meridian, how measured,
135, \idxart{147}.

\idxb{Deimos}, the outer satellite of Mars, \idxart{590},
591.

\idxc{Delisle}, method of determining the solar
parallax, \idxart{682}.

\idxc{Denning}, drawings of Jupiter's red spot,
618.

\idxb{Density} of comets, \idxart{720};
of the earth, determinations of it, \idxart{164}--\idxart{170};
of the moon, \idxart{246};
of a planet, how determined, \idxart{540};
of the sun, \idxart{279}.

\idxb{Detonating} meteors, or ``Bolides,'' \idxart{768}.
%% -----File: 551.png---Folio 540-------

\idxb{Development} of sun spots, \idxart{297}.

\idxb{Dhurmsala} meteorite, ice-coated, \idxart{765}.

\idxb{Diameter} (apparent) as related to distance, \idxart{6};
of a planet, how measured, \idxart{534}.

\idxb{Differential} method of determining a body's place, \idxart{129};
method of determining stellar parallax, \idxart{811}.

\idxb{Diffraction} of an object-glass, \idxart{43}.

\idxb{Dione}, fourth satellite of Saturn, \idxart{643}, \textit{note}.

\idxb{Dip} of the horizon, \idxart{81}.

\idxb{Disc}, spurious, of stars in telescope, \idxart{43}.

\idxb{Discovery} of comets, \idxart{698}.

\idxb{Dissipation} of energy, \idxart{925}.

\idxb{Distance} of the moon, \idxart{239};
of the nebulę, \idxart{896};
and parallax, relation between, \idxart{84};
of a planet in astronomical units, how determined, \idxart{515}--\idxart{518};
of the stars, \idxart{808}--\idxart{815};
of the sun, \idxart{274}, \idxart{275}, also \chapref{CHAPTERXVI}{Chap.~XVI.}

\idxb{Distinctness} of telescopic image, its conditions, \idxart{39}.

\idxb{Distribution} of the nebulę, \idxart{895};
of the stars, \idxart{899};
of the sun spots, \idxart{301}.

\idxb{Disturbing} force, the, \idxart{439}--\idxart{444};
force, diagram of, \idxart{441};
force, its resolution into components, \idxart{445}.

\idxb{Diurnal} aberration, \idxart{226*};
inequality of the tides, \idxart{471};
parallax, \idxart{82}, \idxart{86};
phenomena in various latitudes, \idxart{191}.

\idxb{Divisions} of astronomy, \idxart{2}.

\idxc{Doerfel} proves that a comet moves in a parabola, \idxart{700}.

\idxc{Donati's} comet, \idxart{727}, \idxart{730}, \idxart{747}.

\idxc{Doppler's} principle, \idxart{321}, \textit{note}.

\idxb[Double Stars]{Double} stars, \idxart{866}--\idxart{879};
their colors, \idxart{867};
criterion for distinguishing between these optically and physically double, \idxart{870};
method of measuring them, \idxart{868};
optically and physically double, \idxart{869};
having orbital motion, see \idxsee{Binary Stars}.

\idxc{Draper, H.}, oxygen in the sun, \idxart{316};
photograph of the nebula in Orion, \idxart{893};
photography of stellar spectra, \idxart{859};
memorial, the, \idxart{859}.

\idxb{Duration}, future, of the sun, \idxart{358};
of sun spots, \idxart{300}.

\idxsection*{E.}

\idxb{Earth}, the, her annual motion proved by aberration and stellar parallax, \idxart{174};
approximate dimensions, how measured, \idxart{134}, \idxart{135};
constitution of its interior, \idxart{171};
its dimensions, \hyperref[app:spheroid]{Appendix} and \idxart{145};
its dimensions determined geodetically, \idxart{147}--\idxart{149};
form of, from pendulum experiments, \idxart{152}--\idxart{155};
growth of, by accession of meteoric matter, \idxart{777};
mass compared with that of the sun, \idxart{278};
its mass and density, \idxart{159}--\idxart{170};
its orbit, form of, determined, \idxart{182};
principal facts relating to it, \idxart{132};
proofs of its rotation, \idxart{138}--\idxart{143}.

\idxb{Earth-shine} on the moon, \idxart{254}.

\idxb{Eccentricity} of the earth's orbit, discovered by Hipparchus, \idxart{184};
of the earth's orbit, how determined, \idxart{185};
of the earth's orbit, secular change of, \idxart{198};
of an ellipse defined, \idxart{183}, \idxart{506}.

\idxb{Eclipses}, duration of lunar, \idxart{373};
duration of solar, \idxart{385};
number in a year, \idxart{391}--\idxart{393};
recurrence of, the saros, \idxart{395};
of Jupiter's satellites, \idxart{627}--\idxart{630};
of the moon, \idxart{370}--\idxart{378};
of the sun, \idxart{379}--\idxart{390};
total, of the sun, as showing the solar atmosphere and corona, \idxart{319}, \idxart{323}.

\idxb{Ecliptic}, the, defined, \idxart{175};
obliquity of, \idxart{176};
limits, lunar, \idxart{374}, \idxart{375};
limits, solar, \idxart{386}.

\idxb{Effective} temperature of the sun, \idxart{351}.

\idxb{Ejection} theory of comets and meteors, \idxart{741}.

\idxb{Elbowed} equatorial, the, \idxart{74}.

\idxb{Electrical} registration of observations, \idxart{56}.

\idxb{Electro-dynamic} theory of gravitation, \idxart{602}.

\idxb{Elements}, chemical, not truly elementary, \idxart{318};
chemical, recognized in comets, \idxart{724}, \idxart{725};
chemical, recognized in stars, \idxart{856};
chemical, recognized in sun, \idxart{316}, \idxart{317};
of a planet's orbit, \idxart{505}--\idxart{508}.

\idxc{Elkin}, stellar parallaxes, \idxart{808}, \idxart{814}, \idxart{815}, and Appendix, \hyperref[app:IV]{Table~IV.}

\idxb{Ellipse} defined, \idxart{183};
described as a conic, \idxart{422}, \idxart{423}.

\idxb{Elliptic} comets, their number, \idxart{702};
their orbits, \idxart{703};
recognition of, \idxart{704}.

\idxb{Ellipticity} or oblateness of a planet defined, \idxart{150}.

\idxb{Elongation} of moon or planet defined, \idxart{230}.

\idxb{Enceladus}, the second satellite of Saturn, \idxart{643}, \textit{note}.

\idxc{Encke's} comet, \idxart{710}, \idxart{743}.

\idxc{Encke}, his reduction of the transits of Venus, \idxart{667}.

\idxb{Energy}, the dissipation of, \idxart{925};
and work of solar radiation, \idxart{345}.

\idxb{Enlargement}, apparent, of bodies near horizon, \idxart{4}, \textit{note}, \idxart{88}, \idxart{93}.

\idxb{Envelopes} in the head of a comet, \idxart{713}, \idxart{727}.

\idxb{Epoch} of a planet's orbit defined, \idxart{508}.

\idxb{Epsilon} Lyrę, \idxart{653}, \idxart{866}, \idxart{882}.

\idxb{Equal} altitudes, determination of time, \idxart{115}.
%% -----File: File: 552.png---Folio 541-------

\idxb{Equation}, annual, of moon's motion, \idxart{458};
of the centre, \idxart{189};
of the equinoxes, \idxart{213};
of light, by means of Jupiter's satellites, \idxart{628}--\idxart{630};
of time explained, \idxart{201}--\idxart{204};
expressing the relation between the light of different stellar magnitudes, \idxart{820}.

\idxb[Equator]{Equator}, the celestial, \idxart{16}.

\idxb{Equatorial} acceleration of the sun's rotation, \idxart{283}--\idxart{285};
coudé, Paris, \idxart{74};
parallax, \idxart{85};
telescope, \idxart{72};
telescope used to determine the place of a heavenly body, \idxart{129}.

\idxb{Equinoctial}, the, see \idxsee{Equator}, celestial;
points, or equinoxes, \idxart{17}.

\idxb{Equinoxes}, the, equation of, \idxart{213};
precession of, \idxart{205}--\idxart{212}.

\idxc{Eratosthenes}, his measure of the earth, \idxart{136}.

\idxb{Erecting} eye-piece for telescope, \idxart{45}.

\idxc{Ericsson}, his solar engine, \idxart{345};
experiment upon radiation of molten iron, \idxart{350}.

\idxb{Eruptive} prominences, \idxart{325}.

\idxb{Escapement} of clock, \idxart{50}.

\idxb{Establishment} of a port (harbor) defined, \idxart{463}.

\idxb{Europa}, the second satellite of Jupiter, \idxart{621}.

\idxb{Evection}, the, \idxart{456}.

\idxb{Evolution}, tidal, \idxart{484}, \idxart{916}.

\idxb{Eye-pieces}, telescopic, \idxart{44}.

\idxb{Extinctions}, the method of, in photometry, \idxart{825}.


\idxsection*{F.}

\idxb{Faculę}, solar, \idxart{292}.

\idxb{Fall} of a planet to the sun, time required, \idxart{413}, 3;
of a comet on the sun, probable effect, \idxart{754}.

\idxb{Falling} bodies, eastward deviation, \idxart{138}.

\idxb{Families} (planetary) of comets, \idxart{739}.

\idxc{Faye, H. A.}, his modification of the nebular hypothesis, \idxart{915};
theory of sun spots, \idxart{304}.

\idxb{Flattening}, apparent, of the celestial sphere, 4, \textit{note}.

\idxb{Force}, evidenced not by motion, but by \textit{change} of motion, \idxart{400};
projectile, term carelessly used, \idxart{401};
central, motion under it, \idxart{400}--\idxart{410};
repulsive, action on comets, \idxart{728}--\idxart{733}.

\idxb{Form} of the earth, \idxart{145}--\idxart{155}.

\idxb{Formation} of comets' tails, \idxart{728}.

\idxc{Foucault}, the gyroscope, showing earth's rotation, \idxart{142};
his pendulum experiment, showing earth's rotation, \idxart{139}--\idxart{141};
measures velocity of light, \idxart{690}.

\idxb{Fourteen} hundred and seventy-four line of the spectrum of the corona, \idxart{329}.

\idxc{Fraunhofer} lines in the solar spectrum, \idxart{315}, \idxart{855};
observations on stellar spectra, \idxart{855}.

\idxb{Free} wave, velocity of, \idxart{473}.

\idxb{Frequency}, relative, of solar and lunar eclipses, \idxart{394}.


\idxsection*{G.}

\idxb{Galaxy}, the, \idxart{898}.

\idxc{Galileo}, discovery of Jupiter's satellites, \idxart{621};
discovery of Saturn's rings, \idxart{637};
discovery of phases of Venus, \idxart{567};
use of pendulum in time-keeping, \idxart{50}.

\idxc{Galle}, optical discovery of Neptune, \idxart{654}.

\idxb{Ganymede}, the third satellite of Jupiter, \idxart{621}.

\idxb{Gas} contracting by loss of heat, Lane's law, \idxart{357}.

\idxc{Gauss}, computes the orbit of Ceres, \idxart{592};
determination of the elements of an orbit, \idxart{519};
peculiar form of achromatic object-glass, \idxart{41}.

\idxc{Gay Lussac}, law of gaseous expansion, \idxart{360}, \textit{note}.

\idxb{Geocentric} latitude, \idxart{156};
place of a heavenly body, \idxart{511}.

\idxb{Geodetic} determination of the earth's dimensions, \idxart{147}, \idxart{149}.

\idxb{Genesis} of the solar system, \idxart{908}--\idxart{915};
of star clusters and nebulę, \idxart{924}.

\idxb{Georgium Sidus}, the original name for Uranus, \idxart{645}.

\idxc{Gill}, solar parallax from observations of Mars, \idxart{676};
stellar parallaxes, \idxart{808}.
Appendix, \hyperref[app:IV]{Table~IV.}

\idxb{Globe}, celestial, rectification of, \idxart{33}, \textit{note}.

\idxb{Gnomon}, determination of latitude with it, \idxart{107};
determination of the obliquity of the ecliptic, \idxart{176}.

\idxb{Golden} number, the, \idxart{218}.

\idxb{Gradual} changes in the light of the stars, \idxart{839}.

\idxb{Graduation} errors of a circle, \idxart{69}.

\idxb{Grating} diffraction, \idxart{311}, \textit{note}.

\idxb{Gravitation}, electro-dynamic, theory of, \idxart{602};
law stated, \idxart{161};
nature unknown, \idxart{161};
law extending to the stars, \idxart{872}, \textit{note}, \idxart{873}, \idxart{901}, \textit{note};
Newton's verification of the law by means of the moon's motion, \idxart{419}, \idxart{420}.

\idxb{Gravitational} astronomy defined, \idxart{2};
methods of determining the solar parallax, \idxart{687}--\idxart{689}.

\idxb{Gravity}, increase of, below the earth's surface, \idxart{169};
variation of, between equator and pole, \idxart{152}.
%% -----File: File: 553.png---Folio 542-------

\idxc{Gregorian} calendar, the, and its adoption in England, \idxart{220}, \idxart{221};
telescope, \idxart{48}.

\idxb{Groups}, cometary, \idxart{705};
of stars having common motion, \idxart{803}.

\idxb{Growth} of the earth by meteoric matter, \idxart{777}.

\idxb{Gyroscope}, Foucault's proof of earth's rotation, \idxart{142};
illustrating the precession of the equinoxes, \idxart{210}, \idxart{211}.

\idxsection*{H.}

\idxc{Hall, A.}, discovery of the satellites of Mars, \idxart{590};
on the question whether it is certain that gravitation extends through the stellar universe, \idxart{901}, \textit{note}.

\idxc{Halley}, his comet, \idxart{742};
his computation of cometary orbits, \idxart{700};
his method of determining the sun's parallax, \idxart{679}, \idxart{680};
the moon's secular acceleration, \idxart{459};
proper motions of stars, \idxart{800}.

\idxc{Hansen}, correction of the solar parallax, \idxart{667};
opinion on the form of the moon, \idxart{258}.

\idxc{Harding} discovers Juno, \idxart{593}.

\idxc{Harkness}, observations on the light of meteors, \idxart{776};
observation of the corona spectrum, \idxart{329}.

\idxb{Harmonic} law, Kepler's, \idxart{412}--\idxart{117}.

\idxb{Harton} colliery, density of the earth, \idxart{169}.

\idxb{Harvard} photometry, the, \idxart{827}, \idxart{828}.

\idxb{Harvest} and hunter's moons explained, \idxart{237}.

\idxb{Heat} and light of meteors explained, \idxart{765};
of the moon, \idxart{260};
of the sun, \idxart{338}--\idxart{358};
received by the earth from meteors, \idxart{355}, \idxart{779};
from the stars, \idxart{834}.

\idxb{Height} of lunar mountains, \idxart{270}.

\idxc{Heis}, enumeration of naked-eye stars, \idxart{818}.

\idxb{Heliocentric} place of a planet, \idxart{512}.

\idxb{Heliometer}, the, \idxart{677};
used in determining solar parallax, \idxart{676}, \idxart{683};
used in determining stellar parallax, \idxart{811}, \idxart{815}.

\idxb{Helioscopes}, or solar eye-pieces, \idxart{286}, \idxart{287}.

\idxb{Helium}, an unidentified metal in the solar chromosphere, \idxart{323}.

\idxc{Helmholtz}, contraction theory of solar heat, \idxart{356}.

\idxc{Hencke}, discovers Astręa, the fifth asteroid, \idxart{593}.

\idxc{Henderson}, measures the parallax of $\alpha$~Centauri, \idxart{809}, \idxart{810}.

\idxc{Henry Brothers}, astronomical photography, \idxart{798}.

\idxc{Henry, Prof.\ J.}, heat of sun spots, \idxart{310};
at sun's limb, \idxart{348}.

\idxc{Herschel, Sir John}, astrometry, \idxart{819};
illustration of the planetary system, \idxart{664}.

\idxc{Herschel, Sir W.}, discovery of the sun's motion in space, \idxart{804};
discovery of two satellites of Saturn, \idxart{643};
discovery of Uranus, \idxart{645};
discovery of two satellites of Uranus, \idxart{650};
star-gauges, \idxart{899};
theory of sun spots, \idxart{302};
his reflecting telescope, \idxart{48}.

\idxc{Hevelius}, his view of cometary orbits, \idxart{700}.

\idxc{Hipparchus}, discovers eccentricity of earth's orbit, \idxart{184};
discovers precession, \idxart{205};
his value of the solar parallax, \idxart{671};
the first star-catalogue, \idxart{795}.

\idxc{Holden, E. S.}, on changes in nebulę, \idxart{892}.

\idxb{Horizon}, apparent enlargement of bodies near it, \idxart{4}, \textit{note}, \idxart{88}, \idxart{93};
artificial, \idxart{78};
rational and apparent defined, \idxart{10};
dip of, \idxart{81};
visible, defined, \idxart{11}.

\idxb{Horizontal} parallax, \idxart{83}, \idxart{84};
point of the meridian circle, \idxart{67}.

\idxb{Hour-angle} defined, \idxart{24}.

\idxb{Hour-circle} defined, \idxart{18}.

\idxc{Huggins, W.}, attempts to photograph the solar corona without an eclipse, \idxart{328};
attempted observation of stellar heat, \idxart{834};
observations of stellar spectra, \idxart{856};
photography of stellar spectra, \idxart{859};
spectroscopic observations of T coronę, \idxart{844};
star-motions in line of sight, \idxart{802}.

\idxc{Humboldt, A. von}, classification of the planets, \idxart{549}.

\idxc{Hunt, Sterry}, carbonic acid brought to earth by comets, \idxart{735}.

\idxc{Huyghens}, discovery of Saturn's rings, \idxart{637};
discovery of Saturn's satellite, Titan, \idxart{643};
invention of the pendulum clock, \idxart{50};
his long telescope, \idxart{40}.

\idxb{Hydrogen} in the solar chromosphere and prominences, \idxart{323}--\idxart{325};
bright lines of its spectrum in the nebulę, \idxart{890};
bright lines of its spectrum in temporary stars, \idxart{844};
bright lines of its spectrum in variable stars, \idxart{857}.

\idxb{Hyperbola}, the, described as a conic, \idxart{422}.

\idxb{Hyperbolic} comets, \idxart{702}.

\idxb{Hyperion}, the seventh and last discovered satellite of Saturn, \idxart{643}, \idxart{644}.

\idxb{Hypothesis}, nebular. See \idxsee{Nebular} hypothesis.


\idxsection*{I.}

\idxb{Iapetus}, the outermost satellite of Saturn, \idxart{643}.

\idxc{Ibn Jounis}, use of pendulum in observation, \idxart{50}.
%% -----File: File: 554.png---Folio 543-------

\idxb{Ice}, amount melted by solar radiation, \idxart{344}--\idxart{346}, \idxart{364*}.

\idxb{Illumination} of the moon's disc during a lunar eclipse, \idxart{376}.

\idxb{Image}, telescopic, conditions of distinctness, \idxart{39}.

\idxb{Inequality}, diurnal, of the tides, \idxart{471}.

\idxb{Inferences} deducible from Kepler's laws, \idxart{418}.

\idxb{Inferior} planet, motion of, \idxart{497}.

\idxb{Infinity}, velocity from, \idxart{429}.

\idxb{Influences} of the moon on the earth, \idxart{262}.

\idxb{Intra-Mercurial} planets, \idxart{602}--\idxart{606};
planets,  supposed observations of, during solar eclipse, \idxart{605}.

\idxb{Interior} of the earth, its constitution,  \idxart{171}.

\idxb{Invariable} plane of the solar system, \idxart{531}.

\idxb{Invariability} of the earth's rotation, \idxart{144}.

\idxb{Io}, the first satellite of Jupiter, \idxart{621}.

\idxb{Iron} meteorites, \idxart{758};
in the sun, \idxart{315}.

\idxb{Irradiation} in micrometric measures,  \idxart{256}, \idxart{534}.


\idxsection*{J.}

\idxc{Janssen}, discovery of the spectroscopic  method of observing the solar prominences,  \idxart{323};
solar photography, \idxart{289}.

\idxb{Jets} issuing from the nucleus of a comet,  \idxart{713}, \idxart{727}.

\idxc{Jolly}, observations of the earth's density,  \idxart{170}.

\idxb{Julian} calendar, \idxart{219}.

\idxb{Juno}, discovered by Harding, \idxart{593}.

\idxb{Jupiter}, the planet, \idxart{609}--\idxart{631};
brightness as seen from $\alpha$~Centauri, \idxart{881};
a semi-sun,  \idxart{619}.


\idxsection*{K.}

\idxc{Kant}, proposes the nebular hypothesis,  \idxart{908}.

\idxc{Kepler}, his belief as to cometary orbits, \idxart{700};
his three laws of planetary motion,  \idxart{412}--\idxart{418};
his ``problem,'' \idxart{188};
his ``regular solid'' theory of the planetary distances, \idxart{592}, \textit{note}.

\idxc{Kirchoff}, his fundamental principles of spectrum analysis, \idxart{314}.

\idxsection*{L.}

\idxc{Langley, S. P.}, his Bolometer, \idxart{343};
the color of the sun, \idxart{337};
observations on lunar heat, \idxart{260}, \idxart{261};
on solar heat, \idxart{348};
sun-spot drawings, \idxart{292};
heat in sun spots, \idxart{293}.

\idxc{Lane's} law, rise of temperature consequent on the contraction of a gaseous mass, \idxart{357}.

\idxc{La Place}, his equations relating to the eccentricities and inclinations of the  planetary orbits, \idxart{532};
explanation of  the moon's secular acceleration, \idxart{459},  \idxart{460};
the invariable plane of the solar system, \idxart{531};
the nebular hypothesis, \idxart{901}--\idxart{911}.

\idxc{Lassell}, discovery of the two inner satellites of Uranus, \idxart{650};
independent discovery of Hyperion, \idxart{643}.

\idxb{Latitude} (astronomical) of a place on the earth's surface, \idxart{30}, \idxart{100}, \idxart{156};
astronomical,  geodetic, and geocentric, distinguished, \idxart{156};
determination of, methods used, \idxart{101}--\idxart{107};
at sea, its determination, \idxart{103};
possible variations of it, \idxart{108};
station errors, \idxart{158};
celestial, defined, \idxart{178}, \idxart{179};
and longitude (celestial), conversion into $\alpha$ and $\delta$, \idxart{180}.

\idxb{Law} of angular velocity under central force, \idxart{408};
Bode's, \idxart{488}, \idxart{489};
of Boyle or Mariotte, density of a gas, \idxart{360}, \textit{note};
of Dalton, mixture of gases, \idxart{360}, \textit{note};
of earth's orbital motion, \idxart{186}, \idxart{187};
of  equal areas, \idxart{186}, \idxart{402}--\idxart{406}, \idxart{412};
of Gay Lussac, gaseous expansion, \idxart{360}, \textit{note};
of gravitation, \idxart{161}, \idxart{162}, \idxart{419}, \idxart{872};
Lane's, of temperature in gaseous contraction, \idxart{357};
of linear velocity in angular motion, \idxart{407}.

\idxb{Laws} of Kepler, \idxart{412}--\idxart{418};
motion under a central force, \idxart{400}--\idxart{411}.

\idxb{Leap} year, rule for, \idxart{220}.

\idxb{Length}, of the day, possible changes in it, \idxart{144};
of the year, its invariability, \idxart{526}, \idxart{778}.

\idxb{Leonids}, the, \idxart{780}, \idxart{786}.

\idxc{Lescarbault}, supposed discovery of Vulcan, \idxart{603}.

\idxb{Level} adjustment of the transit instrument, \idxart{60}.

\idxc{Leverrier}, discovery of Neptune, \idxart{653}, \idxart{654};
on an intra-Mercurial planet, \idxart{603};
motion of the perihelion of Mercury's orbit, \idxart{602};
method of determining the solar parallax by planetary perturbations, \idxart{689}.

\idxc{Lexell's} comet, approach to Jupiter, \idxart{718};
recognition of Uranus as a planet, \idxart{645}.

\idxb{Librations} of the moon, \idxart{249}, \idxart{250}, \idxart{251}.

\idxb{Light} of comets, \idxart{722};
of the moon, \idxart{259};
emitted by certain stars, \idxart{835};
received by the earth from certain stars, \idxart{832};
of the sun, \idxart{332}--\idxart{337};
total, of the stars, \idxart{833};
equation of, from Jupiter's satellites, \idxart{628}--\idxart{630};
mechanical equivalent
%% -----File: File: 555.png---Folio 544-------
of, Thomsen, \idxart{776};
time occupied by, in coming from the sun, \idxart{275}, \idxart{629};
velocity of, \idxart{225}, \textit{note}, \idxart{668}, \idxart{690}.

\idxb{Light-curves} of variable stars, \idxart{848}.

\idxb{Light-gathering} power of telescopes, \idxart{38}.

\idxb{Light-ratio}, the, in scale of star-magnitudes, \idxart{819}.

\idxb{Light-year}, the, the unit of stellar distance, \idxart{814}.

\idxb{Limb} of the sun, darkening of, \idxart{337};
of the sun, diminution of heat, \idxart{348}.

\idxb{Limiting} apertures in stellar photometry, \idxart{825}.

\idxb{Linear} and angular dimensions, their relation, \idxart{5};
velocity under central force, its law, \idxart{407}, \idxart{409}.

\idxb{Linné}, lunar crater supposed to have changed, \idxart{269}.

\idxc{Listing}, dimensions of the earth, \idxart{145}.

\idxb{Local} and standard time, \idxart{122}.

\idxc{Lockyer, J. N.}, discovery of the spectroscopic method of observing the solar prominences, \idxart{323};
his ``collision theory'' of variable stars, \idxart{850};
views as to the compound nature of the so-called chemical ``elements,'' \idxart{318};
origin of the Fraunhofer lines, \idxart{320};
theory of sun spots, \idxart{306};
meteoric theory of nebulę, \idxart{894};
meteoric hypothesis, \idxart{926}.

\idxc{Loewy}, peculiar method of determining the refraction, \idxart{95}.

\idxb{Longitude}, arcs of, to determine the earth's dimensions, \idxart{151};
(terrestrial), determination of, \idxart{118}--\idxart{121};
(celestial), \idxart{178}--\idxart{180};
of perihelion, \idxart{505}, \idxart{506}.

\idxb{Luminosity} of bodies at low temperatures, \idxart{737}, \textit{note.}

\idxb{Lunar} distances, \idxart{120}, B;
eclipses, \idxart{370}--\idxart{378};
influences on the earth, \idxart{262};
methods of determining the longitude, \idxart{120};
perturbations, \idxart{448}--\idxart{461};
perturbations used to determine the solar parallax, \idxart{687}.

\idxb{Lyrę}, $\alpha$, see \idxsee{Vega};
$\beta$, variable star, \idxart{847};
$\epsilon$, quadruple star, \idxart{653}, \idxart{866}, \idxart{882}.


\idxsection*{M.}

\idxc{Mädler}, speculations as to a central sun, \idxart{807}.

\idxb{Magnifying} power of a telescope, \idxart{37};
power, highest available, \idxart{43}.

\idxb{Magnitudes} of stars, \idxart{816}--\idxart{822}.

\idxb{Magnitude} of smallest star visible in a given telescope, \idxart{822}.

\idxb{Magnesium} in the nebulę, \idxart{890}, \idxart{894}.

\idxb{Maintenance} of the solar heat, \idxart{353}--\idxart{356}.

\idxb{Mars}, the planet, \idxart{578}--\idxart{591};
observed for solar parallax, \idxart{673}--\idxart{677}.

\idxb{Maskelyne}, his mountain method of determining the earth's density, \idxart{164}.

\idxb{Mass} and weight, distinction between them, \idxart{159}, \idxart{160};
of comets, \idxart{718}, \idxart{719};
of the earth compared with the sun, \idxart{278};
of the earth in terms of the sun as determining the solar parallax, \idxart{689};
of the moon, its determination, \idxart{243};
of a planet, how determined, \idxart{536}--\idxart{539};
of the sun, compared with the earth, \idxart{278};
probable, of shooting stars, \idxart{776}.

\idxb{Mass-brightness} of binary stars, \idxart{879}.

\idxb{Masses} of binary stars, \idxart{877}, \idxart{878}.

\idxc{Mayer, R.}, meteoric theory of the solar heat, \idxart{353}.

\idxb{Maxwell, Clerk}, meteoric theory of Saturn's rings, \idxart{641}.

\idxb{Mazapil}, the meteorite of, \idxart{784}.

\idxb{Mechanical} equivalent of light, \idxart{776}.

\idxb{Mercury}, the planet, \idxart{551}--\idxart{562}.

\idxb{Mercury's} orbit, motion of its perihelion, \idxart{602}.

\idxb{Meridian}, the celestial, defined, \idxart{19};
arc of, how measured, \idxart{147}.

\idxb[Meridian Circle]{Meridian-circle}, the, \idxart{63};
used to determine the place of a heavenly body, \idxart{128}.

\idxb{Meridian} photometer, the, \idxart{828}.

\idxb{Meteors} and shooting stars, \idxart{755}--\idxart{787};
ashes of, \idxart{775};
and comets, their connection, \idxart{785}--\idxart{787};
daily number of, \idxart{771};
detonating, \idxart{768};
effect on the earth's orbital motion, \idxart{778};
effect upon the moon's motion, \idxart{778};
effect upon the transparency of space, \idxart{779};
explanation of their light and heat, \idxart{765};
heat from them, \idxart{355}, \idxart{779};
magnitude of, \idxart{762};
method of observing them, \idxart{764};
their trains, \idxart{766}.

\idxb{Meteoric} growth of the earth, \idxart{777};
showers, \idxart{780}--\idxart{786};
shower of the Bielids, 1872, 1886, \idxart{746};
shower of the Leonids, 1833, 1866--67, \idxart{781};
swarms and rings, \idxart{783};
swarms, special characteristics, \idxart{783};
theory of Saturn's rings, \idxart{641};
theory of the sun's heat, \idxart{353}--\idxart{355};
theory of sun spots, \idxart{306}.

\nblabel{idx:Meteorite}% Typo in Uranolith index entry refers to Meteorite rather than Meteorites
\idxb[Meteorites]{Meteorites}, or uranoliths, or aerolites, \idxart{755}--\idxart{769};
chemical elements in them, \idxart{760};
crust, \idxart{761};
fall of, \idxart{756};
which have fallen in the United States, \idxart{759};
iron, list of, \idxart{758};
number of, \idxart{769};
question of their origin, \idxart{767};
their paths, \idxart{763}.

\idxb{Metonic} cycle, the, \idxart{218}.
%% -----File: File: 556.png---Folio 545-------

\idxc{Michelson}, determination of the velocity of light, \idxart{225}, \textit{note}, \idxart{668}, \idxart{690}.

\idxb{Micrometer}, the filar, \idxart{73}, \idxart{534}, \idxart{867}.

\idxb{Microscope}, the reading, \idxart{64}.

\idxb{Midnight} sun, the, \idxart{191}.

\idxb{Milky Way}, or galaxy, \idxart{898}.

\idxb{Mimas}, the innermost satellite of Saturn, \idxart{643} and \textit{note}.

\idxb{Minor} planets, or asteroids, \idxart{592}--\idxart{601}.

\idxb{Mira}, Omicron Ceti, \idxart{846}.

\idxb{Missing} stars, \idxart{840}.

\idxb{Mohammedan} calendar, \idxart{217}.

\idxc{Monck}, the mass-brightness of binary stars, \idxart{879}.

\idxb{Monocentric} eye-piece for telescope, \idxart{45}.

\idxb{Moon}, the, \idxart{227}--\idxart{272};
her atmosphere, \idxart{255}--\idxart{258};
the, regarded as a clock, \idxart{120};
distance of, etc., \idxart{240};
her heat and temperature, \idxart{260}, \idxart{261};
her light as compared with sunlight, \idxart{259};
influences on the earth, \idxart{262};
mass of, determined, \idxart{243}, \idxart{244};
her motion (apparent), \idxart{228};
her motion relative to the sun, \idxart{241};
her mountains, measurement of their elevation, \idxart{270};
her orbit with reference to the earth, \idxart{238};
her parallax determined, \idxart{239};
perturbations, \idxart{448}--\idxart{461};
her rotation and librations, \idxart{248}--\idxart{251};
her phases, \idxart{253};
her surface character, \idxart{263}--\idxart{270};
culminations for longitude, \idxart{120}, A.

\idxb{Month}, the anomalistic, \idxart{397}, \textit{note};
the nodical, \idxart{397}, \textit{note};
the sidereal, \idxart{229}, \idxart{232};
the synodic, \idxart{229}, \idxart{232};
length of, increased by perturbation, \idxart{453};
slightly shortened by the secular acceleration, \idxart{459}.

\idxb{Motion}, direct and retrograde, of the planets, \idxart{494};
of the solar system in space, \idxart{804}--\idxart{807};
in line of sight, effect on spectrum, \idxart{321};
of stars in line of sight, spectroscopically observed, \idxart{802}.

\idxb{Motions}, proper, of the stars, \idxart{800}--\idxart{803}.

\idxb{Mountains}, lunar, their height, \idxart{270}.

\idxb{Mountain} method of determining the earth's density, \idxart{164}.

\idxb{Multiple} stars, \idxart{882}.

\idxb{Mural} circle, the, \idxart{70}.


\idxsection*{N.}

\idxb{Nadir}, the, defined, \idxart{9};
point of meridian circle, \idxart{67}.

\idxb{Names} of the constellations, \idxart{792};
of Jupiter's satellites, \idxart{621};
of satellites of Mars, \idxart{590};
of the planets, \idxart{487}, \idxart{489};
of Saturn's satellites, \idxart{643}, \textit{note};
of the satellites of Uranus, \idxart{650};
of stars, \idxart{794}.

\idxb{Neap} tide defined, \idxart{463}.

\idxb{Nebula}, the great, in Andromeda, \idxart{886};
the annular, in Lyra, \idxart{888};
of Orion, the, \idxart{886}, \idxart{892}, \idxart{893}.

\idxb{Nebulę}, the, \idxart{886}--\idxart{897};
changes in, \idxart{892};
their distance, \idxart{896};
Lockyer's meteoric theory, \idxart{894};
their nature, \idxart{894};
their number and distribution, \idxart{895};
photographs of, \idxart{893};
planetary, \idxart{888};
spiral, \idxart{888};
their spectra, and chemical elements in them, \idxart{890}, \idxart{891}.

\idxb[Nebular]{Nebular} hypothesis, the, \idxart{908}--\idxart{915};
modifications of the original theory, \idxart{912}, \idxart{913}.

\idxb{Negative} eye-pieces for the telescope, \idxart{44};
shadow of the moon, \idxart{381};
star magnitudes, \idxart{821}.

\idxb{Neptune}, the planet, \idxart{653}--\idxart{661};
anomalous retrograde rotation, in relation to the nebular hypothesis, \idxart{914};
appearance of sun and solar system from it, \idxart{658};
(actual) discovery by Galle, \idxart{654};
theoretical discovery by Leverrier and Adams, \idxart{653}, \idxart{654};
its discovery ``no accident,'' \idxart{655};
the computed elements erroneous, \idxart{655};
its satellite, \idxart{661};
spectrum of, \idxart{660}.

\idxc{Newcomb, S.}, conclusions as to sun's age and duration, \idxart{358}, \idxart{359};
observations on meteors, \idxart{776};
on the moon's secular acceleration, \idxart{461};
on the structure of the heavens, \idxart{900};
his value of the solar parallax, \idxart{667};
velocity of light, \idxart{225}, \textit{note}, \idxart{668}, \idxart{690}.

\idxc{Newton}, Prof. H.~A., daily number of meteors, \idxart{771};
investigation of meteoric orbits, \idxart{767}, \idxart{785};
theory of the constitution of a comet, \idxart{737}.

\idxc{Newton, Sir Isaac}, discovery of gravitation, \idxart{161}, \idxart{419};
verification of the idea of gravitation by means of the moon's motion, \idxart{419}, \idxart{420};
discovery that planetary orbits must be conics, \idxart{421};
computation of a cometary orbit, \idxart{700};
his reflecting telescope, \idxart{48}.

\idxb{Nitrogen}, suspected in the nebulę, \idxart{890}.

\idxb{Node} of an orbit defined, \idxart{233}, \idxart{506}.

\idxb{Nodes} of moon's orbit, their regression, \idxart{455};
of the planetary orbits, their motion, \idxart{527}.

\idxb{Nodical} month, the, \idxart{249}, \idxart{397}, \textit{note}.

\idxc{Nordenskiold}, meteoric ashes, \idxart{775}.

\idxb{Nucleus} of a comet, \idxart{713}, \idxart{716}.

\idxb{Number} of eclipses in a year, \idxart{391};
in a saros, \idxart{398};
of meteors and meteorites, \idxart{759}, \idxart{769}, \idxart{771};
and designation of variable stars, \idxart{852}.

\idxb{Nutation} of the earth's axis, \idxart{214}, \idxart{215}.
%% -----File: File: 557.png---Folio 546-------

\idxsection*{O.}

\idxb{Oberon}, the outer satellite of Uranus, \idxart{650}.

\idxb{Object-glass}, achromatic, \idxart{41};
residual or secondary spectrum of, \idxart{42};
designed for photography, \idxart{42}.

\idxb{Oblateness}, or ellipticity of a spheroid, \idxart{150}.

\idxb{Oblique} sphere, the, \idxart{33}.

\idxb{Obliquity} of the ecliptic defined and measured, \idxart{176};
of the ecliptic, secular change of, \idxart{197}.

\idxb{Occultation}, circle of perpetual, \idxart{33};
of stars, \idxart{399};
of stars used for longitude determination, \idxart{120}~C;
of stars proving absence of lunar atmosphere, \idxart{256}.

\idxc{Olbers}, discovers Pallas and Vesta, \idxart{593}.

\idxc{Olmsted, D.}, his researches on meteors, \idxart{785}.

\idxc{Oppolzer}, effect of meteors on the moon's motion, \idxart{778};
orbit of Tempel's Comet, \idxart{786};
motion of Winnecke's Comet, \idxart{711}, \textit{note}.

\idxb{Orbit} of the earth, its form determined, \idxart{182};
of the earth, effect of meteors upon it, \idxart{778};
of the earth, perturbations, \idxart{197};
of the moon, \idxart{238};
of the moon, its perturbations, \idxart{448}--\idxart{461};
of a planet, determined graphically, \idxart{428}, \idxart{431}, \idxart{432};
planetary, its elements, \idxart{505}--\idxart{510};
planetary, its elements, determination of, \idxart{519}.

\idxb{Orbits} of binary stars, \idxart{873};
of comets, \idxart{700}--\idxart{709};
of planets, diagram, \idxart{489};
of sun and stars in the stellar system, \idxart{904}.

\idxb{Origin} of comets, \idxart{738}--\idxart{741};
of meteorites or aerolites, \idxart{767}.

\idxb{Orthogonal} component of the disturbing force, \idxart{445}, \idxart{455}.

\idxsection*{P.}

\idxb{Pallas}, discovered by Olbers, \idxart{593}.

\idxc{Palisa}, discoverer of sixty-five asteroids, \idxart{593}.

\idxb{Parabola}, the, described as a conic, \idxart{422}, \idxart{423}.

\idxb{Parabolic} comets, their number, \idxart{702};
velocity, the, \idxart{429}.

\idxb[Parallax of the Sun]{Parallax} (diurnal), defined and discussed, \idxart{82}, \idxart{83};
of the moon, determined, \idxart{239};
of the sun, classification of methods, \idxart{669};
of the sun, gravitational methods, \idxart{687}--\idxart{689};
of the sun, history of investigations, \idxart{666}--\idxart{668};
of the sun, method of Aristarchus, \idxart{670};
of the sun, method of Hipparchus, \idxart{671};
of the sun by observations on Mars, \idxart{673}, \idxart{676};
of the sun by transits of Venus, \idxart{678}, \idxart{686};
of the sun by the velocity of light, \idxart{690}--\idxart{692};
of the sun, Ptolemy's value, \idxart{671};
of the stars (annual), \idxart{808}--\idxart{814};
of $\alpha$~Centauri, Henderson, \idxart{809}, \idxart{810};
of 61 Cygni, Bessel, \idxart{809}--\idxart{811};
of $\alpha$~Lyrę, negative, Pond, \idxart{809};
stellar, absolute method, \idxart{810};
stellar, differential method, \idxart{811};
stellar, table of, Appendix, \hyperref[app:IV]{Table IV.}

\idxb{Parallactic} inequality of the moon, \idxart{687};
orbit of a star, \idxart{808}.

\idxb{Parallel} sphere, the, \idxart{32}.

\idxb{Parallels} of declination, \idxart{23}.

\idxc{Peirce, B.}, heat from meteors, \idxart{355};
on the mass of comets, \idxart{719};
theory of sun spots, \idxart{306}.

\idxb{Pendulum}, compensation, \idxart{51};
use in clocks, \idxart{50};
used in determining form of the earth, \idxart{152}--\idxart{155};
free, of Foucault, showing earth's rotation, \idxart{139}--\idxart{141}.

\idxb{Penumbra} of the earth's shadow, \idxart{368};
of the moon's shadow, \idxart{383};
of a sun spot, \idxart{295}.

\idxb{Perigee} and apogee defined, \idxart{238}.

\idxb{Perihelia} of comets, their distribution, \idxart{706}.

\idxb{Perihelion} of earth's orbit defined, \idxart{183};
its motion, \idxart{199};
of Mercury's orbit, its motion, \idxart{602}.

\idxb{Period}, sidereal and synodic, of the moon, \idxart{229}--\idxart{232};
sidereal and synodic, of a planet, defined, \idxart{490};
sidereal, of a planet, determined, \idxart{513}, \idxart{514}.

\idxb{Periodic} comets, \idxart{703}, \idxart{704}, \idxart{738}--\idxart{740};
table of comets of short period, Appendix, \hyperref[app:III]{Table III.}

\idxb{Periodicity} of sun spots, \idxart{307}--\idxart{309}.

\idxb{Persei}, $\beta$, or Algol, \idxart{848}.

\idxb{Perseids}, the, meteoric swarm, \idxart{780}, \idxart{782}, \idxart{783}.

\idxb{Personal} equation, \idxart{114}, \idxart{120}~A, \idxart{121}~B.

\idxb{Perturbations}, lunar, \idxart{448}--\idxart{461};
planetary, \idxart{521}--\idxart{523};
of Mars and Venus by the earth as determining the sun's parallax, \idxart{689}.

\idxc{Peters, C.~H.~F.}, discovers fifty-two asteroids, \idxart{593}.

\idxc{Piazzi}, discovery of Ceres, \idxart{592}.

\idxc{Picard}, measure of earth's diameter, \idxart{136}.

\idxc{Pickering, E.~C.}, his meridian photometer, \idxart{828};
photography of stellar spectra, \idxart{860}, \idxart{862};
photometric observations of the eclipses of Jupiter's satellites, \idxart{630};
the Harvard photometry, \idxart{827}.

\idxb{Phases} of Mercury, Venus, and Mars, \idxart{559}, \idxart{567}, \idxart{582};
of the moon, \idxart{253}.

\idxb{Phobos}, the inner satellite of Mars, \idxart{589}.

\idxb{Photographs} of the moon, \idxart{272};
of the nebulę, \idxart{893};
of the solar corona, \idxart{328};
of the sun's surface and spots, \idxart{289}.
%% -----File: File: 558.png---Folio 547-------

\idxb{Photographic} object-glasses, \idxart{42};
observations of eclipses of Jupiter's satellites, \idxart{630}, and \textit{note};
observations of transit of Venus, \idxart{684}--\idxart{686}.

\idxb{Photography} as a means of photometry, \idxart{829};
solar, \idxart{289};
spectroscopic, motion in line of sight, \idxart{802};
applied to star-charting, \idxart{798};
in determination of stellar parallax, \idxart{812};
of stellar spectra, \idxart{859}--\idxart{863}.

\idxb{Photometer}, the meridian, \idxart{828};
polarization, \idxart{827};
the wedge, \idxart{826}.

\idxb{Photometry}, Harvard, the, \idxart{827}, \idxart{828};
by means of photography, \idxart{829};
by the spectroscope, \idxart{831};
of sunlight, \idxart{332}--\idxart{335};
stellar, \idxart{823}--\idxart{831}.

\idxb{Photosphere of the} sun, its nature, \idxart{291}, \idxart{292}, \idxart{361}.

\idxb{Photo-tachymetrical} determination of the sun's parallax, \idxart{690}--\idxart{692}.

\idxb{Physical} characteristics of comets, \idxart{712};
method of determining sun's parallax, \idxart{690}--\idxart{692}.

\idxb{Planet}, intra-Mercurial, \idxart{602}--\idxart{606};
trans-Neptunian, \idxart{662}.

\idxb{Planets} attending certain stars, \idxart{880}, \idxart{881};
distances and periods, \idxart{489};
enumerated, \idxart{486}, \idxart{487};
relative age, according to nebular hypothesis, \idxart{913}, \idxart{915};
orbits, diagram of, \idxart{489};
orbits, elements of, \idxart{505}, \idxart{510}.

\idxb{Planetoid}. See \idxsee{Asteroid}, \idxart{591}.

\idxb{Planetary} data, tables of, Appendix, \hyperref[app:I]{Table I.};
data, accuracy of, \idxart{653};
nebulę, \idxart{888};
system, facts suggesting the theory of its origin, \idxart{907};
system, Sir J. Herschel's illustration of its dimensions, \idxart{664}.

\idxb{Pleiades}, the, \idxart{884}.

\idxc{Pogson}, his absolute scale of star-magnitudes, \idxart{819}.

\idxb{Pole} of the earth, \idxart{28};
(celestial), defined, \idxart{14};
its altitude equal to the latitude, \idxart{30}, \idxart{100};
its place affected by precession, \idxart{206}, \idxart{207}.

\idxb{Pole-star}, ancient, $\gamma$~Draconis, \idxart{207};
its position and recognition, \idxart{15}.

\idxb{Polar} distance, defined, \idxart{23};
point of meridian circle, \idxart{66}.

\idxb{Position-angle} of a double star, \idxart{868}.

\idxb{Position} of a heavenly body, how determined, \idxart{128}, \idxart{129}.

\idxb{Positive} eye-pieces for telescopes, \idxart{44}.

\idxc{Pouillet}, his pyrheliometer, \idxart{340}.

\idxb{Power}, magnifying, of telescope, \idxart{37}.

\idxc{Poynting}, determination of the earth's density, \idxart{170}.

\idxb{Practical} astronomy defined, \idxart{2}.

\idxb{Precession} of the equinoxes, \idxart{205}--\idxart{212}.

\idxb{Prime} vertical defined, \idxart{19};
vertical instrument, \idxart{62}, \idxart{106}.

\idxb{Priming} and lagging of the tides, \idxart{470}.

\idxc{Pritchard, Prof. C.}, determination of stellar parallax by means of photography, \idxart{812};
stellar photometry, \idxart{826};
Uranometria Oxoniensis, \idxart{826}.

\idxc{Pritchett, C.~W.}, discovery of the great red spot on Jupiter, \idxart{618}.

\idxb{Problem} of three bodies, \idxart{437}--\idxart{461};
of two bodies, \idxart{424}--\idxart{433}.

\idxb{Problems} illustrating Kepler's third law, \idxart{413}.

\idxc{Proctor, R.~A.}, on the origin of comets, \idxart{741};
determination of the rotation period of Mars, \idxart{584}.

\idxb{Projectiles}, deviation caused by earth's rotation, \idxart{143};
their path near the earth, \idxart{435}.

\idxb{Projectile} force, careless use of the term, \idxart{401}.

\idxb{Prominences}, or protuberances, the solar, \idxart{291}, \idxart{323}--\idxart{326}, \idxart{363};
quiescent and eruptive, \idxart{325}, \idxart{326}.

\idxb{Proper} motions of the stars, \idxart{800}, \idxart{803}.

\idxb{Proximity} of a star, indications of it, \idxart{813}.

\idxb{Ptolemaic} system, the, \idxart{500}, \idxart{502}.

\idxc{Ptolemy}, his almagest, \idxart{500}, \idxart{700}, \idxart{795}.

\idxb{Pyrheliometer} of Pouillet, \idxart{340}.


\idxsection*{Q.}

\idxb{Quantity} of the solar radiation in calories, \idxart{338}--\idxart{340};
of sunlight in candle power, \idxart{332}, \idxart{333}.

\idxb{Quiescent} prominences, \idxart{325}.


\idxsection*{R.}

\idxb{Radial} compound of the disturbing force, \idxart{446}.

\idxb{Radian}, the, defined as angular unit, \idxart{5}, \textit{note}.

\idxb{Radiant}, the, in meteoric showers, \idxart{780}.

\idxb{Radius} of curvature of a meridian, \idxart{149}.

\idxc{Ranyard}, peculiar theory of the repulsive force operative in comets' tails, \idxart{733}.

\idxb{Rate} of a clock defined, \idxart{53}.

\idxb{Reading} microscope, the, \idxart{64}.

\idxb{Recognition} of elliptic comets, difficulties, \idxart{704}.

\idxb{Red} spot of Jupiter, the, \idxart{618}.

\idxb{Reduction} of mean star places to apparent and \textit{vice versa}, \idxart{797}.

\idxb{Reflecting} telescope, various forms, \idxart{47}, \idxart{48};
telescopes, large instruments, \idxart{48}.
%% -----File: File: 559.png---Folio 548-------

\idxb{Refraction}, atmospheric, its law, \idxart{89}, \idxart{90};
determination of its amount, \idxart{94}, \idxart{95};
effect of temperature and barometric pressure, \idxart{91};
effect upon form and size of discs of sun and moon near the horizon, \idxart{93};
effect upon time of sunrise and sunset, \idxart{92}.

\idxb{Refracting} telescope (simple), \idxart{36};
telescope, achromatic, \idxart{41}.

\idxb{Refractors} and reflectors compared, \idxart{49}.

\idxc{Reich}, determination of the density of the earth, \idxart{166};
experiments upon falling bodies, \idxart{138}.

\idxb{Relative} motion, law of, \idxart{492};
sizes of the planets, diagram, \idxart{550}.

\idxb{Repulsive} force acting on comets, \idxart{728}, \idxart{731}, \idxart{732}, \idxart{734}.

\idxb{Retardation} of earth's rotation by the tides, \idxart{461}, \idxart{483}.

\idxb{Reticle} used in telescope for pointing, \idxart{46}.

\idxb{Retrograde} revolution of the satellites of Uranus and Neptune, \idxart{652}, \idxart{661}, \idxart{914}.

\idxb{Reversing} layer of the solar atmosphere, \idxart{291}, \idxart{319}, \idxart{320}, \idxart{362}.

\idxb{Reversal} of the spectrum, \idxart{314}.

\idxb{Rhea}, the fifth satellite of Saturn, \idxart{643}, \textit{note}.

\idxb{Rigidity} of the earth, \idxart{171}, \idxart{482}.

\idxb{Right} ascension defined, \idxart{25}, \idxart{27};
ascension determined by transit instrument, \idxart{59}, \idxart{128}, \idxart{129};
sphere, the, \idxart{31}.

\idxb{Rings} of Saturn, \idxart{637}--\idxart{642}.

\idxc{Rosse, Lord}, observations of lunar heat, \idxart{260}, \idxart{261};
his great telescope, \idxart{48};
spiral nebulę, \idxart{888}.

\idxb{Rotation}, distinguished from revolution, \idxart{248}, \idxart{248*};
of the earth, affected by the tides, \idxart{461}, \idxart{483};
of the earth, proofs of, \idxart{138}--\idxart{143};
of planets, how determined, \idxart{543};
period of Jupiter, \idxart{615};
period of Mars, \idxart{584};
period of the moon, \idxart{248}, \idxart{252};
period of Saturn, \idxart{635};
of the sun, \idxart{281}, \idxart{283};
period of Venus, \idxart{570};
periods, see also Appendix, \hyperref[app:I]{Table I.}


\idxsection*{S.}

\idxb{Saros}, the, \idxart{395}--\idxart{398};
number of eclipses in a saros, \idxart{398}.

\idxb{Satellites} of Jupiter, \idxart{621}--\idxart{631};
of Mars, \idxart{590}, \idxart{591};
of Neptune, \idxart{661};
of Saturn, \idxart{643}, \idxart{644};
of Uranus, \idxart{650}--\idxart{652};
general table of, Appendix, \hyperref[app:II]{Table II.}

\idxb{Satellite} orbits, generally circular, \idxart{548}.

\idxb{Saturn}, the planet, \idxart{632}--\idxart{644}.

\idxb{Schehallien}, determination of the earth's mass, \idxart{164}.

\idxc{Schiaparelli}, connection between comets and meteors, \idxart{786};
his map of Mars, \idxart{588}.

\idxc{Schröter}, the rotation of Mercury, \idxart{559};
the rotation of Venus, \idxart{570}.

\idxc{Schwabe}, discovery of the periodicity of sun spots, \idxart{307}.

\idxb{Scintillation} of the stars, \idxart{864}.

\idxb{Sea}, ship's place at, \idxart{103}, \idxart{120}~B, \idxart{121}~A, \idxart{124}--\idxart{126}.

\idxb{Seasons}, the, explained, \idxart{190}, \idxart{192}, \idxart{193};
difference between northern and southern hemispheres, \idxart{194}, \idxart{195}.

\idxc{Secchi}, theories of sun spots, \idxart{303}, \idxart{305};
observations on stellar spectra, \idxart{856}, \idxart{857}.

\idxc{Seidel}, his photometer, \idxart{827}.

\idxb{Secular} acceleration of the moon's mean motion, \idxart{459}--\idxart{461};
changes in the earth's orbit, \idxart{196}--\idxart{200};
perturbations in the planetary system, \idxart{525}--\idxart{529}.

\idxb{Semi-diameter}, augmentation of the moon's, \idxart{88};
correction for, in sextant observations, \idxart{88}.

\idxb{Semi-major} axis of a planet's orbit, defined and discussed, \idxart{505}, \idxart{506};
axis of the planets' orbits, invariable, \idxart{526};
axis as depending on planet's velocity, \idxart{428}--\idxart{430}.

\idxb{Separating} power of a telescope, Dawes, \idxart{43}.

\idxb{Sequences}, method of, in stellar photometry, \idxart{824}.

\idxb{Sextant}, the, described, \idxart{76};
the, used to determine latitude, \idxart{103};
the, used in finding a ship's place at sea, \idxart{103}, \idxart{116}, \idxart{125}, \idxart{126};
the, used in determining time, \idxart{115}, \idxart{116}.

\idxb{Shadow} of the earth, its dimensions, \idxart{367};
of the moon, \idxart{379}, \idxart{380};
of the moon, its velocity over the earth, \idxart{384}.

\idxb{``Sheath''} of the comet of 1882, \idxart{752}.

\idxb{Ship's} place at sea, determination of, \idxart{103}, \idxart{120}~B, \idxart{121}~A, \idxart{124}--\idxart{126}.

\idxb[Shooting Stars]{Shooting Stars}, \idxart{770}, \idxart{787};
ashes of, \idxart{775};
brightness of, \idxart{773};
comparative numbers in morning and evening, \idxart{772};
daily number of, \idxart{771};
elevation of, \idxart{774};
mass of, \idxart{776};
materials of, \idxart{775};
path of, \idxart{774};
showers of, \idxart{780}--\idxart{786};
spectra of, \idxart{775};
velocity of, \idxart{774}.

\idxb{Short-period} comets, \idxart{703};
comets, table of, Appendix, \hyperref[app:III]{Table III.}

\idxb{Showers}, meteoric, \idxart{780}--\idxart{786}.

\idxb{Sidereal} day defined, \idxart{26}, \idxart{110};
month, \idxart{229};
time, \idxart{26}, \idxart{110};
year, its length, \idxart{216}.

\idxb{Signals} used in determining difference of longitude, \idxart{119}.
%% -----File: File: 560.png---Folio 549-------

\idxb{Signs} of the zodiac, \idxart{177}.

\idxb{Single-altitude} method of determining local time, \idxart{116}.

\idxb{Sirius} and its companion, \idxart{875};
its light compared with the sun's, \idxart{334}, \idxart{832}, \idxart{835};
its mass, \idxart{877}.

\idxb{Sky}, apparent distance of, \idxart{6}.

\idxb{Slitless} spectroscope, the, \idxart{860}--\idxart{862}.

\idxb{Solar} constant, the, \idxart{338}--\idxart{340};
eclipses, \idxart{319}, \idxart{323}, \idxart{327}, \idxart{329}, \idxart{387}--\idxart{390}, \idxart{393}, \idxart{398};
eclipses, their rarity, \idxart{398};
engine of Ericsson and Mouchot, \idxart{345};
eye-pieces, \idxart{286}, \idxart{287};
parallax, see \idxsee{Parallax of the Sun};
system, age of, \idxart{922};
time, apparent and mean, defined, \idxart{111}, \idxart{112}.

\idxb{Solstice} defined, \idxart{176}.

\idxc{Sosigenes}, devises the Julian calendar, \idxart{219}.

\idxb{Spectra} of comets, \idxart{724}--\idxart{726};
of meteors, \idxart{775};
of nebulę, \idxart{890}, \idxart{891};
of stars, \idxart{855}--\idxart{863}.

\idxb{Spectroscope}, principles of its construction, \idxart{311}--\idxart{313};
how it shows the solar prominences, \idxart{324};
slitless, \idxart{860}--\idxart{862}.

\idxb{Spectroscopic} measurement of motions in the line of vision, \idxart{321}, \idxart{802}.

\idxb{Spectrum} of the chromosphere and prominences, \idxart{323};
of the corona, \idxart{329};
solar (photosphere), \idxart{312};
solar, compared with iron, \idxart{315};
of sun spots, \idxart{321};
analysis, fundamental principles, \idxart{314};
photometry, \idxart{831}.

\idxb{Sphere}, the celestial, conceptions of it, \idxart{4};
the oblique, \idxart{33};
the parallel, \idxart{32};
the right, \idxart{31}.

\idxb{Spheres}, attraction of, \idxart{162}.

\idxb{Spheroid}, terrestrial, its dimensions, \idxart{145}, Appendix, page~\pageref{app:spheroid}.

\idxb{Spherical} aberration of a lens, \idxart{39};
astronomy, defined, \idxart{3};
shell, its attraction, \idxart{169}.

\idxb{Spider} lines in a reticle, \idxart{46}.

\idxb{Spring} tide defined, etc., \idxart{463}.

\idxb{Spurious} disc of stars in the telescope, \idxart{43}.

\idxb{Stability} of the planetary system, \idxart{530}--\idxart{533}.

\idxb{Standard} and local time, \idxart{122}.

\idxb{Stars}, binary, see \idxsee{Binary Stars};
causes of the difference in their brightness, \idxart{836};
colors of, \idxart{830};
dark, \idxart{836};
designations and names, \idxart{794};
their real diameters, \idxart{837};
distribution of, \idxart{899};
double, see \idxsee{Double Stars};
gradual changes in their light, \idxart{839};
heat from them, \idxart{834};
light compared with sunlight, \idxart{334}, \idxart{832}, \idxart{835};
magnitudes of, \idxart{816}--\idxart{822};
missing, \idxart{840};
nature, as being suns, \idxart{789};
number of, \idxart{790};
parallax and distance, \idxart{808}--\idxart{814};
photography of, \idxart{798};
photometric observations of, \idxart{823}--\idxart{831};
proper motions of, \idxart{800}--\idxart{803};
proximity of, its indications, \idxart{813};
seen by day with telescope, \idxart{38};
shooting, see \idxsee{Shooting Stars};
temporary, \idxart{842}--\idxart{845};
triple and multiple, \idxart{882};
twinkling of, or scintillation, \idxart{864};
variable, see \idxsee{Variable Stars}.

\idxb{Star-atlases}, \idxart{793}.

\idxb{Star-catalogues}, \idxart{795}.

\idxb{Star-charts}, \idxart{798}.

\idxb{Star-clusters}, \idxart{883}, \idxart{884}.

\idxb{Star-gauges}, \idxart{899}.

\idxb{Star-motions}, \idxart{799}--\idxart{803}.

\idxb{Star-places}, how affected by aberration, etc., \idxart{226};
their determination, \idxart{796};
mean and apparent, \idxart{797}.

\idxb{Statical} theory of the tides, \idxart{469}.

\idxb{Station} errors, \idxart{158}.

\idxb{Stellar} spectra, \idxart{855}, \idxart{856};
classification of, \idxart{857}, \idxart{858};
photography of, \idxart{859}--\idxart{863};
system, the hypothetical, \idxart{901}--\idxart{904}.

\idxc{Stone, E.~J.}, attempted observation of stellar heat, \idxart{834}.

\idxb{Stripe}, central, in comets' tails, \idxart{730}.

\idxb{Structure} of the heavens, \idxart{900}--\idxart{904}.

\idxc{Struve, von, F.~G.~W.}, on distribution of stars, \idxart{899}.

\idxc{Struve, von, Ludwig}, investigation of sun's motion in space, \idxart{806}.

\idxc{Struve, von, Otto}, Saturn's rings, \idxart{637}, \idxart{642}.

\idxc{Sumner, Capt.}, his method of determining a ship's place at sea, \idxart{125}, \idxart{126}.

\idxb{Sun}, the, \idxart{273}--\idxart{364};
age and duration of, \idxart{359}, \idxart{922};
apparent annual motion of, \idxart{172}, \idxart{173};
attraction on the earth, its intensity, \idxart{436};
candle power of sunlight, \idxart{332}, \idxart{333};
chemical elements in it, \idxart{315}--\idxart{317};
diameter, surface, and volume, \idxart{276}, \idxart{277};
distance and parallax, \idxart{274}, \idxart{275}, \idxart{665}--\idxart{693};
gravity at its surface, \idxart{280};
heat emission, \idxart{338}--\idxart{357};
light, \idxart{332}--\idxart{336};
mass and density, \idxart{278}, \idxart{279};
its motion in space, \idxart{804}--\idxart{805};
physical constitution, \idxart{360}--\idxart{364};
its temperature, \idxart{349}--\idxart{351};
the central, \idxart{807}.

\idxb{Sun spots}, their development and changes, \idxart{297}, \idxart{298};
distribution on sun's surface, \idxart{301};
general description, \idxart{293}, \idxart{294};
influence on terrestrial conditions, \idxart{309}, \idxart{310};
periodicity of, \idxart{307}--\idxart{309};
their spectrum, \idxart{321};
theories as to their cause and nature, \idxart{302}--\idxart{306}.

\idxb{Sun's way}, apex of, \idxart{804}--\idxart{806}.

\idxb{Sunrise} and sunset affected by refraction, \idxart{92}.
%% -----File: File: 561.png---Folio 550-------

\idxb{Superior} planet, motions of, \idxart{496}.

\idxb{Surface} errors in lenses and mirrors, \idxart{49};
of the moon, \idxart{263}--\idxart{270}.

\idxb{Swarms}, meteoric, \idxart{783}.

\idxc{Swedenborg}, a proposer of the nebular hypothesis, \idxart{908}.

\idxb{System}, planetary, facts suggesting a theory of its origin, \idxart{907};
numerical data, Appendix, \hyperref[app:I]{Table I.};
stellar, \idxart{901}--\idxart{904}.

\idxb{Synodic} month, or revolution, of moon, \idxart{229}--\idxart{231};
period, general definition of, \idxart{490}.

\idxb{Syzygy}, defined, \idxart{230}.


\idxsection*{T.}

\idxb{Tables}, \textit{Appendix}. Greek alphabet, page~\pageref{app:greek};
miscellaneous symbols, page~\pageref{app:misc};
dimensions of the earth, page~\pageref{app:spheroid};
time constants, page~\pageref{app:time};
Table I., elements of solar system, page~\pageref{app:I};
Table II., satellites of the system, page~\pageref{app:II};
Table III., short-period comets, page~\pageref{app:III};
Table IV., parallaxes of stars, page~\pageref{app:IV};
Table V., orbits of binary stars, page~\pageref{app:V};
Table VI., the variable stars, page~\pageref{app:VI}.

\idxb{Tables}, \textit{in body of the book}. The constellations, \idxart{792};
approximate distances and periods of the planets, \idxart{489};
distance of sun corresponding to certain values of the parallax, \idxart{668};
distribution of stars with reference to the galaxy, \idxart{899};
iron meteors seen to fall, \idxart{758};
naked-eye stars north of celestial equator, \idxart{818};
orbits and masses of certain binary stars, \idxart{877};
parallaxes of first-magnitude stars, Elkin, \idxart{815};
proper motions of certain stars, \idxart{800};
signs of the zodiac, \idxart{177};
telescopic aperture required to show stars of given magnitude, \idxart{822};
temporary stars, \idxart{842};
total light from stars of different magnitude, \idxart{833};
velocity of free wave at various depths, \idxart{473}.

\idxb{Tails} or trains of comets, \idxart{713}, \idxart{717}, \idxart{728}--\idxart{736}.

\idxc{Talcott, Capt.}, his zenith telescope, \idxart{105}.

\idxb{Tangential} component of the disturbing force, \idxart{447}.

\idxb{Telegraph} used in determination of longitude, \idxart{121}~B.

\idxb{Telescope}, the, achromatic, \idxart{41};
distinctness of image, \idxart{39};
equatorial, \idxart{72};
eye-pieces, \idxart{44};
the general theory, \idxart{35};
invention of, \idxart{35};
light-gathering power, \idxart{38};
long, of Huyghens, \idxart{40};
magnifying power, \idxart{37};
object-glass, various forms, \idxart{41}, \idxart{42};
reflecting, various forms, \idxart{47}, \idxart{48};
refracting, simple, \idxart{36};
relation of its aperture to the ``magnitude'' of the smallest star visible with it, \idxart{822};
separating or dividing power, \idxart{43}.

\idxb{Telespectroscope}, \idxart{313}.

\idxb{Temperature}, cause of the annual change, \idxart{192}, \idxart{193};
of the moon, \idxart{261};
of the sun, \idxart{349}--\idxart{351}.

\idxb{Temporary} stars, \idxart{842}--\idxart{845}.

\idxb{``Terminator,''} the, on the moon's surface, its form, \idxart{253}.

\idxb{Tethys}, the third satellite of Saturn, \idxart{643}, \textit{note}.

\idxc{Thomsen} of Copenhagen, the mechanical equivalent of light, \idxart{776}.

\idxc{Thomson, Sir W.}, on the temperature of meteors, \idxart{765};
rigidity of the earth, \idxart{171}, \idxart{482}.

\idxb{Three} bodies, the problem of, \idxart{437}, \idxart{438}.

\idxb{Tidal} evolution, \idxart{484}, \idxart{916};
friction, effect on the earth's rotation, \idxart{461}, \idxart{483};
wave, its origin and course, \idxart{476}.

\idxb{Tides}, the, definition of terms relating to them, \idxart{463};
priming and lagging of, \idxart{470};
statical theory of, \idxart{469};
wave theory of, \idxart{472}.

\idxb{Tide-raising} force, the, \idxart{464}--\idxart{467}.

\idxb{Time}, defined as an hour-angle, \idxart{109};
its determination by the sextant, \idxart{115}, \idxart{116};
its determination by the transit instrument, \idxart{113};
equation of, explained and discussed, \idxart{201}--\idxart{204};
sidereal, defined, \idxart{26}, \idxart{110};
solar, apparent, \idxart{111};
solar, mean, \idxart{112};
standard and local, \idxart{122}.

\idxc{Tisserand} on peculiarities of satellite orbits, \idxart{548}.

\idxb{Titan}, the sixth and great satellite of Saturn, \idxart{643}.

\idxb{Titania}, the third satellite of Uranus, \idxart{650}.

\idxc{Todd, Prof. D.~P.}, search for trans-Neptunian planet, \idxart{662}.

\idxb{Torsion} balance, determination of the earth's density, \idxart{165}.

\idxb{Trade} winds, proving rotation of the earth, \idxart{143}.

\idxb{Trains} of meteors, \idxart{766}, \idxart{773}.

\idxb{Transits} of moon across meridian, the interval between them, \idxart{235};
of Mercury, \idxart{561}, \idxart{562};
of Venus, law of recurrence, \idxart{575}--\idxart{577};
of Venus, used for determination of solar parallax, \idxart{678}--\idxart{686}.

\idxb{Transit} circle, see \idxsee{Meridian Circle}, \idxart{63};
instrument, \idxart{59}--\idxart{61};
instrument used in determining time, \idxart{113}.
%% -----File: File: 562.png---Folio 551-------

\idxb{Trans-Neptunian} planet, hypothetical, \idxart{662}.

\idxb{Transparency} of space as affected by meteors, \idxart{779}.

\idxb{Triple} and multiple stars, \idxart{882}.

\idxb{Tropics}, defined, \idxart{176}.

\idxb{Tropical} year, its definition and its length, \idxart{216}.

\idxb{Twilight}, theory and duration of, \idxart{96}, \idxart{97}, \idxart{130}.

\idxb{Twinkling}, or scintillation, of the stars, \idxart{864}.

\idxb{Two} bodies, problem of, \idxart{424}--\idxart{433}.

\idxc[Tycho]{Tycho Brahe} discovers the lunar variation, \idxart{457};
observations of comet of 1577, \idxart{700};
temporary star in Cassiopeia, \idxart{843};
his planetary system, \idxart{504}.


\idxsection*{U.}

\idxb{Umbriel}, the second satellite of Uranus, \idxart{650}.

\idxb{Unit} of stellar distances, the light-year, \idxart{814}.

\idxb{Uranolith.}  See \DPtypo{\idxsee{Meteorite}}{\idxsee{Meteorites}}.

\idxb{Uranometria Nova:} Argelander, \idxart{817};
Oxoniensis, \idxart{826}.

\idxb{Uranus} and Neptune, their anomalous rotation in relation to the nebular hypothesis, \idxart{914};
the planet, \idxart{645}--\idxart{652}.

\idxb{Utility} of astronomy, \idxart{2}.


\idxsection*{V.}

\idxc{Van der Kolk's} theorem, \idxart{434}.

\idxb{Vanishing} point of a system of parallel lines, \idxart{7}.

\idxb{Variable} nebulę, \idxart{889}.

\idxb[Variable Stars]{Variable stars}, \idxart{838}--\idxart{854};
classification of, \idxart{838};
explanation of their variation, \idxart{849}--\idxart{851};
methods of observation, \idxart{854};
their number and designation, \idxart{852};
their range of variation, \idxart{853}.

\idxb{Variation}, the lunar, \idxart{457}.

\idxb[Vega]{Vega}, or $\alpha$~Lyrę, its light compared with the sun's, \idxart{334}, \idxart{832}, \idxart{835};
its spectrum, \idxart{859}.

\idxb{Velocity} of air currents at high elevations, \idxart{773}, \textit{note};
areal, linear and angular, law of, \idxart{407}--\idxart{409};
of earth in her orbit, \idxart{225}, \textit{note}, \idxart{278};
of light, \idxart{225}, \textit{note,} \idxart{690}--\idxart{692};
of the moon's shadow, \idxart{384};
parabolic, or velocity from infinity, \idxart{429};
of planet at any point in its orbit, \idxart{434};
of stellar motions, \idxart{801}.

\idxb{Venus}, the planet, \idxart{563}--\idxart{577};
her atmosphere and its effect upon observations of a transit, \idxart{681};
transits of, used to determine solar parallax,  \idxart{678}--\idxart{686}.

\idxb{Vertical}, angle of the, \idxart{156};
circles defined, \idxart{12}.

\idxc{Vogel}, his classification of stellar spectra, \idxart{858};
star motions in line of sight, \idxart{802}, \idxart{863}.

\idxb{Vulcan}, hypothetical intra-Mercurial
planet, \idxart{603}, \idxart{604}.


\idxsection*{W.}

\idxb{Waste} of solar energy, \idxart{347}.

\idxb{Water}, absence of, on the moon, \idxart{258};
presence of, in atmosphere of planets, \idxart{560}, \idxart{573}, \idxart{589}.

\idxc{Watson, J.~C.} discovers and endows twenty-two asteroids, \idxart{593}, \idxart{601}.

\idxb{Wave-length} of a ray of light, affected by motion in the line of vision---Doppler's principle, \idxart{321}, \textit{note.}

\idxb{Wave-theory} of the tides, \idxart{472}.

\idxb{Weather}, moon's influence on it, \idxart{262}.

\idxb{Wedge} photometer, the, \idxart{826}.

\idxb{Weight}, loss of, between equator and pole, \idxart{152}--\idxart{155};
and mass, distinction between them, \idxart{159}, \idxart{160}.

\idxc{Wilsing}, determination of the earth's density, \idxart{167}.

\idxc{Winnecke's} comet, acceleration of, \idxart{711}.

\idxc{Wolf}, periodicity of the sun spots, \idxart{307}.

\idxc{Worms}, formula for the eastward deviation of a falling body, \idxart{138}.


\idxsection*{Y.}

\idxb{Year}, bissextile, or leap, \idxart{219};
beginning of, \idxart{222};
of confusion, \idxart{219};
eclipse, \idxart{391};
sidereal, tropical, and anomalistic, \idxart{216}, also Appendix, page~\pageref{app:time};
of light, unit of stellar distance, \idxart{814}.


\idxsection*{Z.}

\idxb{Zenith}, the astronomical and geocentric, \idxart{8}.

\idxb{Zenith} distance defined, \idxart{21};
telescope, for determination of latitude, \idxart{105}.

\idxc{Zenker}, theory of a comet's constitution, \idxart{733}.

\idxb{Zero} points of a meridian circle, \idxart{66}, \idxart{67}.

\idxb{Zodiac}, the, and its signs, \idxart{177};
signs of, as affected by precession, \idxart{208}.

\idxb{Zodiacal} light, the, \idxart{607}, \idxart{608}.

\idxc{Zöllner}, albedo of the planets, \idxart{558}, \idxart{572}, \idxart{583}, \idxart{614}, \idxart{636}, \idxart{648}, \idxart{660}, also Appendix, page~\pageref{app:I};
his photometer, \idxart{827};
on the repulsive force acting upon comets, \idxart{732}.

\end{multicols}
} %end group for font size change etc
\chelabel{INDEX}

\clearpage
%% -----File: 563.png---Folio 552-------
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Underfull \hbox (badness 1728) in paragraph at lines 2077--2085
\OT1/cmr/m/it/12 Chronome-ter, and Chrono-graph.\OT1/cmr/m/n/12 ---Modern prac-
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Underfull \hbox (badness 2503) in paragraph at lines 2643--2661
[][]\OT1/cmr/m/n/10.95 The ad-just-ment of the mi-cro-scope for
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Underfull \hbox (badness 1596) in paragraph at lines 2803--2809
\OT1/cmr/bx/n/12 ment. \OT1/cmr/m/n/12 ---Since the tran-sit in-stru-ment and
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[61 <./images/illo034.png (PNG copy)>]
Underfull \hbox (badness 1365) in paragraph at lines 3309--3315
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Underfull \hbox (badness 6063) in paragraph at lines 3815--3829
[][]\OT1/cmr/m/n/12 The es-sen-tial char-ac-ter-is-tic of the
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Underfull \hbox (badness 10000) in paragraph at lines 3916--3927
[] \OT1/cmr/bx/n/12 107.[] \OT1/cmr/m/n/12 Sixth: \OT1/cmr/m/it/12 By the
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Underfull \hbox (badness 10000) in paragraph at lines 3916--3927
\OT1/cmr/m/it/12 Gnomon.\OT1/cmr/m/n/12 ---The an-cients had
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Underfull \hbox (badness 10000) in paragraph at lines 3916--3927
\OT1/cmr/m/n/12 as-tro-nom-i-cal in-stru-ments, \OT1/cmr/m/it/12 the
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Underfull \hbox (badness 10000) in paragraph at lines 4787--4799
[] \OT1/cmr/bx/n/12 134.[] \OT1/cmr/m/n/10.95 An ap-prox-i-mate
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Underfull \hbox (badness 1527) in paragraph at lines 4787--4799
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Underfull \hbox (badness 1484) in paragraph at lines 4787--4799
\OT1/cmr/m/n/10.95 a level plain three rods in
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Underfull \hbox (badness 1728) in paragraph at lines 4787--4799
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Underfull \hbox (badness 10000) in paragraph at lines 4787--4799
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Underfull \hbox (badness 5519) in paragraph at lines 4787--4799
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[]\OT1/cmr/m/n/12 Now $\OML/cmm/m/it/12 R$\OT1/cmr/m/n/12 , the ra-dius of the 
earth, equals
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Underfull \hbox (badness 2828) in paragraph at lines 6318--6330
[] \OT1/cmr/bx/n/12 186.[] To find the Law of the
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Underfull \hbox (badness 2772) in paragraph at lines 6318--6330
\OT1/cmr/bx/n/12 Earth's Mo-tion. \OT1/cmr/m/n/12 ---By com-par-ing the
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[] \OT1/cmr/bx/n/12 209.[] Phys-i-cal Cause of Pre-ces-sion. \OT1/cmr/m/n/12 --
-The
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Underfull \hbox (badness 3029) in paragraph at lines 7005--7011
[]\OT1/cmr/m/n/10.95 235 months equal $6939[] 16[] 31[]$; 19 trop-i-cal years e
qual
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Underfull \hbox (badness 10000) in paragraph at lines 7789--7798
[] \OT1/cmr/bx/n/12 248*.[] Def-i-ni-tion of Ro-ta-tion. \OT1/cmr/m/n/10.95 ---
A
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\OT1/cmr/bx/n/12 tions. \OT1/cmr/m/n/12 ---The craters and moun-
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Underfull \hbox (badness 1043) in paragraph at lines 8340--8372
\OT1/cmr/m/n/12 tains are not the only in-ter-est-
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\OT1/cmr/m/n/10.95 tle ar-rows in []Fig. 88[]. The
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Underfull \hbox (badness 1259) in paragraph at lines 9348--9365
[]\OT1/cmr/m/n/12 Subsequent study fully con-firms this re-mark-able re-sult of

 []

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Underfull \hbox (badness 4391) in paragraph at lines 9725--9768
[] \OT1/cmr/bx/n/12 321.[]  Sun-Spot Spec-trum. \OT1/cmr/m/n/12 ---
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Underfull \hbox (badness 10000) in paragraph at lines 10285--10323
[] \OT1/cmr/bx/n/12 340.[] Pyrhe-liome-ters and Acti-
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Underfull \hbox (badness 6300) in paragraph at lines 10285--10323
\OT1/cmr/bx/n/12 nome-ters. \OT1/cmr/m/n/10.95 ---The in-stru-ments with which
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Underfull \hbox (badness 1590) in paragraph at lines 10285--10323
\OT1/cmr/m/n/10.95 The pyrhe-liome-ter con-sists es-sen-tially of a
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Underfull \hbox (badness 2538) in paragraph at lines 10384--10405
[] \OT1/cmr/bx/n/12 344.[] \OT1/cmr/m/n/12 A less tech-ni-cal state-
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Underfull \hbox (badness 1173) in paragraph at lines 10659--10691
 \OT1/cmr/bx/n/12 356.[] Helmholtz's The-ory of So-lar Con-trac-tion. \OT1/cmr/
m/n/12 ---We
 []

[228] [229] [230] [231] [232] [233


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Underfull \hbox (badness 10000) in paragraph at lines 11428--11448
[] \OT1/cmr/bx/n/12 392.[] Num-ber of Lu-nar
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Underfull \hbox (badness 6575) in paragraph at lines 11806--11825
[] \OT1/cmr/bx/n/12 407.[] Law of Lin-ear Ve-loc-ity. \OT1/cmr/m/n/12 ---
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Underfull \hbox (badness 1067) in paragraph at lines 12205--12215
 \OT1/cmr/bx/n/12 423.[] \OT1/cmr/m/n/12 Philo-soph-i-cally speak-ing there are
 there-fore but \OT1/cmr/m/it/12 two
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Underfull \hbox (badness 10000) in paragraph at lines 12485--12489
[] \OT1/cmr/bx/n/12 435.[] Pro-jec-tiles near the
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Underfull \hbox (badness 2591) in paragraph at lines 12485--12489
\OT1/cmr/bx/n/12 Earth. \OT1/cmr/m/n/10.95 ---A good il-lus-tra-tion of the
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Underfull \hbox (badness 4872) in paragraph at lines 12485--12489
\OT1/cmr/m/n/10.95 prin-ci-ples stated above is ob-tained
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Underfull \hbox (badness 1210) in paragraph at lines 12485--12489
\OT1/cmr/m/n/10.95 by con-sid-er-ing the mo-tion of bod-ies
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Underfull \hbox (badness 1708) in paragraph at lines 13753--13762
 \OT1/cmr/bx/n/12 488.[] Rel-a-tive Dis-tances of Plan-ets from the Sun:
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LaTeX Font Info:    Try loading font information for U+mvs on input line 13780.

(/usr/share/texmf-texlive/tex/latex/marvosym/umvs.fd) [301]
Underfull \hbox (badness 10000) in paragraph at lines 13833--13833
[]|\OT1/cmr/m/n/10 Mean
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Underfull \hbox (badness 3312) in paragraph at lines 13956--13976
[] \OT1/cmr/bx/n/12 494.[] Di-rect and Ret-ro-grade
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Underfull \hbox (badness 4832) in paragraph at lines 14144--14154
 \OT1/cmr/bx/n/12 503.[] Coper-ni-can Sys-tem. \OT1/cmr/m/n/12 ---Copernicus (1
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Underfull \hbox (badness 1019) in paragraph at lines 14924--14928
 \OT1/cmr/bx/n/12 541.[] The Sur-face Grav-ity. \OT1/cmr/m/n/12 ---The force of
 grav-ity on a
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Underfull \hbox (badness 5771) in paragraph at lines 14945--14952
 \OT1/cmr/bx/n/12 542.[] The Planet's Oblate-ness. \OT1/cmr/m/n/12 ---The ``obl
ate-ness'' or
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[327] [328]
Underfull \hbox (badness 10000) in paragraph at lines 15017--15031
 \OT1/cmr/bx/n/12 549.[] Hum-boldt's Clas-si-fi-ca-tion of the Plan-ets. \OT1/c
mr/m/n/12 ---
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