% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% The Project Gutenberg EBook of A History of Mathematics, by Florian Cajori
%                                                                         %
% 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.org                          %
%                                                                         %
%                                                                         %
% Title: A History of Mathematics                                         %
%                                                                         %
% Author: Florian Cajori                                                  %
%                                                                         %
% Release Date: January 24, 2010 [EBook #31061]                           %
% Most recently updated: June 11, 2021                         %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: UTF-8                                           %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS ***  %
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\begin{PGtext}
The Project Gutenberg EBook of A History of Mathematics, by Florian Cajori

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.org


Title: A History of Mathematics

Author: Florian Cajori

Release Date: January 24, 2010 [EBook #31061]
Most recently updated: June 11, 2021

Language: English

Character set encoding: UTF-8     

*** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS ***
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Produced by Andrew D. Hwang, Peter Vachuska, Carl Hudkins
and the Online Distributed Proofreading Team at
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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%

\frontmatter

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\begin{center}
{\LARGE A HISTORY OF}\\[6\Titleskip]
% [** PP: Poor man's gesperrt for one-off use]
{\Huge M\,A\,T\,H\,E\,M\,A\,T\,I\,C\,S}\\[6\Titleskip]
{\footnotesize BY}\\[2\Titleskip]
{\normalsize\scshape FLORIAN CAJORI, Ph.D.}\\[\Titleskip]
\scriptsize\scshape%
Formerly Professor of Applied Mathematics in the Tulane University \\
of Louisiana; now Professor of Physics \\
in Colorado College\normalfont\\[6\Titleskip]
\begin{minipage}{3in}
\scriptsize
\null\hspace*{1em}%
``I am sure that no subject loses more than mathematics
by any attempt to dissociate it from its history.''---\textsc{J.~W.~L.
Glaisher}
\end{minipage}
\vspace*{12\Titleskip}

\normalsize\textgoth{New York}\\[\Titleskip]
THE MACMILLAN COMPANY\\[\Titleskip]
\scriptsize\scshape LONDON: MACMILLAN \&~CO., Ltd.\\[\Titleskip]
\normalsize 1909\\[2\Titleskip]
\scriptsize\itshape All rights reserved
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\scriptsize\scshape Copyright, 1893,\\[\Titleskip]
By MACMILLAN AND CO.

\rule{1in}{0.5pt}\\[\Titleskip]
\upshape Set up and electrotyped January, \Date{1894}.\quad Reprinted March, \\
\Date{1895}; October, \Date{1897}; November, \Date{1901}; January, \Date{1906}; July, \Date{1909}.
\vfil\vfil

{\footnotesize\textgoth{Norwood Press:}}\\
J.~S. Cushing \& Co.---Berwick \& Smith.\\
Norwood, Mass., U.S.A.
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\NChapter{Preface}

\First{An} increased interest in the history of the exact sciences
manifested in recent years by teachers everywhere, and the
attention given to historical inquiry in the mathematical
class-rooms and seminaries of our leading universities, cause
me to believe that a brief general History of Mathematics
will be found acceptable to teachers and students.

The pages treating---necessarily in a very condensed
form---of the progress made during the present century,
are put forth with great diffidence, although I have spent
much time in the effort to render them accurate and
reasonably complete. Many valuable suggestions and criticisms
on the chapter on ``Recent Times'' have been made
by Dr.~E.~W. Davis, of the University of Nebraska. The
proof-sheets of this chapter have also been submitted to
Dr.~J.~E. Davies and Professor C.~A. Van~Velzer, both of the
University of Wisconsin; to Dr.~G.~B. Halsted, of the
University of Texas; Professor L.~M. Hoskins, of the Leland
Stanford~Jr.\ University; and Professor G.~D.~Olds, of Amherst
College,---all of whom have afforded valuable assistance.
I am specially indebted to Professor F.~H.~Loud, of Colorado
College, who has read the proof-sheets throughout. To all
the gentlemen above named, as well as to Dr.~Carlo Veneziani
%% -----File: 006.png---Folio vi-------
of Salt Lake City, who read the first part of my work in
manuscript, I desire to express my hearty thanks. But in
acknowledging their kindness, I trust that I shall not seem
to lay upon them any share in the responsibility for errors
which I may have introduced in subsequent revision of the
text.

\null\hfill{\footnotesize FLORIAN CAJORI.}\hspace*{1em}
\vspace*{2\Titleskip}

\footnotesize\textsc{Colorado College}, December, 1893.
%% -----File: 007.png---Folio vii-------

% Resets font size to \normalsize
\TableofContents

\iffalse
TABLE OF CONTENTS.

                                                   Page
INTRODUCTION  ............            1
ANTIQUITY..............                   5
  The Babylonians...........         5
  The Egyptians............              9
  The Greeks    .............                 16
    Greek Geometry...........                16
      The Ionic School..........                 17
      The School of Pythagoras......              19
      The Sophist School........                    23
      The Platonic School........                   29
      The First Alexandrian School   ....            34
      The Second Alexandrian School    .    . .      54
    Greek Arithmetic..........                      63
  The Romans     ....                     .....  77
MIDDLE AGES ...                       .....        84
  The Hindoos   ...                    ......   84
  The Arabs..............       100
  Europe during the Middle Ages     ....         117
    Introduction of Roman Mathematics .    .        117
    Translation of Arabic Manuscripts ....       124
    The First Awakening and its Sequel   .    .    .      128
MODERN EUROPE    ..........                   138
  The Renaissance    ...              .....     139
  Vieta to Descartes   .    .              .....      156
  Descartes to Newton .    .              .....  183
  Newton to Euler..........     199

%% -----File: 008.png---Folio viii-------

EULER, LAGRANGE, AND LAPLACE      246
The Origin of Modern Geometry      285

RECENT TIMES      291

SYNTHETIC Geometry      293

ANALYTIC  GEOMETRY      307

ALGEBRA      315

ANALYSIS      331

THEORY of FUNCTIONS      347

THEORY of NUMBERS      362

APPLIED MATHEMATICS      373

INDEX      405
\fi
%% -----File: 009.png---Folio IX-------

\pagestyle{fancy}

\NChapter[1]{Books of Reference}%[** PP: `Normal' line spacing]

\Change{The}{\First{The}} following books, pamphlets, and articles have been used
in the preparation of this history. Reference to any of them
is made in the text by giving the respective number. Histories
marked with a star are the only ones of which extensive use
has been made.
\bigskip

\footnotesize
\BookRef{1}\textsc{Günther, S\@.} \Ger{\textit{Ziele und Resultate der neueren Mathe\-ma\-tisch-historischen
Forschung.} Erlangen, 1876.}
\index{Gunther@{Günther, S.}}%

\BookRef{2}\textsc{Cajori, F\@.} \textit{The Teaching and History of Mathematics in the U.~S\@.}
Washington, 1890.

\BookRef{3}*\textsc{Cantor, Moritz.} \Ger{\textit{Vorlesungen über Geschichte der Mathematik.}
Leipzig. Bd.~I., 1880; Bd.~II., 1892.}
\index{Cantor, M.}%

\BookRef{4}\textsc{Epping, J\@.} \Ger{\textit{Astronomisches aus Babylon. Unter Mitwirkung von}
\textsc{P.~J.~R. Strassmaier}. Freiburg, 1889.}
\index{Epping}%
\index{Strassmaier}%

\BookRef{5}\textsc{Bretschneider, C.~A\@.} \Ger{\textit{Die Geometrie und die Geometer vor Euklides.}
Leipzig, 1870.}
\index{Bretschneider}%

\BookRef{6}*\textsc{Gow, James.} \textit{A Short History of Greek Mathematics.} Cambridge,
1884.
\index{Gow}%

\BookRef{7}*\textsc{Hankel, Hermann.} \Ger{\textit{Zur Geschichte der Mathematik im Alterthum
und Mittelalter.} Leipzig, 1874.}

\BookRef{8}*\textsc{Allman, G.~J\@.} \textit{Greek Geometry from Thales to Euclid.} Dublin,
1889.
\index{Allman}%

\BookRef{9}\textsc{De Morgan, A\@.} ``Euclides'' in \textit{Smith's Dictionary of Greek and
Roman Biography and Mythology}.

\BookRef{10}\textsc{Hankel, Hermann.} \Ger{\textit{Theorie der Complexen Zahlensysteme.} Leipzig,
1867.}
\index{Hankel!ref.\ to}%

\BookRef{11}\textsc{Whewell, William.} \textit{History of the Inductive Sciences.}
\index{Whewell}%

\BookRef{12}\textsc{Zeuthen, H.~G\@.} \Ger{\textit{Die Lehre von den Kegelschnitten im Alterthum.}
Kopenhagen, 1886.}
\index{Zeuthen!ref.\ to}%
%% -----File: 010.png---Folio X-------

\BookRef{13}*\textsc{Chasles, M\@.} \Ger{\textit{Geschichte der Geometrie.} Aus dem Fran\-zösi\-schen
übertragen durch \textsc{Dr.~L.~A. Sohncke}. Halle, 1839.}
\index{Chasles}%

\BookRef{14}\textsc{Marie, Maximilien.} \Fre{\textit{Histoire des Sciences Mathématiques et Physiques.}
Tome~I.--XII\@. Paris, 1883--1888.}
\index{Marie, M.}%

\BookRef{15}\textsc{Comte, A\@.} \textit{Philosophy of Mathematics}, translated by \textsc{W.~M. Gillespie}.
\index{Comte}%

\BookRef{16}\textsc{Hankel, Hermann.} \Ger{\textit{Die Entwickelung der Mathematik in den letzten
Jahrhunderten.} Tübingen, 1884.}
\index{Hankel!ref.\ to}%

\BookRef{17}\textsc{Günther, Siegmund} und \textsc{Windelband, W\@.} \Ger{\textit{Geschichte der antiken
Naturwissenschaft und Philosophie.} Nördlingen, 1888.}
\index{Gunther@{Günther, S.}}%

\BookRef{18}\textsc{Arneth, A\@.} \Ger{\textit{Geschichte der reinen Mathematik.} Stuttgart, 1852.}
\index{Arneth,}%

\BookRef{19}\textsc{Cantor, Moritz.} \Ger{\textit{Mathematische Beiträge zum Kulturleben der
Völker.} Halle, 1863.}
\index{Cantor, M.}%

\BookRef{20}\textsc{Matthiessen, Ludwig.} \Ger{\textit{Grundzüge der Antiken und Modernen
Algebra der Litteralen Gleichungen.} Leipzig, 1878.}
\index{Matthiessen}%

\BookRef{21}\textsc{Ohrtmann} und \textsc{Müller}. \Ger{\textit{Fortschritte der Mathematik.}}
\index{Muller@{Müller}}%
\index{Ohrtmann}%

\BookRef{22}\textsc{Peacock, George.} Article ``Arithmetic,'' in \textit{The Encyclopædia of
Pure Mathematics}. London, 1847.

\BookRef{23}\textsc{Herschel, J.~F.~W\@.} Article ``Mathematics,'' in \textit{Edinburgh Encyclopædia}.
\index{Herschel, J.~F.~W.!ref.\ to}%

\BookRef{24}\textsc{Suter, Heinrich.} \Ger{\textit{Geschichte der Mathematischen Wissenschaften.}
Zürich, 1873--75.}
\index{Suter}%

\BookRef{25}\textsc{Quetelet, A\@.} \Fre{\textit{Sciences Mathématiques et Physiques chez les Belges.}
Bruxelles, 1866.}
\index{Quetelet!ref.\ to}%

\BookRef{26}\textsc{Playfair, John.} Article ``Progress of the Mathematical and Physical
Sciences,'' in \textit{Encyclopædia Britannica}, 7th~edition, continued
in the 8th~edition by \textsc{Sir John Leslie}.
\index{Leslie}%
\index{Playfair}%

\BookRef{27}\textsc{De Morgan, A\@.} \textit{Arithmetical Books from the Invention of Printing
to the Present Time.}

\BookRef{28}\textsc{Napier, Mark.} \textit{Memoirs of John Napier of Merchiston.} Edinburgh,
1834.
\index{Napier, M.}%

\BookRef{29}\textsc{Halsted, G.~B\@.} ``Note on the First English Euclid,'' \textit{American
Journal of Mathematics}, Vol.~II., 1879.
\index{Halsted}%

\BookRef{30}\textsc{Madame Perier.} \textit{The Life of Mr.~Paschal.} Translated into
English by W.~A., London, 1744.
\index{Perier, Madame}%

\BookRef{31}\textsc{Montucla, J.~F\@.} \Fre{\textit{Histoire des Mathématiques.} Paris, 1802.}
\index{Montucla}%

\BookRef{32}\textsc{Dühring E\@.} \Ger{\textit{Kritische Geschichte der allgemeinen Principien der
Mechanik.} Leipzig, 1887.}
\index{Duhring@{Dühring, E.}}%

\BookRef{33}\textsc{Brewster, D\@.} \textit{The Memoirs of Newton.} Edinburgh, 1860.

\BookRef{34}\textsc{Ball, W.~W.~R\@.} \textit{A Short Account of the History of Mathematics.}
London, 1888, 2nd~edition, 1893.
\index{Ball, W.~W.~R.}%

\BookRef{35}\textsc{De Morgan, A\@.} ``On the Early History of Infinitesimals,'' in the
\textit{Philosophical Magazine}, November, 1852.
\index{Demorgan@{De Morgan}!ref.\ to}%
%% -----File: 011.png---Folio XI-------

\BookRef{36}\Ger{\textit{Bibliotheca Mathematica}, herausgegeben von \textsc{Gustaf Eneström}},
Stockholm.
\index{Eneström}%

\BookRef{37}\textsc{Günther, Siegmund.} \Ger{\textit{Vermischte Untersuchungen zur Geschichte
der mathematischen Wissenschaften.} Leipzig, 1876.}\index{Gunther@{Günther, S.}}%
%[** PP: Prev. line exactly full; placing \index{} on next line adds vspace.]

\BookRef{38}*\textsc{Gerhardt, C.~I\@.} \Ger{\textit{Geschichte der Mathematik in Deutschland.}
München, 1877.}
\index{Gerhardt}%

\BookRef{39}\textsc{Gerhardt, C.~I\@.} \Ger{\textit{Entdeckung der Differenzialrechnung durch Leibniz.}
Halle, 1848.}

%[** PP: Same Gerhardt, but retaining legitimate variant from original.]
\BookRef{40}\textsc{Gerhardt, K.~I\@.} ``Leibniz in London,'' in \Ger{\textit{Sitzungsberichte der
Königlich Preussischen Academie der Wissenschaften zu Berlin},
Februar, 1891.}

\BookRef{41}\textsc{De Morgan, A\@.} Articles ``Fluxions'' and ``Commercium Epistolicum,''
in the \textit{Penny Cyclopædia}.
\index{Demorgan@{De Morgan}!ref.\ to}%

\BookRef{42}*\textsc{Todhunter, I\@.} \textit{A History of the Mathematical Theory of Probability
from the Time of Pascal to that of Laplace.} Cambridge and
London, 1865.
\index{Todhunter!ref.\ to}%

\BookRef{43}*\textsc{Todhunter, I\@.} \textit{A History of the Theory of Elasticity and of the
Strength of Materials.} Edited and completed by \textsc{Karl Pearson}.
Cambridge, 1886.

\BookRef{44}\textsc{Todhunter, I\@.} ``Note on the History of Certain Formulæ in Spherical
Trigonometry,'' \textit{Philosophical Magazine}, February, 1873.

\BookRef{45}\Ger{\textit{Die Basler Mathematiker, Daniel Bernoulli und Leonhard Euler.}
Basel, 1884.}

\BookRef{46}\textsc{Reiff, R\@.} \Ger{\textit{Geschichte der Unendlichen Reihen.} Tübingen, 1889.}
\index{Reiff}%

\BookRef{47}\textsc{Waltershausen, W.~Sartorius.} \Ger{\textit{Gauss, zum Gedächtniss.} Leipzig,
1856.}
\index{Waltershausen}%

\BookRef{48}\textsc{Baumgart, Oswald.} \Ger{\textit{Ueber das Quadratische Reciprocitätsgesetz.}
Leipzig, 1885.}
\index{Baumgart}%

\BookRef{49}\textsc{Hathaway, A.~S\@.} ``Early History of the Potential,'' \textit{Bulletin of
the N.~Y. Mathematical Society}, I.~3.
\index{Hathaway}%

\BookRef{50}\textsc{Wolf, Rudolf.} \Ger{\textit{Geschichte der Astronomie.} München, 1877.}
\index{Wolf, R.}%

\BookRef{51}\textsc{Arago, D.~F.~J\@.} ``Eulogy on Laplace.'' Translated by \textsc{B.~Powell},
\textit{Smithsonian Report}, 1874.
\index{Arago}%

\BookRef{52}\textsc{Beaumont, M.~Élie~De.} ``Memoir of Legendre.'' Translated by
\textsc{C.~A. Alexander}, \textit{Smithsonian Report}, 1867.
\index{Beaumont}%

\BookRef{53}\textsc{Arago, D.~F.~J\@.} ``Joseph Fourier.'' \textit{Smithsonian Report},
1871.

\BookRef{54}\textsc{Wiener, Christian.} \Ger{\textit{Lehrbuch der Darstellenden Geometrie.} Leipzig,
1884.}
\index{Wiener}%

\BookRef{55}*\textsc{Loria, Gino.} \Ger{\textit{Die Hauptsächlichsten Theorien der Geometrie in
ihrer früheren und heutigen Entwickelung}, ins deutsche über\-tragen
von \textsc{Fritz Schütte}. Leipzig, 1888.}
\index{Loria}%
%% -----File: 012.png---Folio XII-------

\BookRef{56}\textsc{Cayley, Arthur.} Inaugural Address before the British Association,
1883.
\index{Cayley}%

\BookRef{57}\textsc{Spottiswoode, William.} Inaugural Address before the British
Association, 1878.
\index{Spottiswoode!ref.\ to}%

\BookRef{58}\textsc{Gibbs, J.~Willard.} ``Multiple Algebra,'' \textit{Proceedings of the
American Association for the Advancement of Science}, 1886.
\index{Gibbs!ref.\ to}%

\BookRef{59}\textsc{Fink, Karl.} \Ger{\textit{Geschichte der Elementar-Mathematik.} Tübingen,
1890.}
\index{Fink}%

\BookRef{60}\textsc{Wittstein, Armin.} \Ger{\textit{Zur Geschichte des Malfatti'schen Problems.}
Nördlingen, 1878.}
\index{Wittstein}%

\BookRef{61}\textsc{Klein, Felix.} \Ger{\textit{Vergleichende Betrachtungen über neuere geometrische
Forschungen.} Erlangen, 1872.}

\BookRef{62}\textsc{Forsyth, A.~R\@.} \textit{Theory of Functions of a Complex Variable.}
Cambridge, 1893.
\index{Forsyth}%

\BookRef{63}\textsc{Graham, R.~H\@.} \textit{Geometry of Position.} London, 1891.
\index{Graham}%

\BookRef{64}\textsc{Schmidt, Franz.} \Ger{``Aus dem Leben zweier ungarischer Mathematiker
Johann und Wolfgang Bolyai von Bolya.'' \textit{Grunert's
Archiv}, 48:2, 1868.}
\index{Schmidt}%

\BookRef{65}\textsc{Favaro, Anton.} \Ger{``Justus Bellavitis,'' \textit{Zeitschrift für Mathematik
und Physik}, 26:5, 1881.}
\index{Favaro}%

\BookRef{66}\textsc{Dronke, Ad.} \textit{Julius Plücker.} Bonn, 1871.
\index{Dronke}%

\BookRef{67}\textsc{Bauer, Gustav.} \Ger{\textit{Gedächtnissrede auf Otto Hesse.} München,
1882.}
\index{Bauer}%

\BookRef{68}\textsc{Alfred Clebsch.} \Ger{\textit{Versuch einer Darlegung und Würdigung seiner
wissenschaftlichen Leistungen von einigen seiner Freunde.} Leipzig, 1873.}
\index{Clebsch!ref.\ to}%

\BookRef{69}\textsc{Haas, August.} \Ger{\textit{Versuch einer Darstellung der Geschichte des
Krümmungsmasses.} Tübingen, 1881.}
\index{Haas}%

\BookRef{70}\textsc{Fine, Henry~B\@.} \textit{The Number-System of Algebra.} Boston and
New York, 1890.
\index{Fine}%

\BookRef{71}\textsc{Schlegel, Victor.} \Ger{\textit{Hermann Grassmann, sein Leben und seine
Werke.} Leipzig, 1878.}

\BookRef{72}\textsc{Zahn, W.~v.} \Ger{``Einige Worte zum Andenken an Hermann Hankel,''
\textit{Mathematische Annalen}, VII.~4, 1874.}
\index{Zahn}%

\BookRef{73}\textsc{Muir, Thomas.} \textit{A Treatise on Determinants.} 1882.
\index{Muir}%

\BookRef{74}\textsc{Salmon, George.} ``Arthur Cayley,'' \textit{Nature}, 28:21, September,
1883.
\index{Salmon}%

\BookRef{75}\textsc{Cayley, A\@.} ``James Joseph Sylvester,'' \textit{Nature}, 39:10, January,
1889.

\BookRef{76}\textsc{Burkhardt, Heinrich.} \Ger{``Die Anfänge der Gruppentheorie und
Paolo Ruffini,'' \textit{Zeitschrift für Mathematik und Physik}, Supplement,
1892.}
\index{Burkhardt, H.}%
%% -----File: 013.png---Folio XIII-------

\BookRef{77}\textsc{Sylvester, J.~J\@.} \textit{Inaugural Presidential Address to the Mathematical
and Physical Section of the British Association at Exeter.} 1869.
\index{Sylvester!ref.\ to}%

\BookRef{78}\textsc{Valson, C.~A\@.} \Fre{\textit{La Vie et les travaux du Baron Cauchy.} Tome~I.,
II., Paris, 1868.}
\index{Valson}%

\BookRef{79}\textsc{Sachse, Arnold.} \Ger{\textit{Versuch einer Geschichte der Darstellung willkürlicher
Funktionen einer variablen durch trigonometrische
Reihen.} Göttingen, 1879.}
\index{Sachse}%

\BookRef{80}\textsc{Bois-Reymond, Paul du.} \Ger{\textit{Zur Geschichte der Trigonometrischen
Reihen, Eine Entgegnung.} Tübingen.}
\index{Bois-Reymond, P.~du}%

\BookRef{81}\textsc{Poincaré, Henri.} \Fre{\textit{Notice sur les Travaux Scientifiques de Henri
Poincaré.} Paris, 1886.}
\index{Poincaré!ref.\ to}%

\BookRef{82}\textsc{Bjerknes, C.~A\@.} \Fre{\textit{Niels-Henrik Abel, Tableau de sa vie et de son
action scientifique.} Paris, 1885.}
\index{Bjerknes, C. A.}%

\BookRef{83}\textsc{Tucker, R\@.} ``Carl Friedrich Gauss,'' \textit{Nature}, April, 1877.
\index{Tucker}%

\BookRef{84}\textsc{Dirichlet, Lejeune.} \Ger{\textit{Gedächtnissrede auf Carl Gustav Jacob
Jacobi.} 1852.}
\index{Dirichlet!ref.\ to}%

\BookRef{85}\textsc{Enneper, Alfred.} \Ger{\textit{Elliptische Funktionen. Theorie und Geschichte.}
Halle~a/S., 1876.}
\index{Enneper!ref.\ to}%

\BookRef{86}\textsc{Henrici, O\@.} ``Theory of Functions,'' \textit{Nature}, 43:14~and~15, 1891.
\index{Henrici}%

\BookRef{87}\textsc{Darboux, Gaston.} \Fre{\textit{Notice sur les Travaux Scientifiques de M. Gaston
Darboux.} Paris, 1884.}
\index{Darboux}%

\BookRef{88}\textsc{Kummer, E.~E\@.} \Ger{\textit{Gedächtnissrede auf Gustav Peter Lejeune-Diri\-chlet.}
Berlin, 1860.}
\index{Kummer!ref.\ to}%

\BookRef{89}\textsc{Smith, H.~J. Stephen.} ``On the Present State and Prospects of
Some Branches of Pure Mathematics,'' \textit{Proceedings of the London
Mathematical Society}, Vol.~VIII., Nos.\ 104,~105, 1876.
\index{Smith, H.!ref.\ to}%

\BookRef{90}\textsc{Glaisher, J.~W.~L\@.} ``Henry John Stephen Smith,'' \textit{Monthly Notices
of the Royal Astronomical Society}, XLIV.,~4, 1884.

\BookRef{91}\Ger{\textit{Bessel als Bremer Handlungslehrling.} Bremen, 1890.}

\BookRef{92}\textsc{Frantz, J\@.} \Ger{\textit{Festrede aus Veranlassung von Bessel's hundert\-jähr\-igem
Geburtstag.} Königsberg, 1884.}
\index{Frantz}%

\BookRef{93}\textsc{Dziobek, O\@.} \textit{Mathematical Theories of Planetary Motions.}
Translated into English by M.~W. Harrington and W.~J. Hussey.\index{Dziobek}%
%[** PP: Prev. line exactly full; placing \index{} on next line adds vspace.]

\BookRef{94}\textsc{Hermite, Ch.} \Fre{``Discours prononcé devant le président de la République,''
\textit{Bulletin des Sciences Mathématiques}, XIV., Janvier,
1890.}
\index{Hermite!ref.\ to}%

\BookRef{95}\textsc{Schuster, Arthur.} ``The Influence of Mathematics on the Progress
of Physics,'' \textit{Nature}, 25:17, 1882.
\index{Schuster}%

\BookRef{96}\textsc{Kerbedz, E.~de.} \Ita{``Sophie de Kowalevski,'' \textit{Rendiconti del Circolo
Matematico di Palermo}, V., 1891.}
\index{Kerbedz}%

\BookRef{97}\textsc{Voigt, W\@.} \Ger{\textit{Zum Gedächtniss von G.~Kirchhoff.} Göttingen, 1888.}
\index{Voigt}%
%% -----File: 014.png---Folio XIV-------

\BookRef{98}\textsc{Bôcher, Maxime.} ``A Bit of Mathematical History,'' \textit{Bulletin of
the N.~Y. Math.\ Soc.}, Vol.~II., No.~5.
\index{Bocher@{Bôcher}}%

\BookRef{99}\textsc{Cayley, Arthur.} \textit{Report on the Recent Progress of Theoretical
Dynamics.} 1857.
\index{Cayley}%

\BookRef{100}\textsc{Glazebrook, R.~T\@.} \textit{Report on Optical Theories.} 1885.
\index{Glazebrook!ref.\ to}%

\BookRef{101}\textsc{Rosenberger, F\@.} \Ger{\textit{Geschichte der Physik.} Braunschweig, 1887--1890.}
\index{Rosenberger}%

\normalsize

%% -----File: 015.png---Folio 1-------
%% Index cross-references
\index{Alembert, D'|see{D'Alembert}}%
\index{Algebra|see{Notation}}%
\index{Applied mathematics|see{Astronomy, Mechanics}}%
\index{Arithmetic|see{Numbers, Notation}}%
\index{Assumption, tentative|see{Regula falsa}}%
\index{Astronomy|see{Mechanics}}%
\index{Attraction|see{Gravitation, Ellipsoid}}%
\index{Bachet de Méziriac|see{Méziriac}}%
\index{Baune, De|see{De Baune}}%
\index{Byrgius|see{Bürgi}}%
\index{Calculus|see{Differential Calculus}}%
\index{Ceulen, van|see{Ludolph}}%
\index{Complex quantities|see{Imaginaries}}%
\index{Conic sections|see{Geometry}}%
\index{Cube, duplication of|see{Duplication of the cube}}%
\index{Cubic equations|see{Algebra}}%
\index{Curves|see{Cubic curves, Rectification, Geometry, Conic sections}}%
\index{Delian problem|see{Duplication of the cube}}%
\index{Differences, finite|see{Finite differences}}%
\index{Differential calculus|see{Bernoullis, Euler, Lagrange, Laplace, etc}}%
\index{Dyck|see{Groups}}%
\index{Elements (Euclid's)|see{Euclid}}%
\index{Elimination|see{Equations}}%
\index{Elliptic geometry|see{Non-Euclidean geometry}}%
\index{Equations|see{Cubic equations, Algebra, Theory of numbers}}%
\index{Errors, theory of|see{Least squares}}%
\index{Euclidean space|see{Non-Euclidean geometry}}%
\index{Fibonacci|see{Leonardo of Pisa}}%
\index{Force-function|see{Potential}}%
\index{Fractions|see{Arithmetic}}%
\index{Functions|see{Elliptic functions, Abelian functions, Hyperelliptic functions, Theta functions, Beta function, Gamma function, Omega function, Sigma function, Bessel's function, Potential}}%
\index{Geber|see{Gabir ben Aflah}}%
\index{Geometry|see{Curves, Surfaces, Curvature, Quadrature, Rectification, Circle}}%
\index{Guldinus|see{Guldin}}%
\index{Holder@{Hölder, O.}|see{Groups}}%
\index{Hydrodynamics|see{Mechanics}}%
\index{Hydrostatics|see{Mechanics}}%
\index{Hyperbolic geometry|see{Non-Euclidean geometry}}%
\index{Incommensurables|see{Irrationals}}%
\index{Indeterminate analysis|see{Theory of numbers}}%
\index{Indeterminate equations|see{Theory of numbers}}%
\index{Indian mathematics|see{Hindoos}}%
\index{Indian numerals|see{Arabic numerals}}%
\index{Indices|see{Exponents}}%
\index{Infinitesimal calculus|see{Differential calculus}}%
\index{Irrationals|see{Incommensurables}}%
\index{Isoperimetrical figures|see{Calculus of variations}}%
\index{Joachim|see{Rhæticus}}%
\index{Kauffmann|see{Mercator, N.}}%
\index{Kelvin, Lord|see{Thomson, W.}}%
\index{Lucas de Burgo|see{Pacioli}}%
\index{Mechanics|see{Dynamics, Hydrodynamics, Hydrostatics, Graphic statics, Laws of motion, Astronomy, D'Alembert's principle}}%
\index{Moon|see{Astronomy}}%
\index{Muller@{Müller, J.}|see{Regiomontanus}}%
\index{Negative quantities|see{Algebra}}%
\index{Negative roots|see{Algebra}}%
\index{Nicolo of Brescia|see{Tartaglia}}%
\index{Notation|see{Exponents, Algebra}}%
\index{Numerals|see{Apices}}%
\index{Parabola|see{Geometry}}%
\index{Parabolic geometry|see{Non-Euclidean geometry}}%
\index{Perspective|see{Geometry}}%
\index{Physics, mathematical|see{Applied mathematics}}%
\index{Plato Tiburtinus|see{Plato of Tivoli}}%
\index{Ptolemæus|see{Ptolemy}}%
\index{Quadratic equations|see{Algebra, Equations}}%
\index{Quadrature of the circle|see{Circle; also see Circle-squarers, $\pi$}}%
\index{Rectification of curves|see{Curves}}%
\index{Regula aurea|see{Falsa positio}}%
\index{Rheticus|see{Rhæticus}}%
\index{Sacro Bosco|see{Halifax}}%
\index{Series|see{Infinite series, Trigonometric series, Divergent series, Absolutely convergent series, Semi-convergent series, Fourier's series, Uniformly convergent series}}%
\index{Sound, velocity of|see{Acoustics}}%
\index{Squaring the circle|see{Quadrature of the circle}}%
\index{Statics|see{Mechanics}}%
\index{Staudt, von|see{Von Staudt}}%
\index{Stevinus|see{Stevin}}%
\index{Strutt, J.~W.|see{Rayleigh}}%
\index{Tentative assumption|see{Regula falsa}}%
\index{Theory of equations|see{Equations}}%
\index{Theory of functions|see{Functions}}%
\index{Thomson, Sir William|see{Kelvin (Lord)}}%
\index{Trigonometric series|see{Fourier's series}}%
\index{Van Ceulen|see{Ludolph}}%
\index{Von Helmholtz|see{Helmholtz}}%
\index{Wave theory|see{Undulatory theory}}%

\clearpage
\fancyhf{}
\cleardoublepage
\phantomsection
\pdfbookmark[-1]{Main Matter}{Main Matter}
\mainmatter

\IntroChapter{A History of Mathematics}{Introduction}
\index{History of mathematics, its value|(}%
\pagestyle{fancy}
\pagenumbering{arabic}

\First{The} contemplation of the various steps by which mankind
has come into possession of the vast stock of mathematical
knowledge can hardly fail to interest the mathematician. He
takes pride in the fact that his science, more than any other,
is an \emph{exact} science, and that hardly anything ever done in
mathematics has proved to be useless. The chemist smiles
at the childish efforts of alchemists, but the mathematician
finds the geometry of the Greeks and the arithmetic of the
Hindoos as useful and admirable as any research of to-day.
He is pleased to notice that though, in course of its development,
mathematics has had periods of slow growth, yet in
the main it has been pre-eminently a \emph{progressive} science.

The history of mathematics may be instructive as well as
agreeable; it may not only remind us of what we have, but
may also teach us how to increase our store. Says De Morgan,
\index{Demorgan@{De Morgan}!ref.\ to}%
``The early history of the mind of men with regard to mathematics
leads us to point out our own errors; and in this
respect it is well to pay attention to the history of mathematics.''
It warns us against hasty conclusions; it points out
the importance of a good notation upon the progress of the
science; it discourages excessive specialisation on the part of
%% -----File: 016.png---Folio 2-------
\index{p@{$\pi$}: values for!proved to be transcendental}%
investigators, by showing how apparently distinct branches
have been found to possess unexpected connecting links; it
saves the student from wasting time and energy upon problems
which were, perhaps, solved long since; it discourages
him from attacking an unsolved problem by the same method
which has led other mathematicians to failure; it teaches that
fortifications can be taken in other ways than by direct attack,
that when repulsed from a direct assault it is well to reconnoitre
and occupy the surrounding ground and to discover the
secret paths by which the apparently unconquerable position
can be taken.\Reference{1} The importance of this strategic rule may
be emphasised by citing a case in which it has been violated.
An untold amount of intellectual energy has been expended
on the quadrature of the circle, yet no conquest has been made
by direct assault. The circle-squarers have existed in crowds
\index{Circle-squarers}%
ever since the period of Archimedes. After innumerable failures
\index{Archimedes!ref.\ to}%
to solve the problem at a time, even, when investigators
possessed that most powerful tool, the differential calculus,
persons versed in mathematics dropped the subject, while
those who still persisted were completely ignorant of its history
and generally misunderstood the conditions of the problem.
``Our problem,'' says De Morgan, ``is to square the
\index{Demorgan@{De Morgan}!ref.\ to}%
circle with the \emph{old allowance of means}: Euclid's postulates
and nothing more. We cannot remember an instance in which
a question to be solved by a \emph{definite method} was tried by the
best heads, and answered at last, \emph{by that method}, after thousands
of complete failures.'' But progress was made on this
problem by approaching it from a different direction and by
newly discovered paths. Lambert proved in 1761 that the
\index{Lambert!ref.\ to}%
ratio of the circumference of a circle to its diameter is incommensurable.
Some years ago, Lindemann demonstrated that
\index{Lindemann!ref.\ to}%
this ratio is also transcendental and that the quadrature of
the circle, by means of the ruler and compass only, is \emph{impossible}.
%% -----File: 017.png---Folio 3-------
\index{Arabic numerals and notation}%
He thus showed by actual proof that which keen-minded
mathematicians had long suspected; namely, that the
great army of circle-squarers have, for two thousand years,
been assaulting a fortification which is as indestructible as
the firmament of heaven.

Another reason for the desirability of historical study is
the value of historical knowledge to the teacher of mathematics.
The interest which pupils take in their studies may
be greatly increased if the solution of problems and the cold
logic of geometrical demonstrations are interspersed with
historical remarks and anecdotes. A class in arithmetic will
be pleased to hear about the Hindoos and their invention of
\index{Hindoos!ref.\ to}%
the ``Arabic notation''; they will marvel at the thousands
\index{Notation!Arabic notation}%
of years which elapsed before people had even thought of
introducing into the numeral notation that Columbus-egg---the
zero; they will find it astounding that it should have
taken so long to \emph{invent} a notation which they themselves can
now \emph{learn} in a month. After the pupils have learned how to
bisect a given angle, surprise them by telling of the many
futile attempts which have been made to solve, by elementary
geometry, the apparently very simple problem of the trisection
of an angle. When they know how to construct a square
whose area is double the area of a given square, tell them
about the duplication of the cube---how the wrath of Apollo
could be appeased only by the construction of a cubical altar
double the given altar, and how mathematicians long wrestled
with this problem. After the class have exhausted their energies
on the theorem of the right triangle, tell them the legend
about its discoverer---how Pythagoras, jubilant over his great
\index{Pythagoras!ref.\ to}%
accomplishment, sacrificed a hecatomb to the Muses who inspired
him. When the value of mathematical training is
called in question, quote the inscription over the entrance into
the academy of Plato, the philosopher: ``Let no one who is
\index{Plato!ref.\ to}%
%% -----File: 018.png---Folio 4-------
unacquainted with geometry enter here.'' Students in analytical
geometry should know something of Descartes, and, after
\index{Descartes!ref.\ to}%
taking up the differential and integral calculus, they should
become familiar with the parts that Newton, Leibniz, and
\index{Leibniz!ref.\ to}%
\index{Newton!ref.\ to}%
Lagrange played in creating that science. In his historical
\index{Lagrange!ref.\ to}%
talk it is possible for the teacher to make it plain to the
student that mathematics is not a dead science, but a living
one in which steady progress is made.\Reference{2}

The history of mathematics is important also as a valuable
contribution to the history of civilisation. Human progress
is closely identified with scientific thought. Mathematical
and physical researches are a reliable record of intellectual
progress. The history of mathematics is one of the large
\index{History of mathematics, its value|)}%
windows through which the philosophic eye looks into past
ages and traces the line of intellectual development.
%% -----File: 019.png---Folio 5-------
\index{Numerals!Babylonian|(}%


\Chapter{Antiquity}

\Section{The Babylonians}
\index{Babylonians|(}%

\First{The} fertile valley of the Euphrates and Tigris was one of
the primeval seats of human society. Authentic history of
the peoples inhabiting this region begins only with the foundation,
in Chaldæa and Babylonia, of a united kingdom out
of the previously disunited tribes. Much light has been
thrown on their history by the discovery of the art of reading
the \emph{cuneiform} or wedge-shaped system of writing.

In the study of Babylonian mathematics we begin with the
notation of numbers. A vertical wedge~\Glyph{cune1} stood for~$1$, while
\index{Notation!Babylonian numbers|(}%
the characters \Glyph{cune10}~and~\Glyph{cune100} signified $10$ and $100$ respectively.
Grotefend believes the character for~$10$ originally to
have been the picture of two hands, as held in prayer, the
palms being pressed together, the fingers close to each other,
but the thumbs thrust out. In the Babylonian notation two
principles were employed---the additive and multiplicative.
Numbers below 100 were expressed by symbols whose
respective values had to be \emph{added}. Thus, \Glyph{cune2}~stood for~$2$,
\Glyph{cune3}~for~$3$, \Glyph{cune4}~for~$4$, \Glyph{cune23}~for~$23$, \Glyph{cune30}~for~$30$. Here the
symbols of higher order appear always to the left of those of
lower order. In writing the hundreds, on the other hand, a
\emph{smaller} symbol was placed to the left of the~$100$, and was, in
that case, to be \emph{multiplied} by~$100$. Thus, \Glyph{cune1000}~signified
%% -----File: 020.png---Folio 6-------
$10$~times~$100$, or~$1000$. But this symbol for~$1000$ was itself
taken for a new unit, which could take smaller coefficients to
its left. Thus, \Glyph{cune2000} denoted, not $20$~times~$100$, but
$10$~times~$1000$. Of the largest numbers written in cuneiform
symbols, which have hitherto been found, none go as high as
a million.\Reference{3}

If, as is believed by most specialists, the early Sumerians
were the inventors of the cuneiform writing, then they were,
in all probability, also familiar with the notation of numbers.
Most surprising, in this connection, is the fact that Sumerian
inscriptions disclose the use, not only of the above \emph{decimal}
system, but also of a \emph{sexagesimal} one. The latter was used
chiefly in constructing tables for weights and measures. It is
full of historical interest. Its consequential development,
both for integers and fractions, reveals a high degree of
mathematical insight. We possess two Babylonian tablets
which exhibit its use. One of them, probably written between
$2300$ and $1600$~\BC, contains a table of square numbers up to~$60^{2}$.
The numbers $1$,~$4$,~$9$, $16$,~$25$, $36$,~$49$, are given as the
squares of the first seven integers respectively. We have next
$1.4 = 8^{2}$, $1.21 = 9^{2}$, $1.40 = 10^{2}$, $2.1 = 11^{2}$, etc. This remains
unintelligible, unless we assume the sexagesimal scale, which
makes $1.4 = 60 + 4$, $1.21 = 60 + 21$, $2.1 = 2.60 + 1$. The second
tablet records the magnitude of the illuminated portion of the
moon's disc for every day from new to full moon, the whole disc
being assumed to consist of $240$~parts. The illuminated parts
during the first five days are the series $5$,~$10$, $20$,~$40$, $1.20
(=80)$, which is a geometrical progression. From here on
the series becomes an arithmetical progression, the numbers
from the fifth to the fifteenth day being respectively $1.20$, $1.36$,
$1.52$, $\DPtypo{2.8}{1.8}$, $2.24$, $2.40$, $2.56$, $3.12$, $3.28$, $3.44$, $4$. This table not
only exhibits the use of the sexagesimal system, but also indicates
the acquaintance of the Babylonians with progressions.
%% -----File: 021.png---Folio 7-------
\index{Numerals!Babylonian|)}%
\index{Ptolemy!ref.\ to}%
Not to be overlooked is the fact that in the sexagesimal notation
of integers the ``principle of position'' was employed.
Thus, in $1.4$~($=64$), the $1$~is made to stand for~$60$, the unit
of the second order, by virtue of its position with respect to
the~$4$. The introduction of this principle at so early a date
is the more remarkable, because in the decimal notation it
was not introduced till about the fifth or sixth century after
Christ. The principle of position, in its general and systematic
application, requires a symbol for zero. We ask, Did
\index{Zero!(symbol for)}%
the Babylonians possess one? Had they already taken the
gigantic step of representing by a symbol the \emph{absence} of
units? Neither of the above tables answers this question,
for they happen to contain no number in which there was
occasion to use a zero. The sexagesimal system was used also
\index{Sexagesimal system}%
in fractions. Thus, in the Babylonian inscriptions, $\frac{1}{2}$ and~$\frac{1}{3}$
\index{Fractions!Babylonian}%
\index{Fractions!sexagesimal}%
are designated by $30$ and~$20$, the reader being expected, in
his mind, to supply the word ``sixtieths.'' The Greek geometer
Hypsicles and the Alexandrian astronomer Ptolemæus
\index{Hypsicles!ref.\ to}%
borrowed the sexagesimal notation of fractions from the
\index{Notation!Babylonian numbers|)}%
Babylonians and introduced it into Greece. From that time
sexagesimal fractions held almost full sway in astronomical
and mathematical calculations until the sixteenth century,
when they finally yielded their place to the decimal fractions.
It may be asked, What led to the invention of the sexagesimal
system? Why was it that $60$~parts were selected? To
this we have no positive answer. \emph{Ten} was chosen, in the
decimal system, because it represents the number of fingers.
But nothing of the human body could have suggested~$60$.
Cantor offers the following theory: At first the Babylonians
reckoned the year at $360$~days. This led to the division of
\index{Division of the circle}%
the circle into $360$~degrees, each degree representing the daily
\index{Circle!degrees of}%
amount of the supposed yearly revolution of the sun around
the earth. Now they were, very probably, familiar with the
%% -----File: 022.png---Folio 8-------
\index{Musical proportion}%
\index{p@{$\pi$}: values for!Babylonian and Hebrew}%
fact that the radius can be applied to its circumference as a
chord $6$~times, and that each of these chords subtends an arc
measuring exactly $60$~degrees. Fixing their attention upon
these degrees, the division into $60$~parts may have suggested
itself to them. Thus, when greater precision necessitated a
subdivision of the degree, it was partitioned into $60$~minutes.
In this way the sexagesimal notation may have originated.
The division of the day into $24$~hours, and of the hour
into minutes and seconds on the scale of~$60$, is due to the
Babylonians.

It appears that the people in the Tigro-Euphrates basin had
made very creditable advance in arithmetic. Their knowledge
of arithmetical and geometrical progressions has already been
\index{Progressions, first appearance of arithmetical and geometrical}%
alluded to. Iamblichus attributes to them also a knowledge
of proportion, and even the invention of the so-called \emph{musical}
proportion. Though we possess no conclusive proof, we have
nevertheless reason to believe that in practical calculation
they used the \emph{abacus}. Among the races of middle Asia, even
\index{Abacus}%
as far as China, the abacus is as old as fable. Now, Babylon
was once a great commercial centre,---the metropolis of many
nations,---and it is, therefore, not unreasonable to suppose that
her merchants employed this most improved aid to calculation.

In geometry the Babylonians accomplished almost nothing.
\index{Geometry!Babylonian}%
Besides the division of the circumference into $6$~parts by its
radius, and into $360$~degrees, they had some knowledge of
geometrical figures, such as the triangle and quadrangle, which
they used in their auguries. Like the Hebrews (1~Kin.~7:23),
they took $\pi = 3$. Of geometrical demonstrations there is, of
course, no trace. ``As a rule, in the Oriental mind the intuitive
powers eclipse the severely rational and logical.''

The astronomy of the Babylonians has attracted much
\index{Astronomy!Babylonian}%
attention. They worshipped the heavenly bodies from the
earliest historic times. When Alexander the Great, after
%% -----File: 023.png---Folio 9-------
\index{Babylonians|)}%
\index{Calendar}%
the battle of Arbela (331~\BC), took possession of Babylon,
Callisthenes found there on burned brick astronomical records
\index{Callisthenes}%
reaching back as far as 2234~\BC\ Porphyrius says that these
were sent to Aristotle. Ptolemy, the Alexandrian astronomer,
\index{Aristotle!ref.\ to}%
\index{Ptolemy!ref.\ to}%
possessed a Babylonian record of eclipses going back to
747~\BC. Recently Epping and Strassmaier\Reference[]{4} threw considerable
\index{Epping}%
light on Babylonian chronology and astronomy by explaining
two calendars of the years 123~\BC\ and 111~\BC, taken
from cuneiform tablets coming, presumably, from an old
observatory. These scholars have succeeded in giving an
account of the Babylonian calculation of the new and full
moon, and have identified by calculations the Babylonian
names of the planets, and of the twelve zodiacal signs and
twenty-eight normal stars which correspond to some extent
with the twenty-eight \textit{nakshatras} of the Hindoos. We append
part of an Assyrian astronomical report, as translated by
Oppert:---
\medskip

\footnotesize\noindent
``To the King, my lord, thy faithful servant, Mar-Istar.''

``\ldots On the first day, as the new moon's day of the month Thammuz
declined, the moon was again visible over the planet Mercury, as
I had already predicted to my master the King. I erred not.''

\normalsize


\Section{The Egyptians}
\index{Egyptians|(}%

Though there is great difference of opinion regarding the
antiquity of Egyptian civilisation, yet all authorities agree in
the statement that, however far back they go, they find no
uncivilised state of society. ``Menes, the first king, changes
the course of the Nile, makes a great reservoir, and builds the
temple of Phthah at Memphis.'' The Egyptians built the
pyramids at a very early period. Surely a people engaging in
%% -----File: 024.png---Folio 10-------
\index{Rhind papyrus|(}%
enterprises of such magnitude must have known something of
mathematics---at least of practical mathematics.

All Greek writers are unanimous in ascribing, without
envy, to Egypt the priority of invention in the mathematical
sciences. Plato in \textit{Phædrus} says: ``At the Egyptian city
\index{Plato!ref.\ to}%
of Naucratis there was a famous old god whose name was
Theuth; the bird which is called the Ibis was sacred to
him, and he was the inventor of many arts, such as arithmetic
and calculation and geometry and astronomy and draughts
\index{Astronomy!Egyptian}%
\index{Geometry!Egyptian|(}%
and dice, but his great discovery was the use of letters.''

Aristotle says that mathematics had its birth in Egypt,
because there the priestly class had the leisure needful for
the study of it. Geometry, in particular, is said by Herodotus,
Diodorus, Diogenes Laertius, Iamblichus, and other ancient
\index{Diodorus}%
\index{Iamblichus!ref.\ to}%
\index{Laertius}%
writers to have originated in Egypt.\Reference{5} In Herodotus we find
this (II.~c.~109): ``They said also that this king [Sesostris]
divided the land among all Egyptians so as to give each one a
quadrangle of equal size and to draw from each his revenues,
by imposing a tax to be levied yearly. But every one from
whose part the river tore away anything, had to go to him
and notify what had happened; he then sent the overseers,
who had to measure out by how much the land had become
smaller, in order that the owner might pay on what was left,
in proportion to the entire tax imposed. In this way, it
appears to me, geometry originated, which passed thence to
Hellas.''

We abstain from introducing additional Greek opinion
regarding Egyptian mathematics, or from indulging in wild
conjectures. We rest our account on documentary evidence.
A hieratic papyrus, included in the Rhind collection of the
British Museum, was deciphered by Eisenlohr in 1877, and
found to be a mathematical manual containing problems in
arithmetic and geometry. It was written by \textbf{Ahmes} some
\index{Ahmes|(}%
%% -----File: 025.png---Folio 11-------
time before 1700~\BC, and was founded on an older work
believed by Birch to date back as far as 3400~\BC! This
curious papyrus---the most ancient mathematical handbook
known to us---puts us at once in contact with the mathematical
thought in Egypt of three or five thousand years
ago. It is entitled ``Directions for obtaining the Knowledge
of all Dark Things.'' We see from it that the Egyptians
cared but little for theoretical results. Theorems are not
found in it at all. It contains ``hardly any general rules of
procedure, but chiefly mere statements of results intended
possibly to be explained by a teacher to his pupils.''\Reference{6} In
geometry the forte of the Egyptians lay in making constructions
and determining areas. The area of an isosceles
triangle, of which the sides measure $10$~\textit{ruths} and the base
$4$~\textit{ruths}, was erroneously given as $20$ square \DPtypo{ruths}{\textit{ruths}}, or half
the product of the base by one side. The area of an isosceles
trapezoid is found, similarly, by multiplying half the sum
of the parallel sides by one of the non-parallel sides. The
area of a circle is found by deducting from the diameter
$\frac{1}{9}$ of its length and squaring the remainder. Here $\pi$ is
\index{p@{$\pi$}: values for!Egyptian}%
taken $= (\frac{16}{9})^2 = 3.1604 \ldots$, a very fair approximation.\Reference{6} The
papyrus explains also such problems as these,---To mark
out in the field a right triangle whose sides are $10$ and $4$~units;
or a trapezoid whose parallel sides are $6$ and~$4$, and
the non-parallel sides each $20$~units.

Some problems in this papyrus seem to imply a rudimentary
knowledge of proportion.

The base-lines of the pyramids run north and south, and
east and west, but probably only the lines running north and
south were determined by astronomical observations. This,
coupled with the fact that the word \textit{harpedonaptæ}, applied to
Egyptian geometers, means ``rope-stretchers,'' would point to
the conclusion that the Egyptian, like the Indian and Chinese
%% -----File: 026.png---Folio 12-------
geometers, constructed a right triangle upon a given line, by
stretching around three pegs a rope consisting of three parts
in the ratios $3:4:5$, and thus forming a right triangle.\Reference{3} If
this explanation is correct, then the Egyptians were familiar,
2000~years~\BC, with the well-known property of the right
triangle, for the special case at least when the sides are
in the ratio $3:4:5$.

On the walls of the celebrated temple of Horus at Edfu
\index{Edfu}%
have been found hieroglyphics, written about 100~\BC, which
enumerate the pieces of land owned by the priesthood, and
give their areas. The area of any quadrilateral, however
irregular, is there found by the formula $\dfrac{a+b\Fstr}{2} · \dfrac{c+d}{2}$. Thus,
for a quadrangle whose opposite sides are $5$~and~$8$, $20$~and~$15$,
%[** PP: Namely, $113\frac{3}{4}$; see note in text on 028.png below.]
is given the area $113 \frac{1}{2}\, \frac{1}{4}$.\Reference{7} The incorrect formulæ of
Ahmes of 3000~years~\BC\ yield generally closer approximations
than those of the Edfu inscriptions, written 200~years
after Euclid!

The fact that the geometry of the Egyptians consists
chiefly of constructions, goes far to explain certain of its
great defects. The Egyptians failed in two essential points
without which a \emph{science} of geometry, in the true sense of
the word, cannot exist. In the first place, they failed to
construct a rigorously logical system of geometry, resting
upon a few axioms and postulates. A great many of their
rules, especially those in solid geometry, had probably not
been proved at all, but were known to be true merely from
observation or as matters of fact. The second great defect
was their inability to bring the numerous special cases under
a more general view, and thereby to arrive at broader and
more fundamental theorems. Some of the simplest geometrical
truths were divided into numberless special cases
of which each was supposed to require separate treatment.
%% -----File: 027.png---Folio 13-------
\index{Abacus}%
\index{Notation!Egyptian numbers}%

Some particulars about Egyptian geometry can be mentioned
\index{Geometry!Egyptian|)}%
more advantageously in connection with the early
Greek mathematicians who came to the Egyptian priests for
instruction.

%% [** PP: Notes on hieroglyphs:
%% The glyphs for 1, 10, 100, 10,000, 100,000, and 10,000,000 were
%% reproduced from the scanned image. The glyphs for 1000 and 1,000,000
%% were unclear in the scanned image, and were re-created based on
%% images obtained from http://en.wikipedia.org/wiki/Egyptian_numerals
%% on December 30, 2009.]
An insight into Egyptian methods of numeration was obtained
\index{Numerals!Egyptian}%
through the ingenious deciphering of the hieroglyphics
by Champollion, Young, and their successors. The symbols
used were the following: \Glyph{hiero1}~for~$1$, \Glyph{hiero10}~for~$10$, \Glyph{hiero100}~for~$100$,
\Glyph{hierothou}~for~$1000$, \Glyph{hiero10thou}~for~$10,000$, \Glyph{hiero100thou}~for~$100,000$, \Glyph{hieromil}~for~$1,000,000$,
\Glyph{hiero10mil}\ for~$10,000,000$.\Reference{3} The symbol for~$1$ represents a vertical
staff; that for~$10,000$ a pointing finger; that for~$100,000$ a
burbot; that for~$1,000,000$, a man in astonishment. The
significance of the remaining symbols is very doubtful.
The writing of numbers with these hieroglyphics was very
cumbrous. The unit symbol of each order was repeated as
many times as there were units in that order. The principle
employed was the \emph{additive}. Thus, $23$~was written \Glyph{hiero10}\,\Glyph{hiero10}\Glyph{hiero1}\Glyph{hiero1}\Glyph{hiero1}.

Besides the hieroglyphics, Egypt possesses the \emph{hieratic} and
\emph{demotic} writings, but for want of space we pass them by.

Herodotus makes an important statement concerning the
mode of computing among the Egyptians. He says that
they ``\emph{calculate with pebbles} by moving the hand from right
to left, while the Hellenes move it from left to right.''
Herein we recognise again that \emph{instrumental} method of
figuring so extensively used by peoples of antiquity. The
Egyptians used the decimal scale. Since, in figuring, they
moved their hands horizontally, it seems probable that they
used ciphering-boards with vertical columns. In each column
there must have been not more than nine pebbles, for ten
pebbles would be equal to one pebble in the column next
to the left.

The \textit{Ahmes papyrus} contains interesting information on
the way in which the Egyptians employed fractions. Their
\index{Fractions!Egyptian}%
%% -----File: 028.png---Folio 14-------
methods of operation were, of course, radically different from
ours. Fractions were a subject of very great difficulty with
the ancients. Simultaneous changes in both numerator and
denominator were usually avoided. In manipulating fractions
the Babylonians kept the denominators ($60$) constant. The
Romans likewise kept them constant, but equal to~$12$. The
Egyptians and Greeks, on the other hand, kept the numerators
\Addendlabel{14}%
constant, and dealt with variable denominators. Ahmes used
the term ``fraction'' in a restricted sense, for he applied it
only to \emph{unit-fractions}, or fractions having unity for the numerator.
It was designated by writing the denominator and then
placing over it a dot. Fractional values which could not be
expressed by any one unit-fraction were expressed as the
\emph{sum} of two or more of them. Thus, he wrote $\frac{1}{3}\, \frac{1}{15}$ in place
of~$\frac{2}{5}$. The first important problem naturally arising was, how
to represent any fractional value as the sum of unit-fractions.
This was solved by aid of a table, given in the papyrus, in
which all fractions of the form $\dfrac{2\Fstr}{2n+1}$ (where $n$ designates
successively all the numbers up to~$49$) are reduced to the
sum of unit-fractions. Thus, $\frac{2}{7} = \frac{1}{4}\, \frac{1}{28}$; $\frac{2}{99} = \frac{1}{66}\, \frac{1}{198}$. When,
by whom, and how this table was calculated, we do not know.
Probably it was compiled empirically at different times, by
different persons. It will be seen that by repeated application
of this table, a fraction whose numerator exceeds two
can be expressed in the desired form, provided that there
is a fraction in the table having the same denominator that
\emph{it} has. Take, for example, the problem, to divide $5$ by~$21$.
In the first place, $5 = 1 + 2 + 2$. From the table we get
$\frac{2}{21} = \frac{1}{14}\, \frac{1}{42}$. Then $\frac{5}{21} = \frac{1}{21} + (\frac{1}{14}\, \frac{1}{42}) + (\frac{1}{14}\, \frac{1}{42}) = \frac{1}{21} + (\frac{2}{14}\, \frac{2}{42})
= \frac{1}{21}\, \frac{1}{7}\, \frac{1}{21} = \frac{1}{7}\, \frac{2}{21} = \frac{1}{7}\, \frac{1}{14}\, \frac{1}{42}$. The papyrus contains problems
in which it is required that fractions be raised by addition
or multiplication to given whole numbers or to other fractions.
%% -----File: 029.png---Folio 15-------
\index{Notation!in algebra}%
\index{Rhind papyrus|)}%
For example, it is required to increase $\frac{1}{4}\, \frac{1}{8}\, \frac{1}{10}\, \frac{1}{30}\, \frac{1}{45}$ to~$1$.
The common denominator taken appears to be~$45$, for the
numbers are stated as $11 \frac{1}{4}$, $5 \frac{1}{2}\, \frac{1}{8}$, $4 \frac{1}{2}$, $1 \frac{1}{2}$,~$1$. The sum of these
is $23 \frac{1}{2}\, \frac{1}{4}\, \frac{1}{8}$ forty-fifths. Add to this $\frac{1}{9}\, \frac{1}{40}$, and the sum is~$\frac{2}{3}$.
Add~$\frac{1}{3}$, and we have~$1$. Hence the quantity to be added to
the given fraction is $\frac{1}{3}\, \frac{1}{9}\, \frac{1}{40}$.

Having finished the subject of fractions, Ahmes proceeds to
the solution of equations of one unknown quantity. The unknown
\index{Equations!solution of}%
quantity is called `hau' or heap. Thus the problem,
``heap, its~$\frac{1}{7}$, its whole, it makes~$19$,'' \ie~$\dfrac{x}{7} + x = 19$. In this
case, the solution is as follows: $\dfrac{8x}{7} = 19$; $\dfrac{x}{7} = 2 \frac{1}{4}\, \frac{1}{8}$; $x = 16 \frac{1}{2}\, \frac{1}{8}$.
But in other problems, the solutions are effected by various
other methods. It thus appears that the beginnings of algebra
\index{Algebra!Beginnings in Egypt}%
are as ancient as those of geometry.

The principal defect of Egyptian arithmetic was the lack of
a simple, comprehensive symbolism---a defect which not even
the Greeks were able to remove.

The Ahmes papyrus doubtless represents the most advanced
\index{Ahmes|)}%
attainments of the Egyptians in arithmetic and geometry. It
is remarkable that they should have reached so great proficiency
in mathematics at so remote a period of antiquity. But
strange, indeed, is the fact that, during the next two thousand
years, they should have made no progress whatsoever in it.
The conclusion forces itself upon us, that they resemble the
Chinese in the \emph{stationary character}, not only of their government,
but also of their learning. All the knowledge of geometry
which they possessed when Greek scholars visited them,
six centuries~\BC, was doubtless known to them two thousand
years earlier, when they built those stupendous and gigantic
structures---the pyramids. An explanation for this stagnation
of learning has been sought in the fact that their early discoveries
in mathematics and medicine had the misfortune of
%% -----File: 030.png---Folio 16-------
being entered upon their sacred books and that, in after ages,
it was considered heretical to augment or modify anything
therein. Thus the books themselves closed the gates to
progress.

\Section{The Greeks}
\index{Greeks|(}%

%[** PP: A unit with this heading is present in ToC, but not in the text]
\SpecialSection{Greek Geometry}

About the seventh century~\BC\ an active commercial intercourse
sprang up between Greece and Egypt. Naturally there
arose an interchange of ideas as well as of merchandise.
Greeks, thirsting for knowledge, sought the Egyptian priests
for instruction. Thales, Pythagoras, \OE{}nopides, Plato, Democritus,
\index{Democritus!ref.\ to}%
\index{OEnopides@{\OE{}nopides}!ref.\ to}%
\index{Plato!ref.\ to}%
\index{Pythagoras!ref.\ to}%
\index{Thales!ref.\ to}%
Eudoxus, all visited the land of the pyramids.
\index{Eudoxus!ref.\ to}%
Egyptian ideas were thus transplanted across the sea and
there stimulated Greek thought, directed it into new lines,
and gave to it a basis to work upon. Greek culture, therefore,
is not primitive. Not only in mathematics, but also in mythology
and art, Hellas owes a debt to older countries. To
Egypt Greece is indebted, among other things, for its elementary
geometry. But this does not lessen our admiration for
the Greek mind. From the moment that Hellenic philosophers
applied themselves to the study of Egyptian geometry,
this science assumed a radically different aspect. ``Whatever
we Greeks receive, we improve and perfect,'' says Plato. The
Egyptians carried geometry no further than was absolutely
\index{Egyptians|)}%
necessary for their practical wants. The Greeks, on the other
hand, had within them a strong speculative tendency. They
felt a craving to discover the reasons for things. They found
pleasure in the contemplation of \emph{ideal} relations, and loved
science \emph{as} science.

Our sources of information on the history of Greek geometry
before Euclid consist merely of scattered notices in ancient
writers. The early mathematicians, Thales and Pythagoras,
%% -----File: 031.png---Folio 17-------
left behind no written records of their discoveries. A full
history of Greek geometry and astronomy during this period,
\index{Geometry!Greek|(}%
written by Eudemus, a pupil of Aristotle, has been lost. It
\index{Aristotle!ref.\ to}%
\index{Eudemus}%
was well known to Proclus, who, in his commentaries on
\index{Proclus!ref.\ to}%
Euclid, gives a brief account of it. This abstract constitutes
\index{Euclid!ref.\ to}%
our most reliable information. We shall quote it frequently
under the name of \textit{Eudemian Summary}.
\index{Eudemian Summary}%

\Subsection{The Ionic School}
\index{Ionic School|(}%

To \textbf{Thales} of Miletus (640--546~\BC), one of the ``seven wise
\index{Thales}%
men,'' and the founder of the Ionic school, falls the honour of
having introduced the study of geometry into Greece. During
middle life he engaged in commercial pursuits, which took
him to Egypt. He is said to have resided there, and to have
studied the physical sciences and mathematics with the Egyptian
priests. Plutarch declares that Thales soon excelled his
masters, and amazed King Amasis by measuring the heights
of the pyramids from their shadows. According to Plutarch,
this was done by considering that the shadow cast by a vertical
staff of known length bears the same ratio to the shadow
of the pyramid as the height of the staff bears to the height
of the pyramid. This solution presupposes a knowledge of
proportion, and the Ahmes papyrus actually shows that the
\index{Ahmes!ref.\ to}%
\index{Proportion}%
rudiments of proportion were known to the Egyptians. According
to Diogenes Laertius, the pyramids were measured by
\index{Diogenes Laertius}%
Thales in a different way; viz.~by finding the length of the
shadow of the pyramid at the moment when the shadow of a
staff was equal to its own length.

The \textit{Eudemian Summary} ascribes to Thales the invention
of the theorems on the equality of vertical angles, the equality
of the angles at the base of an isosceles triangle, the bisection
of a circle by any diameter, and the congruence of two
%% -----File: 032.png---Folio 18-------
triangles having a side and the two adjacent angles equal respectively.
The last theorem he applied to the measurement
of the distances of ships from the shore. Thus Thales was
\index{Thales}%
the first to apply theoretical geometry to practical uses. The
theorem that all angles inscribed in a semicircle are right
angles is attributed by some ancient writers to Thales, by
others to Pythagoras. Thales was doubtless familiar with
\index{Pythagoras!ref.\ to}%
other theorems, not recorded by the ancients. It has been
inferred that he knew the sum of the three angles of a triangle
to be equal to two right angles, and the sides of equiangular
triangles to be proportional.\Reference{8} The Egyptians must
have made use of the above theorems on the straight line, in
some of their constructions found in the Ahmes papyrus, but
\index{Ahmes!ref.\ to}%
it was left for the Greek philosopher to give these truths,
which others saw, but did not formulate into words, an
explicit, abstract expression, and to put into scientific language
and subject to proof that which others merely felt to
be true. Thales may be said to have created the geometry
of lines, essentially abstract in its character, while the Egyptians
studied only the geometry of surfaces and the rudiments
of solid geometry, empirical in their character.\Reference{8}

With Thales begins also the study of scientific astronomy.
\index{Astronomy!Greek}%
He acquired great celebrity by the prediction of a solar eclipse
in 585~\BC. Whether he predicted the day of the occurrence,
or simply the year, is not known. It is told of him that
while contemplating the stars during an evening walk, he fell
into a ditch. The good old woman attending him exclaimed,
``How canst thou know what is doing in the heavens, when
thou seest not what is at thy feet?''

The two most prominent pupils of Thales were \textbf{Anaximander}
\index{Anaximander}%
(b.~611~\BC) and \textbf{Anaximenes} (b.~570~\BC). They studied
\index{Anaximenes}%
chiefly astronomy and physical philosophy. Of \textbf{Anaxagoras}, a
\index{Anaxagoras}%
pupil of Anaximenes, and the last philosopher of the Ionic
%% -----File: 033.png---Folio 19-------
\index{Circle-squarers}%
\index{Pythagorean School|(}%
school, we know little, except that, while in prison, he passed
his time attempting to square the circle. This is the first
\index{Circle}%
time, in the history of mathematics, that we find mention of
the famous problem of the quadrature of the circle, that rock
upon which so many reputations have been destroyed. It
turns upon the determination of the exact value of~$\pi$. Approximations
to~$\pi$ had been made by the Chinese, Babylonians,
\index{Babylonians!ref.\ to}%
\index{Chinese}%
Hebrews, and Egyptians. But the invention of a method to
\index{Egyptians}%
\index{Hebrews}%
find its \emph{exact} value, is the knotty problem which has engaged
the attention of many minds from the time of Anaxagoras
down to our own. Anaxagoras did not offer any solution of
it, and seems to have luckily escaped paralogisms.

About the time of Anaxagoras, but isolated from the Ionic
school, flourished \textbf{\OE{}nopides} of Chios. Proclus ascribes to him
\index{OEnopides@{\OE{}nopides}}%
\index{Proclus!ref.\ to}%
the solution of the following problems: From a point without,
to draw a perpendicular to a given line, and to draw an angle
on a line equal to a given angle. That a man could gain a
reputation by solving problems so elementary as these, indicates
that geometry was still in its infancy, and that the
Greeks had not yet gotten far beyond the Egyptian constructions.

The Ionic school lasted over one hundred years. The
\index{Ionic School|)}%
progress of mathematics during that period was slow, as
compared with its growth in a later epoch of Greek history.
A new impetus to its progress was given by Pythagoras.
\index{Pythagoras|(}%

\Subsection{The School of Pythagoras}

\textbf{Pythagoras} (580?--500?~\BC) was one of those figures which
impressed the imagination of succeeding times to such an
extent that their real histories have become difficult to be
discerned through the mythical haze that envelops them. The
following account of Pythagoras excludes the most doubtful
%% -----File: 034.png---Folio 20-------
statements. He was a native of Samos, and was drawn by
the fame of Pherecydes to the island of Syros. He then
\index{Pherecydes}%
visited the ancient Thales, who incited him to study in Egypt.
\index{Thales!ref.\ to}%
He sojourned in Egypt many years, and may have visited
Babylon. On his return to Samos, he found it under the
tyranny of Polycrates. Failing in an attempt to found a
school there, he quitted home again and, following the current
of civilisation, removed to Magna Græcia in South Italy. He
settled at Croton, and founded the famous Pythagorean school.
This was not merely an academy for the teaching of philosophy,
mathematics, and natural science, but it was a brotherhood,
the members of which were united for life. This brotherhood
had observances approaching masonic peculiarity. They were
forbidden to divulge the discoveries and doctrines of their
school. Hence we are obliged to speak of the Pythagoreans
as a body, and find it difficult to determine to whom each
particular discovery is to be ascribed. The Pythagoreans
themselves were in the habit of referring every discovery back
to the great founder of the sect.

This school grew rapidly and gained considerable political
ascendency\DPnote{** PP: [sic]}. But the mystic and secret observances, introduced
in imitation of Egyptian usages, and the aristocratic
tendencies of the school, caused it to become an object of
suspicion. The democratic party in Lower Italy revolted and
destroyed the buildings of the Pythagorean school. Pythagoras
fled to Tarentum and thence to Metapontum, where he was
murdered.

Pythagoras has left behind no mathematical treatises, and
our sources of information are rather scanty. Certain it is
that, in the Pythagorean school, mathematics was the principal
study. Pythagoras raised mathematics to the rank of a science.
Arithmetic was courted by him as fervently as geometry. In
\index{Arithmetic!Pythagoreans}%
fact, arithmetic is the foundation of his philosophic system.
%% -----File: 035.png---Folio 21-------
\index{Regular solids}%

The \textit{Eudemian Summary} says that ``Pythagoras changed
\index{Eudemian Summary}%
the study of geometry into the form of a liberal education,
for he examined its principles to the bottom, and investigated
its theorems in an immaterial and intellectual manner.'' His
geometry was connected closely with his arithmetic. He was
especially fond of those geometrical relations which admitted
of arithmetical expression.

Like Egyptian geometry, the geometry of the Pythagoreans
is much concerned with areas. To Pythagoras is ascribed the
important theorem that the square on the hypotenuse of a
right triangle is equal to the sum of the squares on the other
two sides. He had probably learned from the Egyptians the
truth of the theorem in the special case when the sides are
$3$,~$4$,~$5$, respectively. The story goes, that Pythagoras was so
jubilant over this discovery that he sacrificed a hecatomb. Its
authenticity is doubted, because the Pythagoreans believed in
the transmigration of the soul and opposed, therefore, the
shedding of blood. In the later traditions of the Neo-Pythagoreans
this objection is removed by replacing this bloody sacrifice
by that of ``an ox made of flour''! The proof of the law
of three squares, given in Euclid's \textit{Elements}, I.~47, is due to
\index{Euclid!ref.\ to}%
Euclid himself, and not to the Pythagoreans. What the Pythagorean
method of proof was has been a favourite topic for
conjecture.

The theorem on the sum of the three angles of a triangle,
presumably known to Thales, was proved by the Pythagoreans
\index{Thales!ref.\ to}%
after the manner of Euclid. They demonstrated also that the
plane about a point is completely filled by six equilateral
triangles, four squares, or three regular hexagons, so that it
is possible to divide up a plane into figures of either kind.

From the equilateral triangle and the square arise the solids,
namely the tetraedron, octaedron, icosaedron,\DPnote{[** PP: Original spellings]} and the cube.
These solids were, in all probability, known to the Egyptians,
%% -----File: 036.png---Folio 22-------
\index{Irrationals}%
\index{Star-polygons}%
excepting, perhaps, the icosaedron. In Pythagorean philosophy,
they represent respectively the four elements of the
physical world; namely, fire, air, water, and earth. Later
another regular solid was discovered, namely the dodecaedron,
which, in absence of a fifth element, was made to represent
the universe itself. Iamblichus states that Hippasus, a Pythagorean,
\index{Hippasus}%
\index{Iamblichus!ref.\ to}%
perished in the sea, because he boasted that he first
divulged ``the sphere with the twelve pentagons.'' The star-shaped
pentagram was used as a symbol of recognition by the
Pythagoreans, and was called by them Health.

Pythagoras called the sphere the most beautiful of all solids,
and the circle the most beautiful of all plane figures. The
treatment of the subjects of proportion and of irrational
\index{Proportion}%
quantities by him and his school will be taken up under the
head of arithmetic.

According to Eudemus, the Pythagoreans invented the problems
\index{Eudemus}%
concerning the application of areas, including the cases
of defect and excess, as in Euclid, VI.~28,~29.
\index{Euclid!ref.\ to}%

They were also familiar with the construction of a polygon
equal in area to a given polygon and similar to another given
polygon. This problem depends upon several important and
somewhat advanced theorems, and testifies to the fact that
the Pythagoreans made no mean progress in geometry.

Of the theorems generally ascribed to the Italian school,
some cannot be attributed to Pythagoras himself, nor to his
earliest successors. The progress from empirical to reasoned
solutions must, of necessity, have been slow. It is worth
noticing that on the circle no theorem of any importance was
discovered by this school.

Though politics broke up the Pythagorean fraternity, yet
the school continued to exist at least two centuries longer.
Among the later Pythagoreans, Philolaus and Archytas are
\index{Philolaus}%
the most prominent. \textbf{Philolaus} wrote a book on the Pythagorean
%% -----File: 037.png---Folio 23-------
\index{Pythagoras|)}%
\index{Pythagorean School|)}%
doctrines. By him were first given to the world the
teachings of the Italian school, which had been kept secret
for a whole century. The brilliant \textbf{Archytas} of Tarentum
\index{Archytas}%
(428--347~\BC), known as a great statesman and general, and
universally admired for his virtues, was the only great geometer
among the Greeks when Plato opened his school. Archytas
\index{Plato!ref.\ to}%
was the first to apply geometry to mechanics and to treat
\index{Mechanics!Greek}%
the latter subject methodically. He also found a very ingenious
mechanical solution to the problem of the duplication of
\index{Duplication of the cube|(}%
the cube. His solution involves clear notions on the generation
of cones and cylinders. This problem reduces itself to
finding two mean proportionals between two given lines.
These mean proportionals were obtained by Archytas from
the section of a half-cylinder. The doctrine of proportion
\index{Proportion}%
was advanced through him.

There is every reason to believe that the later Pythagoreans
exercised a strong influence on the study and development of
mathematics at Athens. The Sophists acquired geometry from
Pythagorean sources. Plato bought the works of Philolaus,
and had a warm friend in Archytas.

\Subsection{The Sophist School}
\index{Sophist School|(}%

After the defeat of the Persians under Xerxes at the battle
of Salamis, 480~\BC, a league was formed among the Greeks
to preserve the freedom of the now liberated Greek cities on
the islands and coast of the Ægæan Sea. Of this league
Athens soon became leader and dictator. She caused the
separate treasury of the league to be merged into that of
Athens, and then spent the money of her allies for her own
aggrandisement. Athens was also a great commercial centre.
Thus she became the richest and most beautiful city of antiquity.
All menial work was performed by slaves. The
%% -----File: 038.png---Folio 24-------
\index{Pythagoras!ref.\ to}%
\index{Trisection of angles}%
citizen of Athens was well-to-do and enjoyed a large amount
of leisure. The government being purely democratic, every
citizen was a politician. To make his influence felt among
his fellow-men he must, first of all, be educated. Thus there
arose a demand for teachers. The supply came principally
from Sicily, where Pythagorean doctrines had spread. These
teachers were called \emph{Sophists}, or ``wise men.'' Unlike the
Pythagoreans, they accepted pay for their teaching. Although
rhetoric was the principal feature of their instruction, they
also taught geometry, astronomy, and philosophy. Athens
\index{Astronomy!Greek}%
soon became the headquarters of Grecian men of letters, and
of mathematicians in particular. The home of mathematics
among the Greeks was first in the Ionian Islands, then in
Lower Italy, and during the time now under consideration,
at Athens.

The geometry of the circle, which had been entirely
\index{Circle|(}%
neglected by the Pythagoreans, was taken up by the Sophists.
Nearly all their discoveries were made in connection with
their innumerable attempts to solve the following three
famous problems:---

(1)~To trisect an arc or an angle.

(2)~To ``double the cube,'' \ie~to find a cube whose \emph{volume}
is double that of a given cube.

(3)~To ``square the circle,'' \ie~to find a square or some
other rectilinear figure exactly equal in area to a given circle.

These problems have probably been the subject of more
discussion and research than any other problems in mathematics.
The bisection of an angle was one of the easiest
problems in geometry. The trisection of an angle, on the
other hand, presented unexpected difficulties. A right angle
had been divided into three equal parts by the Pythagoreans.
But the general problem, though easy in appearance, transcended
the power of \emph{elementary} geometry. Among the first
%% -----File: 039.png---Folio 25-------
\index{Dinostratus!ref.\ to}%
to wrestle with it was \textbf{Hippias of Elis}, a contemporary of
\index{Hippias of Elis}%
Socrates, and born about 460~\BC. Like all the later geometers,
he failed in effecting the trisection by means of a ruler
and compass only. Proclus mentions a man, Hippias, presumably
Hippias of Elis, as the inventor of a transcendental curve
which served to divide an angle not only into three, but into
any number of equal parts. This same curve was used later
by Deinostratus and others for the quadrature of the circle.
On this account it is called the \emph{quadratrix}.
\index{Quadratrix}%

The Pythagoreans had shown that the diagonal of a square
is the side of another square having double the area of the
original one. This probably suggested the problem of the
duplication of the cube, \ie~to find the edge of a cube having
double the volume of a given cube. Eratosthenes ascribes to
\index{Eratosthenes!ref.\ to}%
this problem a different origin. The Delians were once suffering
from a pestilence and were ordered by the oracle to
double a certain cubical altar. Thoughtless workmen simply
constructed a cube with edges twice as long, but this did not
pacify the gods. The error being discovered, Plato was consulted
on the matter. He and his disciples searched eagerly
for a solution to this ``Delian Problem.'' \textbf{Hippocrates of Chios}
\index{Hippocrates of Chios}%
(about 430~\BC), a talented mathematician, but otherwise slow
and stupid, was the first to show that the problem could be
reduced to finding two mean proportionals between a given
line and another twice as long. For, in the proportion $a:x
=x:y=y:2a$, since $x^2 = ay$ and $y^2 = 2ax$ and $x^4 = a^2y^2$, we
have $x^4 = 2a^3x$ and $x^3 = 2a^3$. But he failed to find the two
mean proportionals. His attempt to square the circle was
also a failure; for though he made himself celebrated by
squaring a lune, he committed an error in attempting to apply
\index{Lune, squaring of}%
this result to the squaring of the circle.

In his study of the quadrature and duplication-problems,
\index{Duplication of the cube|)}%
Hippocrates contributed much to the geometry of the circle.
%% -----File: 040.png---Folio 26-------
\index{Irrationals}%

The subject of similar figures was studied and partly
developed by Hippocrates. This involved the theory of
proportion. Proportion had, thus far, been used by the
\index{Proportion}%
Greeks only in numbers. They never succeeded in uniting
the notions of numbers and magnitudes. The term ``number''
was used by them in a restricted sense. What we call
irrational numbers was not included under this notion. Not
even rational fractions were called numbers. They used the
\index{Fractions!Greek}%
word in the same sense as we use ``integers.'' Hence numbers
were conceived as \emph{discontinuous}, while magnitudes were
\emph{continuous}. The two notions appeared, therefore, entirely
distinct. The chasm between them is exposed to full view
in the statement of Euclid that ``incommensurable magnitudes
\index{Euclid!ref.\ to}%
do not have the same ratio as numbers.'' In Euclid's
\textit{Elements} we find the theory of proportion of magnitudes
developed and treated independent of that of numbers. The
transfer of the theory of proportion from numbers to magnitudes
(and to lengths in particular) was a difficult and
important step.

Hippocrates added to his fame by writing a geometrical
text-book, called the \textit{Elements}. This publication shows that
the Pythagorean habit of secrecy was being abandoned;
secrecy was contrary to the spirit of Athenian life.

The Sophist \textbf{Antiphon}, a contemporary of Hippocrates, introduced
\index{Antiphon}%
the \emph{process} of exhaustion for the purpose of solving
\index{Exhaustion, method of}%
the problem of the quadrature. He did himself credit by
remarking that by inscribing in a circle a square, and on its
sides erecting isosceles triangles with their vertices in the
circumference, and on the sides of these triangles erecting
new triangles, etc., one could obtain a succession of regular
polygons of $8$, $16$, $32$, $64$~sides, and so on, of which each
approaches nearer to the circle than the previous one, until
the circle is finally \emph{exhausted}. Thus is obtained an inscribed
%% -----File: 041.png---Folio 27-------
polygon whose sides coincide with the circumference. Since
there can be found squares equal in area to any polygon,
there also can be found a square equal to the last polygon
inscribed, and therefore equal to the circle itself. \textbf{Bryson
\index{Bryson of Heraclea}%
of Heraclea}, a contemporary of Antiphon, advanced the problem
\index{Antiphon!ref.\ to}%
of the quadrature considerably by circumscribing polygons
at the same time that he inscribed polygons. He erred,
however, in assuming that the area of a circle was the arithmetical
mean between circumscribed and inscribed polygons.
Unlike Bryson and the rest of Greek geometers, Antiphon
seems to have believed it possible, by continually doubling
the sides of an inscribed polygon, to obtain a polygon coinciding
with the circle. This question gave rise to lively
disputes in Athens. If a polygon can coincide with the
circle, then, says Simplicius, we must put aside the notion
that magnitudes are divisible \textit{ad infinitum}. Aristotle always
\index{Aristotle!ref.\ to}%
supported the theory of the infinite divisibility, while Zeno,
\index{Zeno}%
the Stoic, attempted to show its absurdity by proving that
if magnitudes are infinitely divisible, motion is impossible.
Zeno argues that Achilles could not overtake a tortoise; for
\index{Achilles and tortoise, paradox of}%
while he hastened to the place where the tortoise had been
when he started, the tortoise crept some distance ahead, and
while Achilles reached that second spot, the tortoise again
moved forward a little, and so on. Thus the tortoise was
always in advance of Achilles. Such arguments greatly confounded
Greek geometers. No wonder they were deterred
by such paradoxes from introducing the idea of infinity into
\index{Infinity}%
their geometry. It did not suit the rigour of their proofs.

The process of Antiphon and Bryson gave rise to the cumbrous
but perfectly rigorous ``method of exhaustion.'' In
determining the ratio of the areas between two curvilinear
plane figures, say two circles, geometers first inscribed or
circumscribed similar polygons, and then by increasing indefinitely
%% -----File: 042.png---Folio 28-------
the number of sides, nearly exhausted the spaces
between the polygons and circumferences. From the theorem
that similar polygons inscribed in circles are to each
other as the squares on their diameters, geometers may have
divined the theorem attributed to Hippocrates of Chios that
\index{Hippocrates of Chios}%
the circles, which differ but little from the last drawn polygons,
must be to each other as the squares on their diameters.
But in order to exclude all vagueness and possibility of doubt,
later Greek geometers applied reasoning like that in Euclid,
XII.~2, as follows: Let $C$~and~$c$, $D$~and~$d$ be respectively the
circles and diameters in question. Then if the proportion
$D^2 : d^2 = C : c$ is not true, suppose that $D^2 : d^2 = C : c'$. If~$c' < c$,
then a polygon~$p$ can be inscribed in the circle~$c$ which comes
\index{Circle|)}%
nearer to it in area than does~$c'$. If~$P$ be the corresponding
polygon in~$C$, then $P : p = D^2 : d^2 = C : c'$, and $P : C = p : c'$.
Since~$p > c'$, we have~$P > C$, which is absurd. Next they
proved by this same method of \textit{reductio ad absurdum} the
\index{Reductio ad absurdum}%
falsity of the supposition that~$c' > c$. Since~$c'$ can be neither
larger nor smaller than~$c$, it must be equal to it,~\textsc{q.e.d.}
Hankel refers this Method of Exhaustion back to Hippocrates
\index{Exhaustion, method of}%
\index{Hankel!ref.\ to}%
\index{Method of exhaustion}%
of Chios, but the reasons for assigning it to this early
writer, rather than to Eudoxus, seem insufficient.
\index{Eudoxus!ref.\ to}%

Though progress in geometry at this period is traceable only
at Athens, yet Ionia, Sicily, Abdera in Thrace, and Cyrene
produced mathematicians who made creditable contributions
to the science. We can mention here only \textbf{Democritus of
\index{Democritus}%
Abdera} (about 460--370~\BC), a pupil of Anaxagoras, a friend
\index{Anaxagoras!ref.\ to}%
of Philolaus, and an admirer of the Pythagoreans. He
\index{Philolaus!ref.\ to}%
visited Egypt and perhaps even Persia. He was a successful
geometer and wrote on incommensurable lines, on geometry,
on numbers, and on perspective. None of these works are
extant. He used to boast that in the construction of plane
figures with proof no one had yet surpassed him, not even
%% -----File: 043.png---Folio 29-------
\index{Pythagoras!ref.\ to}%
\index{Sophist School|)}%
the so-called harpedonaptæ (``rope-stretchers'') of Egypt. By
this assertion he pays a flattering compliment to the skill
and ability of the Egyptians.

\Subsection{The Platonic School}
\index{Platonic School|(}%

During the Peloponnesian War (431--404~\BC) the progress
of geometry was checked. After the war, Athens sank into
the background as a minor political power, but advanced more
and more to the front as the leader in philosophy, literature,
and science. Plato was born at Athens in 429~\BC, the year
\index{Plato|(}%
of the great plague, and died \Change{in~348.}{in 348~\BC.} He was a pupil and
near friend of Socrates, but it was not from him that he
acquired his taste for mathematics. After the death of Socrates,
Plato travelled extensively. In Cyrene he studied
mathematics under Theodorus. He went to Egypt, then to
\index{Theodorus!ref.\ to}%
Lower Italy and Sicily, where he came in contact with the
Pythagoreans. Archytas of Tarentum and Timæus of Locri
\index{Archytas!ref.\ to}%
\index{Timæus of Locri}%
became his intimate friends. On his return to Athens, about
389~\BC, he founded his school in the groves of the \textit{Academia},
and devoted the remainder of his life to teaching and writing.

Plato's physical philosophy is partly based on that of the
Pythagoreans. Like them, he sought in arithmetic and
\index{Arithmetic!Platonists}%
geometry the key to the universe. When questioned about
the occupation of the Deity, Plato answered that ``He geometrises
continually.'' Accordingly, a knowledge of geometry
is a necessary preparation for the study of philosophy. To
show how great a value he put on mathematics and how
necessary it is for higher speculation, Plato placed the inscription
over his porch, ``Let no one who is unacquainted with
geometry enter here.'' Xenocrates, a successor of Plato as
\index{Xenocrates}%
teacher in the Academy, followed in his master's footsteps, by
declining to admit a pupil who had no mathematical training,
%% -----File: 044.png---Folio 30-------
with the remark, ``Depart, for thou hast not the grip of
philosophy.'' Plato observed that geometry trained the mind
for correct and vigorous thinking. Hence it was that the
\textit{Eudemian Summary} says, ``He filled his writings with mathematical
\index{Eudemian Summary}%
discoveries, and exhibited on every occasion the remarkable
connection between mathematics and philosophy.''

With Plato as the head-master, we need not wonder that
the Platonic school produced so large a number of mathematicians.
Plato did little real original work, but he made
valuable improvements in the logic and methods employed
in geometry. It is true that the Sophist geometers of the
previous century were rigorous in their proofs, but as a rule
they did not reflect on the inward nature of their methods.
They used the axioms without giving them explicit expression,
\index{Axioms (of geometry)}%
and the geometrical concepts, such as the point, line, surface,
etc., without assigning to them formal definitions. The Pythagoreans
called a point ``unity in position,'' but this is a
statement of a philosophical theory rather than a definition.
Plato objected to calling a point a ``geometrical fiction.'' He
defined a point as the ``beginning of a line'' or as ``an indivisible
line,'' and a line as ``length without breadth.'' He called
the point, line, surface, the `boundaries' of the line, surface,
solid, respectively. Many of the definitions in Euclid are to
\index{Euclid!ref.\ to}%
be ascribed to the Platonic school. The same is probably
true of Euclid's axioms. Aristotle refers to Plato the axiom
that ``equals subtracted from equals leave equals.''

One of the greatest achievements of Plato and his school is
the invention of \emph{analysis} as a method of proof. To be sure,
\index{Analysis!(in synthetic geometry)}%
this method had been used unconsciously by Hippocrates and
\index{Hippocrates of Chios}%
others; but Plato, like a true philosopher, turned the instinctive
logic into a conscious, legitimate method.

The terms \emph{synthesis} and \emph{analysis} are used in mathematics
\index{Synthesis}%
in a more special sense than in logic. In ancient mathematics
%% -----File: 045.png---Folio 31-------
\index{Circle}%
\index{Trisection of angles}%
they had a different meaning from what they now have. The
oldest definition of mathematical analysis as opposed to synthesis
\index{Synthesis}%
is that given in Euclid, XIII.~5, which in all probability
\index{Euclid!ref.\ to}%
was framed by Eudoxus: ``Analysis is the obtaining of the
\index{Eudoxus!ref.\ to}%
thing sought by assuming it and so reasoning up to an
admitted truth; synthesis is the obtaining of the thing
sought by reasoning up to the inference and proof of it.''
The analytic method is not conclusive, unless all operations
involved in it are known to be reversible. To remove all
doubt, the Greeks, as a rule, added to the analytic process
a synthetic one, consisting of a reversion of all operations
occurring in the analysis. Thus the aim of analysis was to
aid in the discovery of synthetic proofs or solutions.

Plato is said to have solved the problem of the duplication
\index{Duplication of the cube}%
of the cube. But the solution is open to the very same objection
which he made to the solutions by Archytas, Eudoxus,
\index{Archytas!ref.\ to}%
and Menæchmus. He called their solutions not geometrical,
\index{Menæchmus!ref.\ to}%
but mechanical, for they required the use of other instruments
than the ruler and compasses. He said that thereby ``the good
of geometry is set aside and destroyed, for we again reduce it
to the world of sense, instead of elevating and imbuing it with
the eternal and incorporeal images of thought, even as it is
employed by God, for which reason He always is God.'' These
objections indicate either that the solution is wrongly attributed
to Plato or that he wished to show how easily non-geometric
solutions of that character can be found. It is now
generally admitted that the duplication problem, as well as
the trisection and quadrature problems, cannot be solved by
means of the ruler and compass only.

Plato gave a healthful stimulus to the study of stereometry,
\index{Plato|)}%
\index{Stereometry}%
which until his time had been entirely neglected. The sphere
and the regular solids had been studied to some extent, but
\index{Regular solids}%
the prism, pyramid, cylinder, and cone were hardly known to
%% -----File: 046.png---Folio 32-------
\index{Astronomy!Greek}%
\index{Deinostratus.!see{Dinostratus}}%
\index{Duplication of the cube}%
exist. All these solids became the subjects of investigation
by the Platonic school. One result of these inquiries was
epoch-making. \textbf{Menæchmus}, an associate of Plato and pupil
\index{Menæchmus}%
\index{Plato!ref.\ to}%
of Eudoxus, invented the conic sections, which, in course of
\index{Conic sections!Greek}%
\index{Eudoxus}%
\index{Eudoxus!ref.\ to}%
only a century, raised geometry to the loftiest height which
it was destined to reach during antiquity. Menæchmus cut
three kinds of cones, the `right-angled,' `acute-angled,' and
`obtuse-angled,' by planes at right angles to a side of the
cones, and thus obtained the three sections which we now call
the parabola, ellipse, and hyperbola. Judging from the two
very elegant solutions of the ``Delian Problem'' by means of
intersections of these curves, Menæchmus must have succeeded
well in investigating their properties.

Another great geometer was \textbf{Dinostratus}, the brother of
\index{Dinostratus}%
Menæchmus and pupil of Plato. Celebrated is his mechanical
solution of the quadrature of the circle, by means of the \emph{quadratrix}
\index{Quadratrix}%
of Hippias.

Perhaps the most brilliant mathematician of this period was
\textbf{Eudoxus}. He was born at Cnidus about 408~\BC, studied under
Archytas, and later, for two months, under Plato. He was
\index{Archytas!ref.\ to}%
imbued with a true spirit of scientific inquiry, and has been
called the father of scientific astronomical observation. From
the fragmentary notices of his astronomical researches, found
in later writers, Ideler and Schiaparelli succeeded in reconstructing
\index{Ideler}%
\index{Schiaparelli}%
the system of Eudoxus with its celebrated representation
of planetary motions by ``concentric spheres.'' Eudoxus
\index{Concentric spheres of Eudoxus}%
had a school at Cyzicus, went with his pupils to Athens, visiting
Plato, and then returned to Cyzicus, where he died 355~\BC.
The fame of the academy of Plato is to a large extent
due to Eudoxus's pupils of the school at Cyzicus, among
whom are Menæchmus, Dinostratus, Athenæus, and Helicon.
\index{Athenæus}%
\index{Helicon}%
Diogenes Laertius describes Eudoxus as astronomer, physician,
\index{Diogenes Laertius}%
legislator, as well as geometer. The \textit{Eudemian Summary}
\index{Eudemian Summary}%
%% -----File: 047.png---Folio 33-------
\index{Method of exhaustion!ref.\ to}%
\index{Proportion}%
\index{Stereometry}%
says that Eudoxus ``first increased the number of general
\index{Eudoxus}%
theorems, added to the three proportions three more, and
raised to a considerable quantity the learning, begun by Plato,
\index{Plato!ref.\ to}%
on the subject of the section, to which he applied the analytical
method.'' By this `section' is meant, no doubt, the
``golden section'' (\textit{sectio aurea}), which cuts a line in extreme
\index{Golden section}%
\index{Sectio aurea}%
\index{Section, the golden}%
and mean ratio. The first five propositions in Euclid~XIII. relate
\index{Euclid!ref.\ to}%
to lines cut by this section, and are generally attributed
to Eudoxus. Eudoxus added much to the knowledge of solid
geometry. He proved, says Archimedes, that a pyramid is
exactly one-third of a prism, and a cone one-third of a cylinder,
having equal base and altitude. The proof that spheres are
to each other as the cubes of their radii is probably due to
him. He made frequent and skilful use of the method of
exhaustion, of which he was in all probability the inventor.
\index{Exhaustion, method of}%
A scholiast on Euclid, thought to be Proclus, says further that
\index{Proclus!ref.\ to}%
Eudoxus practically invented the whole of Euclid's fifth book.
Eudoxus also found two mean proportionals between two
given lines, but the method of solution is not known.

Plato has been called a maker of mathematicians. Besides
the pupils already named, the \textit{Eudemian Summary} mentions
\index{Eudemian Summary}%
the following: \textbf{Theætetus} of Athens, a man of great
\index{Theætetus}%
natural gifts, to whom, no doubt, Euclid was greatly indebted
in the composition of the 10th~book,\Reference[]{8} treating of incommensurables;
\textbf{Leodamas} of Thasos; \textbf{Neocleides} and his pupil \textbf{Leon},
\index{Leodamas}%
\index{Leon}%
\index{Neocleides}%
who added much to the work of their predecessors, for Leon
wrote an \textit{Elements} carefully designed, both in number and
utility of its proofs; \textbf{Theudius of Magnesia}, who composed a
\index{Theudius}%
very good book of \textit{Elements} and generalised propositions,
which had been confined to particular cases; \textbf{Hermotimus of
\index{Hermotimus}%
Colophon}, who discovered many propositions of the \textit{Elements}
and composed some on \emph{loci}; and, finally, the names of \textbf{Amyclas
\index{Amyclas}%
of Heraclea}, \textbf{Cyzicenus of Athens}, and \textbf{Philippus of Mende}.
\index{Cyzicenus}%
\index{Philippus}%
%% -----File: 048.png---Folio 34-------
\index{Conic sections!Greek}%

A skilful mathematician of whose life and works we have
no details is \textbf{Aristæus}, the elder, probably a senior contemporary
\index{Aristæus}%
of Euclid. The fact that he wrote a work on conic
\index{Euclid!ref.\ to}%
sections tends to show that much progress had been made in
their study during the time of Menæchmus. Aristæus wrote
\index{Menæchmus!ref.\ to}%
also on regular solids and cultivated the analytic method.
\index{Regular solids}%
His works contained probably a summary of the researches
of the Platonic school.\Reference{8}

\textbf{Aristotle} (384--322~\BC), the systematiser of deductive logic,
\index{Aristotle}%
though not a professed mathematician, promoted the science
of geometry by improving some of the most difficult definitions.
His \textit{Physics} contains passages with suggestive hints
of the principle of virtual velocities. About his time there
\index{Virtual velocities}%
appeared a work called \textit{Mechanica}, of which he is regarded
by some as the author. Mechanics was totally neglected by
\index{Mechanics!Greek}%
the Platonic school.
\index{Platonic School|)}%

\Subsection{The First Alexandrian School}
\index{Alexandrian School!(first)|(}%

In the previous pages we have seen the birth of geometry
in Egypt, its transference to the Ionian Islands, thence to
Lower Italy and to Athens. We have witnessed its growth
in Greece from feeble childhood to vigorous manhood, and
now we shall see it return to the land of its birth and there
derive new vigour.

During her declining years, immediately following the
Peloponnesian War, Athens produced the greatest scientists
and philosophers of antiquity. It was the time of Plato
\index{Plato!ref.\ to}%
and Aristotle. In 338~\BC, at the battle of Chæronea, Athens
was beaten by Philip of Macedon, and her power was broken
forever. Soon after, Alexander the Great, the son of Philip,
started out to conquer the world. In eleven years he built
up a great empire which broke to pieces in a day. Egypt
%% -----File: 049.png---Folio 35-------
fell to the lot of Ptolemy Soter. Alexander had founded
the seaport of Alexandria, which soon became ``the noblest
of all cities.'' Ptolemy made Alexandria the capital. The
history of Egypt during the next three centuries is mainly
the history of Alexandria. Literature, philosophy, and art
were diligently cultivated. Ptolemy created the university
of Alexandria. He founded the great Library and built laboratories,
museums, a zoölogical garden, and promenades. Alexandria
soon became the great centre of learning.

Demetrius Phalereus was invited from Athens to take
charge of the Library, and it is probable, says Gow, that
\index{Gow}%
\textbf{Euclid} was invited with him to open the mathematical school.
\index{Euclid|(}%
Euclid's greatest activity was during the time of the first
Ptolemy, who reigned from 306 to 283~\BC. Of the life of
Euclid, little is known, except what is added by Proclus to
\index{Proclus!ref.\ to}%
the \textit{Eudemian Summary}. Euclid, says Proclus, was younger
\index{Eudemian Summary}%
than Plato and older than Eratosthenes and Archimedes, the
\index{Archimedes!ref.\ to}%
\index{Eratosthenes!ref.\ to}%
\index{Plato!ref.\ to}%
latter of whom mentions him. He was of the Platonic sect, and
well read in its doctrines. He collected the \textit{Elements}, put
in order much that Eudoxus had prepared, completed many
\index{Eudoxus!ref.\ to}%
things of Theætetus, and was the first who reduced to unobjectionable
\index{Theætetus!ref.\ to}%
demonstration the imperfect attempts of his predecessors.
When Ptolemy once asked him if geometry could
not be mastered by an easier process than by studying the
\textit{Elements}, Euclid returned the answer, ``There is no royal
road to geometry.'' Pappus states that Euclid was distinguished
\index{Pappus!ref.\ to}%
by the fairness and kindness of his disposition, particularly
toward those who could do anything to advance
the mathematical sciences. Pappus is evidently making a
contrast to Apollonius, of whom he more than insinuates the
\index{Apollonius!ref.\ to}%
opposite character.\Reference{9} A pretty little story is related by Stobæus:\Reference[]{6}
``A youth who had begun to read geometry with
Euclid, when he had learnt the first proposition, inquired,
%% -----File: 050.png---Folio 36-------
\index{Exhaustion, method of}%
\index{Method of exhaustion!ref.\ to}%
`What do I get by learning these things?' So Euclid called
his slave and said, `Give him threepence, since he must
make gain out of what he learns.'\,'' These are about all the
personal details preserved by Greek writers. Syrian and
Arabian writers claim to know much more, but they are unreliable.
At one time Euclid of Alexandria was universally
confounded with Euclid of Megara, who lived a century
earlier.

The fame of Euclid has at all times rested mainly upon his
book on geometry, called the \textit{Elements}. This book was so far
\index{Elements (Euclid's)|(}%
superior to the \textit{Elements} written by Hippocrates, Leon, and
Theudius, that the latter works soon perished in the struggle
for existence. The Greeks gave Euclid the special title of
``the author of the \textit{Elements}.'' It is a remarkable fact in the
history of geometry, that the \textit{Elements} of Euclid, written two
thousand years ago, are still regarded by many as the best
introduction to the mathematical sciences. In England they
are used at the present time extensively as a text-book in
schools. Some editors of Euclid have, however, been inclined
to credit him with more than is his due. They would have
us believe that a finished and unassailable system of geometry
sprang at once from the brain of Euclid, ``an armed Minerva
from the head of Jupiter.'' They fail to mention the earlier
eminent mathematicians from whom Euclid got his material.
Comparatively few of the propositions and proofs in the
\textit{Elements} are his own discoveries. In fact, the proof of the
``Theorem of Pythagoras'' is the only one directly ascribed to
\index{Pythagoras!ref.\ to}%
him. Allman conjectures that the substance of Books~I., II.,~IV.
\index{Allman}%
comes from the Pythagoreans, that the substance of Book~VI.
is due to the Pythagoreans and Eudoxus, the latter contributing
\index{Eudoxus!ref.\ to}%
the doctrine of proportion as applicable to incommensurables
\index{Incommensurables}%
\index{Proportion}%
and also the Method of Exhaustions (Book~XII.),
that Theætetus contributed much toward Books X.~and~XIII.,
\index{Theætetus!ref.\ to}%
%% -----File: 051.png---Folio 37-------
that the principal part of the original work of Euclid himself
is to be found in Book~X\@.\Reference{8} Euclid was the greatest systematiser
of his time. By careful selection from the material
before him, and by logical arrangement of the propositions
selected, he built up, from a few definitions and axioms, a
\index{Axioms (of geometry)}%
proud and lofty structure. It would be erroneous to believe
that he incorporated into his \textit{Elements} all the elementary
theorems known at his time. Archimedes, Apollonius, and
\index{Apollonius!ref.\ to}%
\index{Archimedes!ref.\ to}%
even he himself refer to theorems not included in his \textit{Elements},
as being well-known truths.

The text of the \textit{Elements} now commonly used is Theon's
\index{Theon of Alexandria!ref.\ to}%
edition. Theon of Alexandria, the father of Hypatia, brought
\index{Hypatia!ref.\ to}%
out an edition, about 700~years after Euclid, with some alterations
in the text. As a consequence, later commentators,
especially Robert Simson, who laboured under the idea that
\index{Simson!ref.\ to}%
Euclid must be absolutely perfect, made Theon the scapegoat
for all the defects which they thought they could discover
in the text as they knew it. But among the manuscripts sent
by Napoleon~I. from the Vatican to Paris was found a copy of
the \textit{Elements} believed to be anterior to Theon's recension.
Many variations from Theon's version were noticed therein,
but they were not at all important, and showed that Theon
generally made only verbal changes. The defects in the
\textit{Elements} for which Theon was blamed must, therefore, be
due to Euclid himself. The \textit{Elements} has been considered as
offering models of scrupulously rigorous demonstrations. It
is certainly true that in point of rigour it compares favourably
with its modern rivals; but when examined in the light of
strict mathematical logic, it has been pronounced by C.~S.
\index{Logic}%
Peirce to be ``riddled with fallacies.'' The results are correct
\index{Peirce, C.~S.!ref.\ to}%
only because the writer's experience keeps him on his guard.

At the beginning of our editions of the \textit{Elements}, under
the head of definitions, are given the assumptions of such
%% -----File: 052.png---Folio 38-------
\index{Non-Euclidean geometry}%
\index{Prime numbers}%
notions as the point, line, etc., and some verbal explanations.
Then follow three postulates or demands, and twelve axioms.
\index{Axioms (of geometry)}%
The term `axiom' was used by Proclus, but not by Euclid.
\index{Proclus!ref.\ to}%
He speaks, instead, of `common notions'---common either
to all men or to all sciences. There has been much controversy
among ancient and modern critics on the postulates and
axioms. An immense preponderance of manuscripts and the
testimony of Proclus place the `axioms' about \emph{right angles}
and \emph{parallels} (Axioms 11 and~12) among the postulates.\BiReference{9}{10}
\index{Parallels}%
This is indeed their proper place, for they are really \emph{assumptions},
and not \emph{common notions} or axioms. The postulate
about \emph{parallels} plays an important rôle in the history of non-Euclidean
geometry. The only postulate which Euclid missed
was the one of superposition, according to which figures
can be moved about in space without any alteration in form
or magnitude.

The \textit{Elements} contains thirteen books by Euclid, and two,
of which it is supposed that Hypsicles and Damascius are
\index{Damascius!ref.\ to}%
\index{Hypsicles!ref.\ to}%
the authors. The first four books are on plane geometry.
The fifth book treats of the theory of proportion as applied
\index{Proportion}%
to magnitudes in general. The sixth book develops the
geometry of similar figures. The seventh, eighth, ninth
books are on the theory of numbers, or on arithmetic. In the
\index{Arithmetic!Euclid}%
ninth book is found the proof to the theorem that the number
of primes is infinite. The tenth book treats of the theory of
incommensurables. The next three books are on stereometry.
\index{Incommensurables}%
\index{Stereometry}%
The eleventh contains its more elementary theorems; the
twelfth, the metrical relations of the pyramid, prism, cone,
cylinder, and sphere. The thirteenth treats of the regular
polygons, especially of the triangle and pentagon, and then uses
them as faces of the five regular solids; namely, the tetraedron,
\index{Regular solids}%
octaedron, icosaedron, cube, and dodecaedron. The regular
solids were studied so extensively by the Platonists that they
%% -----File: 053.png---Folio 39-------
received the name of ``Platonic figures.'' The statement of
\index{Platonic figures}%
Proclus that the whole aim of Euclid in writing the \textit{Elements}
\index{Proclus!ref.\ to}%
was to arrive at the construction of the regular solids, is
obviously wrong. The fourteenth and fifteenth books, treating
of solid geometry, are apocryphal.

A remarkable feature of Euclid's, and of all Greek geometry
before Archimedes is that it eschews mensuration. Thus the
\index{Archimedes!ref.\ to}%
theorem that the area of a triangle equals half the product
of its base and its altitude is foreign to Euclid.

Another extant book of Euclid is the \textit{Data}. It seems to
\index{Data (Euclid's)}%
have been written for those who, having completed the \textit{Elements},
wish to acquire the power of solving new problems
proposed to them. The \textit{Data} is a course of practice in \emph{analysis}.
\index{Analysis!(in synthetic geometry)}%
It contains little or nothing that an intelligent student
could not pick up from the \textit{Elements} itself. Hence it contributes
\index{Elements (Euclid's)|)}%
little to the stock of scientific knowledge. The following
are the other extant works generally attributed to Euclid:
\textit{Phænomena}, a work on spherical geometry and astronomy;
\index{Astronomy!Greek}%
\textit{Optics}, which develops the hypothesis that light proceeds
\index{Optics}%
from the eye, and not from the object seen; \textit{Catoptrica}, containing
propositions on reflections from mirrors; \textit{De~Divisionibus},
a treatise on the division of plane figures into parts
having to one another a given ratio; \textit{Sectio Canonis}, a work
on musical intervals. His treatise on \textit{Porisms} is lost; but
\index{Porisms}%
much learning has been expended by Robert Simson and
\index{Simson!ref.\ to}%
M.~Chasles in restoring it from numerous notes found in the
\index{Chasles!ref.\ to}%
writings of Pappus. The term `porism' is vague in meaning.
\index{Pappus!ref.\ to}%
\Addendlabel{39}%
The aim of a porism is not to state some property or truth,
like a theorem, nor to effect a construction, like a problem,
but to find and bring to view a thing which necessarily exists
with given numbers or a given construction, as, to find the
centre of a given circle, or to find the G.C.D. of two given
numbers.\Reference{6} His other lost works are \textit{Fallacies}, containing
%% -----File: 054.png---Folio 40-------
exercises in detection of fallacies; \textit{Conic Sections}, in four
\index{Conic sections!Greek}%
books, which are the foundation of a work on the same subject
by Apollonius; and \textit{Loci on a Surface}, the meaning of
\index{Apollonius!ref.\ to}%
which title is not understood. Heiberg believes it to mean
``loci which are surfaces.''

The immediate successors of Euclid in the mathematical
\index{Euclid|)}%
school at Alexandria were probably \textbf{Conon}, \textbf{Dositheus}, and
\index{Conon}%
\index{Dositheus}%
\textbf{Zeuxippus}, but little is known of them.
\index{Zeuxippus}%

\textbf{Archimedes} (287?--212~\BC), the greatest mathematician of
\index{Archimedes|(}%
antiquity, was born in Syracuse. Plutarch calls him a relation
of King Hieron; but more reliable is the statement of
Cicero, who tells us he was of low birth. Diodorus says he
\index{Diodorus}%
visited Egypt, and, since he was a great friend of Conon and
Eratosthenes, it is highly probable that he studied in Alexandria.
\index{Eratosthenes!ref.\ to}%
This belief is strengthened by the fact that he had
the most thorough acquaintance with all the work previously
done in mathematics. He returned, however, to Syracuse,
where he made himself useful to his admiring friend and
patron, King Hieron, by applying his extraordinary inventive
genius to the construction of various war-engines, by which
he inflicted much loss on the Romans during the siege of
Marcellus. The story that, by the use of mirrors reflecting
the sun's rays, he set on fire the Roman ships, when they
came within bow-shot of the walls, is probably a fiction. The
city was taken at length by the Romans, and Archimedes
perished in the indiscriminate slaughter which followed. According
to tradition, he was, at the time, studying the diagram
to some problem drawn in the sand. As a Roman soldier
approached him, he called out, ``Don't spoil my circles.''
The soldier, feeling insulted, rushed upon him and killed
him. No blame attaches to the Roman general Marcellus,
who admired his genius, and raised in his honour a tomb
bearing the figure of a sphere inscribed in a cylinder. When
%% -----File: 055.png---Folio 41-------
\index{Conic sections!Greek}%
Cicero was in Syracuse, he found the tomb buried under
rubbish.

Archimedes was admired by his fellow-citizens chiefly for
his mechanical inventions; he himself prized far more highly
his discoveries in pure science. He declared that ``every kind
of art which was connected with daily needs was ignoble and
vulgar.'' Some of his works have been lost. The following
are the extant books, arranged approximately in chronological
order: 1.~Two books on \textit{Equiponderance of Planes} or \textit{Centres
of Plane Gravities}, between which is inserted his treatise on
the \textit{Quadrature of the Parabola}; 2.~Two books on the \textit{Sphere}
and \textit{Cylinder}; 3.~The \textit{Measurement of the Circle}; 4.~\textit{On Spirals};
\index{Circle}%
5.~\textit{Conoids} and \textit{Spheroids}; 6.~The \textit{Sand-Counter}; 7.~Two books
on \textit{Floating Bodies}; 8.~Fifteen \textit{Lemmas}.

In the book on the \textit{Measurement of the Circle}, Archimedes
proves first that the area of a circle is equal to that of a
right triangle having the length of the circumference for its
base, and the radius for its altitude. In this he assumes that
there exists a straight line equal in length to the circumference---an
assumption objected to by some ancient critics, on
the ground that it is not evident that a straight line can equal
a curved one. The finding of such a line was the next problem.
He first finds an upper limit to the ratio of the circumference
to the diameter, or~$\pi$. To do this, he starts with an
\index{p@{$\pi$}: values for!Archimedean}%
equilateral triangle of which the base is a tangent and the
vertex is the centre of the circle. By successively bisecting
the angle at the centre, by comparing ratios, and by taking the
irrational square roots always a little too small, he finally
arrived at the conclusion that $\pi < 3 \frac{1}{7}$. Next he finds a lower
limit by inscribing in the circle regular polygons of $6$, $12$, $24$,
$48$, $96$~sides, finding for each successive polygon its perimeter,
which is, of course, always less than the circumference. Thus
he finally concludes that ``the circumference of a circle exceeds
%% -----File: 056.png---Folio 42-------
\index{Curves!quadrature of}%
\index{Quadrature of curves}%
three times its diameter by a part which is less than $\frac{1}{7}$
but more than $\frac{10}{71}$ of the diameter.'' This approximation is
exact enough for most purposes.

The \textit{Quadrature of the Parabola} contains two solutions to
\index{Parabola}%
the problem---one mechanical, the other geometrical. The
method of exhaustion is used in both.
\index{Exhaustion, method of}%
\index{Method of exhaustion!ref.\ to}%

Archimedes studied also the ellipse and accomplished its
quadrature, but to the hyperbola he seems to have paid less attention.
It is believed that he wrote a book on conic sections.

Of all his discoveries Archimedes prized most highly those
in his \textit{Sphere} and \textit{Cylinder}. In it are proved the new
theorems, that the surface of a sphere is equal to four times
a great circle; that the surface of a segment of a sphere is
equal to a circle whose radius is the straight line drawn from
the vertex of the segment to the circumference of its basal
circle; that the volume and the surface of a sphere are $\frac{2}{3}$~of
the volume and surface, respectively, of the cylinder circumscribed
about the sphere. Archimedes desired that the figure
to the last proposition be inscribed on his tomb. This was
ordered done by Marcellus.

The spiral now called the ``spiral of Archimedes,'' and
described in the book \textit{On Spirals}, was discovered by Archimedes,
\index{Spirals}%
and not, as some believe, by his friend Conon.\Reference{3} His
\index{Conon!ref.\ to}%
treatise thereon is, perhaps, the most wonderful of all his
works. Nowadays, subjects of this kind are made easy by
the use of the infinitesimal calculus. In its stead the ancients
used the method of exhaustion. Nowhere is the fertility of
his genius more grandly displayed than in his masterly use of
this method. With Euclid and his predecessors the method
\index{Euclid!ref.\ to}%
of exhaustion was only the means of proving propositions
which must have been seen and believed before they were
proved. But in the hands of Archimedes it became an instrument
of discovery.\Reference{9}
%% -----File: 057.png---Folio 43-------

By the word `conoid,' in his book on \textit{Conoids and
Spheroids}, is meant the solid produced by the revolution
of a parabola or a hyperbola about its axis. Spheroids
are produced by the revolution of an ellipse, and are long
or flat, according as the ellipse revolves around the major
or minor axis. The book leads up to the cubature of these
solids.

We have now reviewed briefly all his extant works on geometry.
His arithmetical treatise and problems will be considered
later. We shall now notice his works on mechanics.
\index{Mechanics!Greek}%
Archimedes is the author of the first sound knowledge on this
subject. Archytas, Aristotle, and others attempted to form
\index{Archytas!ref.\ to}%
\index{Aristotle!ref.\ to}%
the known mechanical truths into a science, but failed. Aristotle
knew the property of the lever, but could not establish
its true mathematical theory. The radical and fatal defect
in the speculations of the Greeks, says Whewell, was ``that
\index{Whewell}%
though they had in their possession facts and ideas, \emph{the ideas
were not distinct and appropriate to the facts}.'' For instance,
Aristotle asserted that when a body at the end of a lever is
moving, it may be considered as having two motions; one in
the direction of the tangent and one in the direction of the
radius; the former motion is, he says, \emph{according to nature}, the
latter \emph{contrary to nature}. These inappropriate notions of
`natural' and `unnatural' motions, together with the habits
of thought which dictated these speculations, made the perception
of the true grounds of mechanical properties impossible.\Reference{11}
It seems strange that even after Archimedes had
entered upon the right path, this science should have remained
absolutely stationary till the time of Galileo---a period of
\index{Galileo!ref.\ to}%
nearly two thousand years.

The proof of the property of the lever, given in his \textit{Equiponderance
of Planes}, holds its place in text-books to this day.
His estimate of the efficiency of the lever is expressed in the
%% -----File: 058.png---Folio 44-------
\index{Statics}%
saying attributed to him, ``Give me a fulcrum on which to
rest, and I will move the earth.''

While the \textit{Equiponderance} treats of solids, or the equilibrium
of solids, the book on \textit{Floating Bodies} treats of hydrostatics.
\index{Hydrostatics}%
His attention was first drawn to the subject of
specific gravity when King Hieron asked him to test whether
a crown, professed by the maker to be pure gold, was not
alloyed with silver. The story goes that our philosopher was
in a bath when the true method of solution flashed on his
mind. He immediately ran home, naked, shouting, ``I have
found it!'' To solve the problem, he took a piece of gold and
a piece of silver, each weighing the same as the crown. According
to one author, he determined the volume of water
displaced by the gold, silver, and crown respectively, and
calculated from that the amount of gold and silver in the
crown. According to another writer, he weighed separately
the gold, silver, and crown, while immersed in water, thereby
determining their loss of weight in water. From these data
he easily found the solution. It is possible that Archimedes
solved the problem by both methods.

After examining the writings of Archimedes, one can well
understand how, in ancient times, an `Archimedean problem'
came to mean a problem too deep for ordinary minds to solve,
and how an `Archimedean proof' came to be the synonym for
unquestionable certainty. Archimedes wrote on a very wide
range of subjects, and displayed great profundity in each. He
is the Newton of antiquity.

\textbf{Eratosthenes}, eleven years younger than Archimedes, was a
\index{Eratosthenes}%
native of Cyrene. He was educated in Alexandria under
Callimachus the poet, whom he succeeded as custodian of
the Alexandrian Library. His many-sided activity may be
inferred from his works. He wrote on \textit{Good and Evil}, \textit{Measurement
of the Earth}, \textit{Comedy}, \textit{Geography}, \textit{Chronology}, \textit{Constellations},
%% -----File: 059.png---Folio 45-------
and the \textit{Duplication of the Cube}. He was also a
\index{Duplication of the cube}%
philologian and a poet. He measured the obliquity of the
ecliptic and invented a device for finding prime numbers.
\index{Prime numbers}%
Of his geometrical writings we possess only a letter to
Ptolemy Euergetes, giving a history of the duplication problem
and also the description of a very ingenious mechanical
contrivance of his own to solve it. In his old age he lost
his eyesight, and on that account is said to have committed
suicide by voluntary starvation.

About forty years after Archimedes flourished \textbf{Apollonius of
\index{Apollonius|(}%
\index{Archimedes!ref.\ to}%
Perga}, whose genius nearly equalled that of his great predecessor.
He incontestably occupies the second place in distinction
among ancient mathematicians. Apollonius was
born in the reign of Ptolemy Euergetes and died under
Ptolemy Philopator, who reigned 222--205~\BC. He studied at
Alexandria under the successors of Euclid, and for some time,
also, at Pergamum, where he made the acquaintance of that
Eudemus to whom he dedicated the first three books of his
\index{Eudemus}%
\textit{Conic Sections}. The brilliancy of his great work brought him
\index{Conic sections!Greek|(}%
the title of the ``Great Geometer.'' This is all that is known
of his life.

His \textit{Conic Sections} were in eight books, of which the first
four only have come down to us in the original Greek. The
next three books were unknown in Europe till the middle of
the seventeenth century, when an Arabic translation, made
about~1250, was discovered. The eighth book has never been
found. In~1710 Halley of Oxford published the Greek text
\index{Halley}%
of the first four books and a Latin translation of the remaining
three, together with his conjectural restoration of the
eighth book, founded on the introductory lemmas of Pappus.
\index{Pappus!ref.\ to}%
The first four books contain little more than the substance
of what earlier geometers had done. Eutocius tells us that
\index{Eutocius!ref.\ to}%
Heraclides, in his life of Archimedes, accused Apollonius of
\index{Archimedes|)}%
\index{Heraclides}%
%% -----File: 060.png---Folio 46-------
having appropriated, in his \textit{Conic Sections}, the unpublished
discoveries of that great mathematician. It is difficult to
believe that this charge rests upon good foundation. Eutocius
\index{Eutocius!ref.\ to}%
quotes Geminus as replying that neither Archimedes nor
\index{Geminus!ref.\ to}%
Apollonius claimed to have invented the conic sections, but
that Apollonius had introduced a real improvement. While
the first three or four books were founded on the works of
Menæchmus, Aristæus, Euclid, and Archimedes, the remaining
\index{Aristæus!ref.\ to}%
\index{Euclid!ref.\ to}%
\index{Menæchmus!ref.\ to}%
ones consisted almost entirely of new matter. The first three
books were sent to Eudemus at intervals, the other books
\index{Eudemus}%
(after Eudemus's death) to one Attalus. The preface of the
\index{Attalus}%
second book is interesting as showing the mode in which
Greek books were `published' at this time. It reads thus:
``I have sent my son Apollonius to bring you (Eudemus) the
second book of my Conics. Read it carefully and communicate
it to such others as are worthy of it. If Philonides, the
\index{Philonides}%
geometer, whom I introduced to you at Ephesus, comes into
the neighbourhood of Pergamum, give it to him also.''\Reference{12}

The first book, says Apollonius in his preface to it, ``contains
the mode of producing the three sections and the conjugate
hyperbolas and their principal characteristics, more fully
and generally worked out than in the writings of other
authors.'' We remember that Menæchmus, and all his successors
down to Apollonius, considered only sections of \emph{right}
cones by a plane perpendicular to their sides, and that the
three sections were obtained each from a different cone.
Apollonius introduced an important generalisation. He produced
all the sections from one and the same cone, whether
right or scalene, and by sections which may or may not be
perpendicular to its sides. The old names for the three curves
were now no longer applicable. Instead of calling the three
curves, sections of the `acute-angled,' `right-angled,' and
`obtuse-angled' cone, he called them \emph{ellipse}, \emph{parabola}, and
%% -----File: 061.png---Folio 47-------
\emph{hyperbola}, respectively. To be sure, we find the words `parabola'
and `ellipse' in the works of Archimedes, but they are
\index{Archimedes!ref.\ to}%
probably only interpolations. The word `ellipse' was applied
because $y^2 < px$, $p$~being the parameter; the word `parabola'
was introduced because $y^2 = px$, and the term `hyperbola'
because $y^2 > px$.

The treatise of Apollonius rests on a unique property of
conic sections, which is derived directly from the nature of
the cone in which these sections are found. How this property
forms the key to the system of the ancients is told in a masterly
way by M.~Chasles.\Reference{13} ``Conceive,'' says he, ``an oblique
\index{Chasles!ref.\ to}%
cone on a circular base; the straight line drawn from its
summit to the centre of the circle forming its base is called
the \emph{axis} of the cone. The plane passing through the axis,
perpendicular to its base, cuts the cone along two lines and
determines in the circle a diameter; the triangle having this
diameter for its base and the two lines for its sides, is called
\emph{the triangle through the axis}. In the formation of his conic
sections, Apollonius supposed the cutting plane to be perpendicular
to the plane of the triangle through the axis. The
points in which this plane meets the two sides of this triangle
are the \emph{vertices} of the curve; and the straight line which joins
these two points is a diameter of it. Apollonius called this
diameter \textit{latus transversum}. At one of the two vertices of the
curve erect a perpendicular (\textit{latus rectum}) to the plane of the
triangle through the axis, of a certain length, to be determined
as we shall specify later, and from the extremity of this perpendicular
draw a straight line to the other vertex of the
curve; now, through any point whatever of the diameter of
the curve, draw at right angles an \emph{ordinate}: the square of this
ordinate, comprehended between the diameter and the curve,
will be equal to the rectangle constructed on the portion of
the ordinate comprised between the diameter and the straight
%% -----File: 062.png---Folio 48-------%[** Quote continues]
line, and the part of the diameter comprised between the first
vertex and the foot of the ordinate. Such is the characteristic
property which Apollonius recognises in his conic sections and
which he uses for the purpose of inferring from it, by adroit
transformations and deductions, nearly all the rest. It plays,
as we shall see, in his hands, almost the same rôle as the
equation of the second degree with two variables (abscissa and
ordinate) in the system of analytic geometry of Descartes.
\index{Descartes!ref.\ to}%

``It will be observed from this that the diameter of the
curve and the perpendicular erected at one of its extremities
suffice to construct the curve. These are the two elements
which the ancients used, with which to establish their theory
of conics. The perpendicular in question was called by them
\textit{latus erectum}; the moderns changed this name first to that of
\textit{latus rectum}, and afterwards to that of \emph{parameter}.''
\index{Latus rectum}%
\index{Parameter}%

The first book of the \textit{Conic Sections} of Apollonius is almost
wholly devoted to the generation of the three principal conic
sections.

The second book treats mainly of asymptotes, axes, and
diameters.

The third book treats of the equality or proportionality
of triangles, rectangles, or squares, of which the component
parts are determined by portions of transversals, chords,
asymptotes, or tangents, which are frequently subject to a
great number of conditions. It also touches the subject of
foci of the ellipse and hyperbola.

In the fourth book, Apollonius discusses the harmonic division
of straight lines. He also examines a system of two
conics, and shows that they cannot cut each other in more
than four points. He investigates the various possible relative
positions of two conics, as, for instance, when they have one
or two points of contact with each other.

The fifth book reveals better than any other the giant
%% -----File: 063.png---Folio 49-------
\index{Maxima and minima}%
\index{Quadrature of curves}%
intellect of its author. Difficult questions of \emph{maxima and
minima}, of which few examples are found in earlier works, are
here treated most exhaustively. The subject investigated is,
to find the longest and shortest lines that can be drawn from
a given point to a conic. Here are also found the germs of
the subject of \emph{evolutes} and \emph{centres of osculation}.
\index{Centres of osculation}%
\index{Evolutes}%

The sixth book is on the similarity of conics.

The seventh book is on conjugate diameters.

The eighth book, as restored by Halley, continues the subject
of conjugate diameters.

It is worthy of notice that Apollonius nowhere introduces
the notion of \emph{directrix} for a conic, and that, though he incidentally
\index{Directrix}%
discovered the \emph{focus} of an ellipse and hyperbola, he did
\index{Focus}%
not discover the focus of a parabola.\Reference{6} Conspicuous in his
geometry is also the absence of technical terms and symbols,
which renders the proofs long and cumbrous.

The discoveries of Archimedes and Apollonius, says M. Chasles,\Reference[]{13}
\index{Archimedes!ref.\ to}%
\index{Chasles!ref.\ to}%
marked the most brilliant epoch of ancient geometry.
Two questions which have occupied geometers of all periods
may be regarded as having originated with them. The first
of these is the quadrature of curvilinear figures, which gave
birth to the infinitesimal calculus. The second is the theory
of conic sections, which was the prelude to the theory of
geometrical curves of all degrees, and to that portion of
\index{Curves!quadrature of}%
geometry which considers only the forms and situations
of figures, and uses only the intersection of lines and surfaces
and the ratios of rectilineal distances. These two great
divisions of geometry may be designated by the names of
\emph{Geometry of Measurements} and \emph{Geometry of Forms and Situations},
or, Geometry of Archimedes and of Apollonius.

Besides the \textit{Conic Sections}, Pappus ascribes to Apollonius
\index{Conic sections!Greek|)}%
\index{Pappus!ref.\ to}%
the following works: \textit{On Contacts}, \textit{Plane Loci}, \textit{Inclinations},
\textit{Section of an Area}, \textit{Determinate Section}, and gives lemmas
%% -----File: 064.png---Folio 50-------
\index{Duplication of the cube}%
\index{Trisection of angles}%
from which attempts have been made to restore the lost
originals. Two books on \textit{De Sectione Rationis} have been
found in the Arabic. The book on \textit{Contacts}, as restored by
Vieta, contains the so-called ``Apollonian Problem'': Given
\index{Apollonian Problem}%
\index{Vieta!ref.\ to}%
three circles, to find a fourth which shall touch the three.

Euclid, Archimedes, and Apollonius brought geometry to
\index{Archimedes!ref.\ to}%
\index{Euclid!ref.\ to}%
as high a state of perfection as it perhaps could be brought
without first introducing some more general and more powerful
method than the old method of exhaustion. A briefer symbolism,
a Cartesian geometry, an infinitesimal calculus, were
needed. The Greek mind was not adapted to the invention of
general methods. Instead of a climb to still loftier heights
we observe, therefore, on the part of later Greek geometers, a
descent, during which they paused here and there to look
around for details which had been passed by in the hasty
ascent.\Reference{3}

Among the earliest successors of Apollonius was \textbf{Nicomedes}.
\index{Apollonius|)}%
\index{Nicomedes}%
Nothing definite is known of him, except that he invented the
\emph{conchoid} (``mussel-like''). He devised a little machine by
\index{Conchoid}%
which the curve could be easily described. With aid of the
conchoid he duplicated the cube. The curve can also be used
for trisecting angles in a way much resembling that in the
eighth lemma of Archimedes. Proclus ascribes this mode of
\index{Proclus!ref.\ to}%
trisection to Nicomedes, but Pappus, on the other hand, claims
\index{Pappus!ref.\ to}%
it as his own. The conchoid was used by Newton in constructing
\index{Newton!ref.\ to}%
curves of the third degree.

About the time of Nicomedes, flourished also \textbf{Diocles}, the
\index{Diocles}%
inventor of the \emph{cissoid} (``ivy-like''). This curve he used for
\index{Cissoid}%
finding two mean proportionals between two given straight
lines.

About the life of \textbf{Perseus} we know as little as about that of
\index{Perseus}%
Nicomedes and Diocles. He lived some time between 200 and
100~\BC. From Heron and Geminus we learn that he wrote a
\index{Geminus!ref.\ to}%
\index{Heron the Elder!ref.\ to}%
%% -----File: 065.png---Folio 51-------
\index{Astronomy!Greek}%
\index{Isoperimetrical figures}%
work on the \emph{spire}, a sort of anchor-ring surface described by
Heron as being produced by the revolution of a circle around
one of its chords as an axis. The sections of this surface
yield peculiar curves called \emph{spiral sections}, which, according to
Geminus, were thought out by Perseus. These curves appear
to be the same as the \textit{Hippopede} of Eudoxus.
\index{Eudoxus!ref.\ to}%
\index{Hippopede}%

Probably somewhat later than Perseus lived \textbf{Zenodorus}. He
\index{Zenodorus}%
wrote an interesting treatise on a new subject; namely, \emph{isoperimetrical
figures}. Fourteen propositions are preserved by
Pappus and Theon. Here are a few of them: Of isoperimetrical,
\index{Theon of Alexandria!ref.\ to}%
regular polygons, the one having the largest number of
angles has the greatest area; the circle has a greater area than
any regular polygon of equal periphery; of all isoperimetrical
polygons of $n$~sides, the regular is the greatest; of all solids
having surfaces equal in area, the sphere has the greatest
volume.

\textbf{Hypsicles} (between 200 and 100~\BC) was supposed to be
\index{Hypsicles}%
the author of both the fourteenth and fifteenth books of
Euclid, but recent critics are of opinion that the fifteenth
book was written by an author who lived several centuries
after Christ. The fourteenth book contains seven elegant
theorems on \emph{regular solids}. A treatise of Hypsicles on \textit{Risings}
\index{Regular solids}%
is of interest because it is the first Greek work giving the
division of the circumference into $360$~degrees after the fashion
of the Babylonians.
\index{Babylonians!ref.\ to}%

\textbf{Hipparchus} of Nicæa in Bithynia was the greatest astronomer
\index{Hipparchus}%
of antiquity. He established inductively the famous
theory of epicycles and eccentrics. As might be expected, he
\index{Epicycles}%
was interested in mathematics, not \textit{per se}, but only as an aid
to astronomical inquiry. No mathematical writings of his
are extant, but Theon of Alexandria informs us that Hipparchus
originated the science of \emph{trigonometry}, and that he calculated
\index{Trigonometry}%
a ``table of chords'' in twelve books. Such calculations
%% -----File: 066.png---Folio 52-------
\index{Circle}%
must have required a ready knowledge of arithmetical and
algebraical operations.

About 100~\BC\ flourished \textbf{Heron the Elder} of Alexandria.
\index{Heron the Elder}%
He was the pupil of Ctesibius, who was celebrated for his
\index{Ctesibius}%
ingenious mechanical inventions, such as the hydraulic organ,
the water-clock, and catapult. It is believed by some that
Heron was a son of Ctesibius. He exhibited talent of the
same order as did his master by the invention of the eolipile
and a curious mechanism known as ``Heron's fountain.''
Great uncertainty exists concerning his writings. Most authorities
believe him to be the author of an important \textit{Treatise
on the Dioptra}, of which there exist three manuscript copies,
quite dissimilar. But M.~Marie\Reference[]{14} thinks that the \textit{Dioptra} is
\index{Marie, M.}%
the work of \textit{Heron the Younger}, who lived in the seventh or
eighth century after Christ, and that \textit{Geodesy}, another book
supposed to be by Heron, is only a corrupt and defective copy
of the former work. \textit{Dioptra} contains the important formula
for finding the area of a triangle expressed in terms of its
sides; its derivation is quite laborious and yet exceedingly
ingenious. ``It seems to me difficult to believe,'' says Chasles,
\index{Chasles!ref.\ to}%
``that so beautiful a theorem should be found in a work so
ancient as that of Heron the Elder, without that some Greek
geometer should have thought to cite it.'' Marie lays great
stress on this silence of the ancient writers, and argues from
it that the true author must be Heron the Younger or some
writer much more recent than Heron the Elder. But no reliable
evidence has been found that there actually existed a
second mathematician by the name of Heron.

``Dioptra,'' says Venturi, were instruments which had great
\index{Venturi}%
resemblance to our modern theodolites. The book \textit{Dioptra} is
a treatise on geodesy containing solutions, with aid of these
instruments, of a large number of questions in geometry, such
as to find the distance between two points, of which one only
%% -----File: 067.png---Folio 53-------
is accessible, or between two points which are visible but both
inaccessible; from a given point to draw a perpendicular to a
line which cannot be approached; to find the difference of
level between two points; to measure the area of a field without
entering it.

Heron was a practical surveyor. This may account for the
fact that his writings bear so little resemblance to those of
the Greek authors, who considered it degrading the science
to apply geometry to surveying. The character of his geometry
is not Grecian, but decidedly Egyptian. This fact is the
more surprising when we consider that Heron demonstrated
his familiarity with Euclid by writing a commentary on the
\index{Euclid!ref.\ to}%
\textit{Elements}.\Reference{21} Some of Heron's formulas point to an old Egyptian
origin. Thus, besides the above exact formula for the
area of a triangle in terms of its sides, Heron gives the formula
$\dfrac{a_1 + a_2\Fstr}{2} × \dfrac{b}{2}$, which bears a striking likeness to the formula
$\dfrac{a_1 + a_2}{2} × \dfrac{b_1 + b_2}{2}$ for finding the area of a quadrangle,
found in the Edfu inscriptions. There are, moreover, points
\index{Edfu}%
of resemblance between Heron's writings and the ancient
Ahmes papyrus. Thus Ahmes used unit-fractions exclusively;
\index{Ahmes!ref.\ to}%
Heron uses them oftener than other fractions. Like Ahmes
and the priests at Edfu, Heron divides complicated figures
into simpler ones by drawing auxiliary lines; like them,
he shows, throughout, a special fondness for the isosceles
trapezoid.

The writings of Heron satisfied a practical want, and for
that reason were borrowed extensively by other peoples. We
find traces of them in Rome, in the Occident during the Middle
Ages, and even in India.

\textbf{Geminus} of Rhodes (about 70~\BC) published an astronomical
\index{Geminus}%
work still extant. He wrote also a book, now lost, on the
\textit{Arrangement of Mathematics}, which contained many valuable
%% -----File: 068.png---Folio 54-------
notices of the early history of Greek mathematics. Proclus
\index{Proclus!ref.\ to}%
and Eutocius quote it frequently. \textbf{Theodosius} of Tripolis is
\index{Eutocius!ref.\ to}%
\index{Theodosius}%
the author of a book of little merit on the geometry of the
sphere. \textbf{Dionysodorus} of Amisus in Pontus applied the intersection
\index{Dionysodorus}%
of a parabola and hyperbola to the solution of a problem
which Archimedes, in his \textit{Sphere and Cylinder}, had left
\index{Archimedes!ref.\ to}%
incomplete. The problem is ``to cut a sphere so that its segments
shall be in a given ratio.''

We have now sketched the progress of geometry down to
the time of Christ. Unfortunately, very little is known of
the history of geometry between the time of Apollonius and
\index{Apollonius!ref.\ to}%
the beginning of the Christian era. The names of quite a
number of geometers have been mentioned, but very few of
their works are now extant. It is certain, however, that there
were no mathematicians of real genius from Apollonius to
Ptolemy, excepting Hipparchus and perhaps Heron.
\index{Heron the Elder!ref.\ to}%
\index{Hipparchus!ref.\ to}%
\index{Ptolemy!ref.\ to}%

\Subsection{The Second Alexandrian School}
\index{Alexandrian School!(first)|)}%
\index{Alexandrian School!(second)|(}%

The close of the dynasty of the Lagides which ruled Egypt
from the time of Ptolemy Soter, the builder of Alexandria,
for 300~years; the absorption of Egypt into the Roman Empire;
the closer commercial relations between peoples of the
East and of the West; the gradual decline of paganism and
spread of Christianity,---these events were of far-reaching
influence on the progress of the sciences, which then had their
home in Alexandria. Alexandria became a commercial and
intellectual emporium. Traders of all nations met in her
busy streets, and in her magnificent Library, museums, lecture-halls,
scholars from the East mingled with those of the
West; Greeks began to study older literatures and to compare
them with their own. In consequence of this interchange
of ideas the Greek philosophy became fused with Oriental
%% -----File: 069.png---Folio 55-------
\index{Conic sections!Greek}%
\index{Ptolemy!ref.\ to}%
philosophy. Neo-Pythagoreanism and Neo-Platonism were the
names of the modified systems. These stood, for a time, in
opposition to Christianity. The study of Platonism and
Pythagorean mysticism led to the revival of the theory of
numbers. Perhaps the dispersion of the Jews and their
\index{Numbers!theory of numbers}%
introduction to Greek learning helped in bringing about this
revival. The theory of numbers became a favourite study.
\index{Theory of numbers}%
This new line of mathematical inquiry ushered in what we
may call a new school. There is no doubt that even now
geometry continued to be one of the most important studies
in the Alexandrian course. This Second Alexandrian School
may be said to begin with the Christian era. It was made
famous by the names of Claudius Ptolemæus, Diophantus,
\index{Diophantus!ref.\ to}%
Pappus, Theon of Smyrna, Theon of Alexandria, Iamblichus,
\index{Pappus!ref.\ to}%
\index{Theon of Alexandria!ref.\ to}%
\index{Theon of Smyrna}%
Porphyrius, and others.
\index{Porphyrius}%

By the side of these we may place \textbf{Serenus} of Antissa, as
\index{Serenus}%
having been connected more or less with this new school.
He wrote on sections of the cone and cylinder, in two books,
one of which treated only of the triangular section of the cone
through the apex. He solved the problem, ``given a cone
(cylinder), to find a cylinder (cone), so that the section of
both by the same plane gives similar ellipses.'' Of particular
interest is the following theorem, which is the foundation
of the modern theory of
harmonics: If from $D$ we
\index{Harmonics}%
% [Illustration]
\begin{wrapfigure}{r}{2.25in} %[* PP: Width-dependent line break]
\hfill\Graphic{2.25in}{069a}
\end{wrapfigure}
draw~$DF$, cutting the triangle
$ABC$, and choose
$H$ on it, so that $DE : DF
= EH : HF$, and if we draw
the line~$AH$, then every transversal through $D$, such as
$DG$, will be divided by $AH$ so that $DK : DG = KJ : JG$.
\textbf{Menelaus} of Alexandria (about 98~\AD) was the author of
\index{Menelaus}%
\textit{Sphærica}, a work extant in Hebrew and Arabic, but not
%% -----File: 070.png---Folio 56-------
\index{Astronomy!Greek}%
\index{Ptolemaic System}%
\index{Spherical trigonometry}%
in Greek. In it he proves the theorems on the congruence
of spherical triangles, and describes their properties in
much the same way as Euclid treats plane triangles. In
it are also found the theorems that the sum of the three
sides of a spherical triangle is less than a great circle, and
that the sum of the three angles exceeds two right angles.
Celebrated are two theorems of his on plane and spherical
triangles. The one on plane triangles is that, ``if the three
sides be cut by a straight line, the product of the three segments
which have no common extremity is equal to the
product of the other three.'' The illustrious Carnot makes
\index{Carnot, Lazare!ref.\ to}%
this proposition, known as the `lemma of Menelaus,' the base
of his theory of transversals. The corresponding theorem for
spherical triangles, the so-called `regula sex quantitatum,' is
obtained from the above by reading ``chords of three segments
doubled,'' in place of ``three segments.''

\textbf{Claudius Ptolemæus}, a celebrated astronomer, was a native of
Egypt. Nothing is known of his personal history except that
he flourished in Alexandria in 139~\AD\ and that he made the
earliest astronomical observations recorded in his works, in
125~\AD, the latest in 151~\AD. The chief of his works are
the \textit{Syntaxis Mathematica} (or the \textit{Almagest}, as the Arabs call
\index{Almagest|(}%
it) and the \textit{Geographica}, both of which are extant. The
former work is based partly on his own researches, but mainly
on those of Hipparchus. Ptolemy seems to have been not
\index{Hipparchus!ref.\ to}%
\index{Ptolemy|(}%
so much of an independent investigator, as a corrector and
improver of the work of his great predecessors. The \textit{Almagest}
forms the foundation of all astronomical science down to
Copernicus. The fundamental idea of his system, the ``Ptolemaic
\index{Copernicus}%
System,'' is that the earth is in the centre of the universe,
and that the sun and planets revolve around the earth.
Ptolemy did considerable for mathematics. He created, for
astronomical use, a \emph{trigonometry} remarkably perfect in form.
\index{Trigonometry}%
%% -----File: 071.png---Folio 57-------
\index{Fractions!sexagesimal}%
\index{Sexagesimal system}%
The foundation of this science was laid by the illustrious
Hipparchus.

The \textit{Almagest} is in 13~books. Chapter~9 of the first book
shows how to calculate tables of chords. The circle is divided
into $360$~degrees, each of which is halved. The diameter is
divided into $120$~divisions; each of these into $60$~parts, which
are again subdivided into $60$~smaller parts. In Latin, these
parts were called \textit{partes minutæ primæ} and \textit{partes minutæ
secundæ}. Hence our names, `minutes' and `seconds.'\Reference{3} The
sexagesimal method of dividing the circle is of Babylonian
origin, and was known to Geminus and Hipparchus. But
\index{Geminus!ref.\ to}%
Ptolemy's method of calculating chords seems original with
him. He first proved the proposition, now appended to
Euclid~VI.~(D), that ``the rectangle contained by the diagonals
\index{Euclid!ref.\ to}%
of a quadrilateral figure inscribed in a circle is equal
to both the rectangles contained by its opposite sides.'' He
then shows how to find from the chords of two arcs the
chords of their sum and difference, and from the chord of any
arc that of its half. These theorems he applied to the calculation
of his tables of chords. The proofs of these theorems
are very pretty.

Another chapter of the first book in the \textit{Almagest} is devoted
to \emph{trigonometry}, and to \emph{spherical} trigonometry in particular.
\index{Trigonometry!spherical}%
Ptolemy proved the `lemma of Menelaus,' and also the `regula
\index{Menelaus!ref.\ to}%
sex quantitatum.' Upon these propositions he built up his
trigonometry. The fundamental theorem of plane trigonometry,
that two sides of a triangle are to each other as the chords
of double the arcs measuring the angles opposite the two
sides, was not stated explicitly by him, but was contained
implicitly in other theorems. More complete are the propositions
in spherical trigonometry.

The fact that trigonometry was cultivated not for its own
sake, but to aid astronomical inquiry, explains the rather
%% -----File: 072.png---Folio 58-------
startling fact that spherical trigonometry came to exist in a
developed state earlier than plane trigonometry.

The remaining books of the \textit{Almagest} are on astronomy.
Ptolemy has written other works which have little or no bearing
on mathematics, except one on geometry. Extracts from
this book, made by Proclus, indicate that Ptolemy did not
\index{Proclus!ref.\ to}%
regard the parallel-axiom of Euclid as self-evident, and that
\index{Euclid!ref.\ to}%
Ptolemy was the first of the long line of geometers from
ancient time down to our own who toiled in the vain attempt
to prove it.

Two prominent mathematicians of this time were Nicomachus
\index{Nicomachus!ref.\ to}%
and Theon of Smyrna. Their favourite study was theory
\index{Theon of Smyrna}%
of numbers. The investigations in this science culminated
later in the algebra of Diophantus. But no important geometer
appeared after Ptolemy for 150~years. The only occupant
\index{Ptolemy|)}%
of this long gap was \textbf{Sextus Julius Africanus}, who wrote an
\index{Sextus Julius Africanus}%
unimportant work on geometry applied to the art of war,
entitled \textit{Cestes}.

\textbf{Pappus}, probably born about 340~\AD, in Alexandria, was
\index{Pappus|(}%
the last great mathematician of the Alexandrian school. His
genius was inferior to that of Archimedes, Apollonius, and
Euclid, who flourished over 500~years earlier. But living,
as he did, at a period when interest in geometry was declining,
he towered above his contemporaries ``like the peak
of Teneriffa above the Atlantic.'' He is the author of a \textit{Commentary
on the Almagest}, a \textit{Commentary on Euclid's Elements},
\index{Almagest|)}%
a \textit{Commentary on the Analemma of Diodorus},---a writer of
\index{Diodorus}%
whom nothing is known. All these works are lost. Proclus,
probably quoting from the \textit{Commentary on Euclid}, says that
Pappus objected to the statement that an angle equal to a
right angle is always itself a right angle.

The only work of Pappus still extant is his \textit{Mathematical
Collections}. This was originally in eight books, but the first
%% -----File: 073.png---Folio 59-------
and portions of the second are now missing. The \textit{Mathematical
Collections} seems to have been written by Pappus to supply
the geometers of his time with a succinct analysis of the most
difficult mathematical works and to facilitate the study of
them by explanatory lemmas. But these lemmas are selected
very freely, and frequently have little or no connection with the
subject on hand. However, he gives very accurate summaries
of the works of which he treats. The \textit{Mathematical Collections}
is invaluable to us on account of the rich information it
gives on various treatises by the foremost Greek mathematicians,
which are now lost. Mathematicians of the last century
considered it possible to restore lost works from the \textit{résumé}
by Pappus alone.

We shall now cite the more important of those theorems in
the \textit{Mathematical Collections} which are supposed to be original
with Pappus. First of all ranks the elegant theorem re-discovered
by \textit{Guldin},\DPnote{** PP: [sic], not boldface} over 1000~years later, that the volume
\index{Guldin!ref.\ to}%
generated by the revolution of a plane curve which lies wholly
on one side of the axis, equals the area of the curve multiplied
by the circumference described by its centre of gravity.
Pappus proved also that the centre of gravity of a triangle is
that of another triangle whose vertices lie upon the sides of
the first and divide its three sides in the same ratio. In the
fourth book are new and brilliant propositions on the quadratrix
\index{Quadratrix}%
which indicate an intimate acquaintance with curved
surfaces. He generates the quadratrix as follows: Let a
spiral line be drawn upon a right circular cylinder; then the
perpendiculars to the axis of the cylinder drawn from each
point of the spiral line form the surface of a screw. A plane
passed through one of these perpendiculars, making any convenient
angle with the base of the cylinder, cuts the screw-surface
in a curve, the orthogonal projection of which upon
the base is the \emph{quadratrix}. A second mode of generation is
%% -----File: 074.png---Folio 60-------
\index{Problem of Pappus}%
\index{Spirals}%
no less admirable: If we make the spiral of Archimedes the
base of a right cylinder, and imagine a cone of revolution
having for its axis the side of the cylinder passing through
the initial point of the spiral, then this cone cuts the cylinder
in a curve of double curvature. The perpendiculars to the
axis drawn through every point in this curve form the surface
of a screw which Pappus here calls the \emph{plectoidal surface}. A
\index{Plectoidal surface}%
plane passed through one of the perpendiculars at any convenient
angle cuts that surface in a curve whose orthogonal
projection upon the plane of the spiral is the required \emph{quadratrix}.
\index{Quadratrix}%
Pappus considers curves of double curvature still further.
He produces a \emph{spherical spiral} by a point moving uniformly
along the circumference of a great circle of a sphere, while
the great circle itself revolves uniformly around its diameter.
He then finds the area of that portion of the surface of the
sphere determined by the spherical spiral, ``a complanation
which claims the more lively admiration, if we consider that,
although the entire surface of the sphere was known since
Archimedes' time, to measure portions thereof, such as spherical
triangles, was then and for a long time afterwards an
unsolved problem.''\Reference{3} A question which was brought into
prominence by Descartes and Newton is the ``problem of
\index{Descartes!ref.\ to}%
\index{Newton!ref.\ to}%
Pappus.'' Given several straight lines in a plane, to find the
locus of a point such that when perpendiculars (or, more
generally, straight lines at given angles) are drawn from it to
the given lines, the product of certain ones of them shall be in
a given ratio to the product of the remaining ones. It is
worth noticing that it was Pappus who first found the focus
\index{Focus}%
of the parabola, suggested the use of the directrix, and propounded
\index{Directrix}%
the theory of the involution of points. He solved
\index{Involution of points}%
the problem to draw through three points lying in the same
straight line, three straight lines which shall form a triangle
inscribed in a given circle.\Reference{3} From the \textit{Mathematical Collections}
%% -----File: 075.png---Folio 61-------
many more equally difficult theorems might be quoted which
are original with Pappus as far as we know. It ought to be
remarked, however, that he is known in three instances to
have copied theorems without giving due credit, and that he
may have done the same thing in other cases in which we
have no data by which to ascertain the real discoverer.

About the time of Pappus lived \textbf{Theon} of Alexandria. He
\index{Pappus|)}%
\index{Theon of Alexandria}%
brought out an edition of Euclid's \textit{Elements} with notes, which
\index{Elements (Euclid's)}%
\index{Euclid!ref.\ to}%
he probably used as a text-book in his classes. His commentary
on the \textit{Almagest} is valuable for the many historical notices,
and especially for the specimens of Greek arithmetic which it
contains. Theon's daughter \textbf{Hypatia}, a woman celebrated for
\index{Hypatia}%
her beauty and modesty, was the last Alexandrian teacher of
reputation, and is said to have been an abler philosopher and
mathematician than her father. Her notes on the works of
Diophantus and Apollonius have been lost. Her tragic death
\index{Apollonius!ref.\ to}%
\index{Diophantus!ref.\ to}%
in 415~\AD\ is vividly described in Kingsley's \textit{Hypatia}.

From now on, mathematics ceased to be cultivated in
Alexandria. The leading subject of men's thoughts was
Christian theology. Paganism disappeared, and with it pagan
learning. The Neo-Platonic school at Athens struggled on a
century longer. Proclus, Isidorus, and others kept up the
\index{Isidorus of Seville!ref.\ to}%
\index{Proclus}%
``golden chain of Platonic succession.'' \textbf{Proclus}, the successor
of Syrianus, at the Athenian school, wrote a commentary on
Euclid's \textit{Elements}. We possess only that on the first book,
which is valuable for the information it contains on the
history of geometry. \textbf{Damascius} of Damascus, the pupil of
\index{Damascius}%
Isidorus, is now believed to be the author of the fifteenth
book of Euclid. Another pupil of Isidorus was \textbf{Eutocius} of
\index{Eutocius}%
Ascalon, the commentator of Apollonius and Archimedes.
\index{Archimedes!ref.\ to}%
\textbf{Simplicius} wrote a commentary on Aristotle's \textbf{De C\oe{}lo}. In
\index{Aristotle!ref.\ to}%
\index{Simplicius}%
the year~529, Justinian, disapproving heathen learning, finally
closed by imperial edict the schools at Athens.
%% -----File: 076.png---Folio 62-------
\index{Alexandrian School!(second)|)}%

As a rule, the geometers of the last 500~years showed
a lack of creative power. They were commentators rather
than discoverers.

The principal characteristics of ancient geometry are:---

(1)~A wonderful clearness and definiteness of its concepts
and an almost perfect logical rigour of its conclusions.

(2)~A complete want of general principles and methods.
Ancient geometry is decidedly \emph{special}. Thus the Greeks
\index{Geometry!Greek|)}%
possessed no general method of drawing tangents. ``The
\index{Tangents!in geometry}%
determination of the tangents to the three conic sections did
not furnish any rational assistance for drawing the tangent to
any other new curve, such as the conchoid, the cissoid, etc.''\Reference{15}
In the demonstration of a theorem, there were, for the ancient
geometers, as many different cases requiring separate proof
as there were different positions for the lines. The greatest
geometers considered it necessary to treat all possible cases
independently of each other, and to prove each with equal
fulness. To devise methods by which the various cases could
all be disposed of by one stroke, was beyond the power of the
ancients. ``If we compare a mathematical problem with a
huge rock, into the interior of which we desire to penetrate,
then the work of the Greek mathematicians appears to us like
that of a vigorous stonecutter who, with chisel and hammer,
begins with indefatigable perseverance, from without, to
crumble the rock slowly into fragments; the modern mathematician
appears like an excellent miner, who first bores
through the rock some few passages, from which he then bursts
it into pieces with one powerful blast, and brings to light the
treasures within.''\Reference{16}
%% -----File: 077.png---Folio 63-------
\index{Finger-reckoning}%

\SpecialSection{Greek Arithmetic}
\index{Arithmetic!Greek|(}%

Greek mathematicians were in the habit of discriminating
between the \emph{science} of numbers and the \emph{art} of calculation.
The former they called \textit{arithmetica}, the latter \textit{logistica}. The
drawing of this distinction between the two was very natural
and proper. The difference between them is as marked as
that between theory and practice. Among the Sophists the
art of calculation was a favourite study. Plato, on the other
\index{Plato!ref.\ to}%
hand, gave considerable attention to philosophical arithmetic,
but pronounced calculation a vulgar and childish art.

In sketching the history of Greek calculation, we shall first
give a brief account of the Greek mode of counting and of
writing numbers. Like the Egyptians and Eastern nations,
the earliest Greeks counted on their fingers or with pebbles.
In case of large numbers, the pebbles were probably arranged
in parallel vertical lines. Pebbles on the first line
represented units, those on the second tens, those on the third
hundreds, and so on. Later, frames came into use, in which
strings or wires took the place of lines. According to tradition,
Pythagoras, who travelled in Egypt and, perhaps, in
\index{Pythagoras!ref.\ to}%
India, first introduced this valuable instrument into Greece.
The \emph{abacus}, as it is called, existed among different peoples and
\index{Abacus}%
at different times, in various stages of perfection. An abacus
is still employed by the Chinese under the name of \emph{Swan-pan}.
We possess no specific information as to how the Greek abacus
looked or how it was used. Boethius says that the Pythagoreans
\index{Boethius!ref.\ to}%
used with the abacus certain nine signs called \emph{apices},
\index{Apices of Boethius!ref.\ to}%
which resembled in form the nine ``Arabic numerals.'' But
the correctness of this assertion is subject to grave doubts.

The oldest Grecian numerical symbols were the so-called
\emph{Herodianic signs} (after Herodianus, a Byzantine grammarian of
\index{Herodianic signs}%
about 200~\AD, who describes them). These signs occur frequently
%% -----File: 078.png---Folio 64-------
\index{Notation!Greek numbers}%
in Athenian inscriptions and are, on that account, now
generally called \emph{Attic}. For some unknown reason these symbols
were afterwards replaced by the \emph{alphabetic numerals}, in
\index{Numerals!Greek}%
which the letters of the Greek alphabet were used, together
with three strange and antique letters \textgreek{\stigma},~\Glyph{qoppa}, and~\Glyph{sampi}, and the
symbol~$\mathbf{M}$. This change was decidedly for the worse, for the
old Attic numerals were less burdensome on the memory, inasmuch
as they contained fewer symbols and were better adapted
to show forth analogies in numerical operations. The following
table shows the Greek alphabetic numerals and their
respective values:---
\medskip

\footnotesize
\setlength{\TmpLen}{0.055555\linewidth}%
\noindent\begin{tabular}{*{18}{@{}c@{}}}
\TE{$\alpha$} & \TE{$\beta$} & \TE{$\gamma$} & \TE{$\delta$} &
\TE{$\epsilon$} & \TE{\textgreek{\stigma}} &
\TE{$\zeta$} & \TE{$\eta$} & \TE{$\theta$} & \TE{$\iota$} &
\TE{$\kappa$} & \TE{$\lambda$} & \TE{$\mu$} &
\TE{$\nu$} & \TE{$\xi$} & \TE{o} & \TE{$\pi$} & \TE{\Glyph{qoppa}} \\
1&2&3&4&5&6&7&8&9&10&20&30&40&50&60&70&80&90
\end{tabular}

\setlength{\TmpLen}{0.076923\linewidth}%
\noindent\begin{tabular}{*{13}{@{}c@{}}}
\TE{$\rho$} & \TE{$\sigma$} & \TE{$\tau$} & \TE{$\upsilon$} &
\TE{$\phi$} & \TE{$\chi$} & \TE{$\psi$} & \TE{$\omega$} &
\TE{\Glyph{sampi}} &
\TE{\textgreek{\katwtonos}$\alpha$} &
\TE{\textgreek{\katwtonos}$\beta$ } &
\TE{\textgreek{\katwtonos}$\gamma$} & \TE{etc.} \\
100&200&300&400&500&600&700&800&900&1000&2000&3000
\end{tabular}

\noindent\begin{tabular}{@{}*{4}{c}@{}}
$\mathbf{M}$&$\overset{\beta}{\mathbf{M}}$&$\overset{\gamma}{\mathbf{M}}$&etc.\\
10,000&20,000&30,000
\end{tabular}
\medskip

\normalsize
It will be noticed that at $1000$, the alphabet is begun over
again, but, to prevent confusion, a stroke is now placed before
the letter and generally somewhat below it. A horizontal line
drawn over a number served to distinguish it more readily
from words. The coefficient for~$\mathbf{M}$ was sometimes placed
before or behind instead of over the~$\mathbf{M}$. Thus $43,678$ was
written $\overline{\delta\text{M\textgreek{\katwtonos}}\gamma\chi o\eta}$. It is to be observed that the Greeks had no
zero.

Fractions were denoted by first writing the numerator
\index{Fractions!Greek}%
marked with an accent, then the denominator marked with
two accents and written twice. Thus, $\iota \gamma' \kappa \theta'' \kappa \theta'' = \frac{13}{29}$. In case
of fractions having unity for the numerator, the $\alpha'$ was omitted
and the denominator was written only once. Thus $\mu \delta'' = \frac{1}{44}$.
%% -----File: 079.png---Folio 65-------
\index{Sexagesimal system}%

Greek writers seldom refer to calculation with alphabetic
numerals. Addition, subtraction, and even multiplication were
probably performed on the abacus. Expert mathematicians
may have used the symbols. Thus Eutocius, a commentator
\index{Eutocius!ref.\ to}%
of the sixth century after Christ, gives a great many multiplications
of which the following is a specimen:\Reference[]{6}---

\begin{wrapfigure}[8]{l}{2.125in}\footnotesize
\vspace*{-0.0625in}\begin{tabular}{@{}c@{\;}c@{\;}c@{\;}c@{}c@{}r@{\;}r@{\;}r@{}}
\multicolumn{4}{l}{$\quad\overline{\sigma\,\xi\,\epsilon}$} &
  $\mspace{18mu}$ & \multicolumn{2}{l}{\quad 2 6 5} \\
\multicolumn{4}{l}{$\quad\overline{\sigma\,\xi\,\epsilon}$} &
                  & \multicolumn{2}{l}{\quad 2 6 5} \\\cline{1-4}\cline{6-8}
$\overset{\delta}{\mathbf{M}}$ & $\overset{\alpha}{\mathbf{M}}$ & \textgreek{\katwtonos}$\beta$ & \textgreek{\katwtonos}$\alpha$ &
                  & 40000, & 12000, & 1000 \\
$\overset{\alpha}{\mathbf{M}}$ & \textgreek{\katwtonos}$\beta$  & \textgreek{\katwtonos}$\overline{\gamma\chi}$ & $\overline{\tau}$ &
                  & 12000, &  3600, &  300 \\
\phantom{$\overset{\alpha}{\mathbf{M}}$} & \textgreek{\katwtonos}$\alpha$ & $\overline{\tau}$ & $\overline{\kappa\epsilon}$ &
                  &  1000, &   300, &   25 \\\cline{1-4}\cline{6-8}
\multicolumn{4}{c}{$\overset{\zeta}{\mathbf{M}}\enskip\overline{\sigma\,\kappa\,\epsilon}$} &
                  & 70225\phantom{,}
\end{tabular}
\end{wrapfigure}
The operation is explained sufficiently by the modern numerals appended.
In case of mixed numbers, the process was still more clumsy. Divisions
are found in Theon of Alexandria's commentary on the \textit{Almagest}. As
\index{Theon of Alexandria!ref.\ to}%
might be expected, the process is long and tedious.

We have seen in geometry that the more advanced mathematicians
frequently had occasion to extract the square root.
\index{Square root}%
Thus Archimedes in his \textit{Mensuration of the Circle} gives a
\index{Archimedes!ref.\ to}%
large number of square roots. He states, for instance, that
$\sqrt{3} < \frac{1351}{780}$ and $\sqrt{3} > \frac{265}{153}$, but he gives no clue to the method
by which he obtained these approximations. It is not improbable
that the earlier Greek mathematicians found the
square root by trial only. Eutocius says that the method of
extracting it was given by Heron, Pappus, Theon, and other
\index{Heron the Elder!ref.\ to}%
\index{Pappus!ref.\ to}%
commentators on the \textit{Almagest}. Theon's is the only ancient
method known to us. It is the same as the one used nowadays,
except that sexagesimal fractions are employed in place
\index{Fractions!Greek}%
\index{Fractions!sexagesimal}%
of our decimals. What the mode of procedure actually was
when sexagesimal fractions were not used, has been the subject
of conjecture on the part of numerous modern writers.\Reference{17}

Of interest, in connection with arithmetical symbolism, is
the \textit{Sand-Counter} (Arenarius), an essay addressed by \textbf{Archimedes}
\index{Arenarius}%
\index{Sand-counter}%
%% -----File: 080.png---Folio 66-------
to Gelon, king of Syracuse. In it Archimedes shows
that people are in error who think the sand cannot be counted,
or that if it can be counted, the number cannot be expressed
by arithmetical symbols. He shows that the number of grains
in a heap of sand not only as large as the whole earth, but as
large as the entire universe, can be arithmetically expressed.
Assuming that $10,000$ grains of sand suffice to make a little
solid of the magnitude of a poppy-seed, and that the diameter
of a poppy-seed be not smaller than $\frac{1}{40}$~part of a finger's
breadth; assuming further, that the diameter of the universe
(supposed to extend to the sun) be less than $10,000$ diameters
of the earth, and that the latter be less than $1,000,000$ stadia,
Archimedes finds a number which would exceed the number
of grains of sand in the sphere of the universe. He goes on
even further. Supposing the universe to reach out to the fixed
stars, he finds that the sphere, having the distance from the
earth's centre to the fixed stars for its radius, would contain
a number of grains of sand less than $1000$ myriads of the
eighth octad. In our notation, this number would be $10^{63}$ or
$1$~with $63$~ciphers after it. It can hardly be doubted that one
object which Archimedes had in view in making this calculation
was the improvement of the Greek symbolism. It is not
known whether he invented some short notation by which to
represent the above number or not.

We judge from fragments in the second book of Pappus that
\index{Pappus!ref.\ to}%
Apollonius proposed an improvement in the Greek method of
\index{Apollonius!ref.\ to}%
writing numbers, but its nature we do not know. Thus we
see that the Greeks never possessed the boon of a clear, comprehensive
symbolism. The honour of giving such to the world,
once for all, was reserved by the irony of fate for a nameless
Indian of an unknown time, and we know not whom to thank
for an invention of such importance to the general progress of
intelligence.\Reference{6}
%% -----File: 081.png---Folio 67-------
\index{Arithmetic!Pythagoreans|(}%
\index{Fractions!sexagesimal}%
\index{Sexagesimal system}%

Passing from the subject of \textit{logistica} to that of \textit{arithmetica},
our attention is first drawn to the science of numbers of
\textbf{Pythagoras}. Before founding his school, Pythagoras studied
\index{Pythagoras|(}%
for many years under the Egyptian priests and familiarised
himself with Egyptian mathematics and mysticism. If he
ever was in Babylon, as some authorities claim, he may have
learned the sexagesimal notation in use there; he may have
picked up considerable knowledge on the theory of proportion,
\index{Proportion}%
and may have found a large number of interesting astronomical
observations. Saturated with that speculative spirit then
pervading the Greek mind, he endeavoured to discover some
principle of homogeneity in the universe. Before him, the
philosophers of the Ionic school had sought it in the matter
of things; Pythagoras looked for it in the structure of things.
He observed various numerical relations or analogies between
numbers and the phenomena of the universe. Being convinced
that it was in numbers and their relations that he was to find
the foundation to true philosophy, he proceeded to trace the
origin of all things to numbers. Thus he observed that musical
strings of equal length stretched by weights having the
proportion of $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, produced intervals which were an octave,
a fifth, and a fourth. Harmony, therefore, depends on musical
proportion; it is nothing but a mysterious numerical relation.
Where harmony is, there are numbers. Hence the
order and beauty of the universe have their origin in numbers.
There are seven intervals in the musical scale, and also seven
planets crossing the heavens. The same numerical relations
which underlie the former must underlie the latter. But where
numbers are, there is harmony. Hence his spiritual ear discerned
in the planetary motions a wonderful `harmony of the
spheres.' The Pythagoreans invested particular numbers with
extraordinary attributes. Thus \emph{one} is the essence of things;
it is an absolute number; hence the origin of all numbers and
%% -----File: 082.png---Folio 68-------
so of all things. \emph{Four} is the most perfect number, and was in
some mystic way conceived to correspond to the human soul.
Philolaus believed that $5$ is the cause of color, $6$ of cold, $7$ of
\index{Philolaus!ref.\ to}%
mind and health and light, $8$ of love and friendship.\Reference{6} In
Plato's works are evidences of a similar belief in religious
\index{Plato!ref.\ to}%
relations of numbers. Even Aristotle referred the virtues to
\index{Aristotle!ref.\ to}%
\index{Numbers!amicable}%
\index{Numbers!excessive}%
\index{Numbers!heteromecic}%
\index{Numbers!perfect}%
\index{Numbers!defective}%
numbers.

Enough has been said about these mystic speculations to
show what lively interest in mathematics they must have
created and maintained. Avenues of mathematical inquiry
were opened up by them which otherwise would probably
have remained closed at that time.

The Pythagoreans classified numbers into odd and even.
They observed that the sum of the series of odd numbers
from $1$ to $2n + 1$ was always a complete square, and that
by addition of the even numbers arises the series $2$, $6$, $12$, $20$,
in which every number can be decomposed into two factors
differing from each other by unity. Thus, $6 = 2·3$, $12 = 3·4$,
etc. These latter numbers were considered of sufficient importance
to receive the separate name of \emph{heteromecic} (not
equilateral).\Reference{7} Numbers of the form $\dfrac{n(n+1)}{2}$ were called \emph{triangular},
because they could always be arranged thus,~\smash[t]{\TallGlyph{tendots}}.
Numbers which were equal to the sum of all their possible
factors, such as $6$, $28$, $496$, were called \emph{perfect}; those exceeding
that sum, \emph{excessive}; and those which were less, \emph{defective}.
\emph{Amicable} numbers were those of which each was the sum of
the factors in the other. Much attention was paid by the
Pythagoreans to the subject of proportion. The quantities
\index{Proportion}%
$a$, $b$, $c$, $d$ were said to be in \emph{arithmetical} proportion when
$a - b = c - d$; in \emph{geometrical} proportion, when $a : b = c : d$; in
\emph{harmonic} proportion, when $a - b : b - c = a : c$. It is probable
that the Pythagoreans were also familiar with the \emph{musical}
%% -----File: 083.png---Folio 69-------
\index{Irrationals}%
proportion $a : \dfrac{a + b\Fstr}{2} = \dfrac{2ab}{a+b} : b$. Iamblichus says that Pythagoras
\index{Iamblichus!ref.\ to}%
introduced it from Babylon.

In connection with arithmetic, Pythagoras made extensive
investigations into geometry. He believed that an arithmetical
\index{Geometry!Greek}%
fact had its analogue in geometry, and \textit{vice versa}. In
connection with his theorem on the right triangle he devised
a rule by which integral numbers could be found, such
that the sum of the squares of two of them equalled the
square of the third. Thus, take for one side an odd number
$(2n + 1)$; then $\dfrac{(2n + 1)^{2} - 1}{2} = 2n^{2} + 2n =$ the other side, and
$(2n^2 + 2n + 1) =$ hypotenuse. If $2n + 1 = 9$, then the other
two numbers are $40$~and~$41$. But this rule only applies to
cases in which the hypotenuse differs from one of the sides
by~$1$. In the study of the right triangle there doubtless arose
questions of puzzling subtlety. Thus, given a number equal
to the side of an isosceles right triangle, to find the number
which the hypotenuse is equal to. The side may have been
taken equal to $1$, $2$, $\frac{3}{2}$, $\frac{6}{5}$, or any other number, yet in every
instance all efforts to find a number exactly equal to the
hypotenuse must have remained fruitless. The problem may
have been attacked again and again, until finally ``some rare
genius, to whom it is granted, during some happy moments,
to soar with eagle's flight above the level of human thinking,''
grasped the happy thought that this problem cannot be
solved. In some such manner probably arose the theory of
\emph{irrational quantities}, which is attributed by Eudemus to the
\index{Eudemus}%
Pythagoreans. It was indeed a thought of extraordinary
boldness, to assume that straight lines could exist, differing
from one another not only in length,---that is, in quantity,---but
also in a quality, which, though real, was absolutely
invisible.\Reference{7} Need we wonder that the Pythagoreans saw in
%% -----File: 084.png---Folio 70-------
\index{Parabola}%
irrationals a deep mystery, a symbol of the unspeakable?
We are told that the one who first divulged the theory of
irrationals, which the Pythagoreans kept secret, perished in
consequence in a shipwreck. Its discovery is ascribed to
Pythagoras, but we must remember that all important Pythagorean
\index{Pythagoras|)}%
discoveries were, according to Pythagorean custom,
referred back to him. The first incommensurable ratio known
seems to have been that of the side of a square to its diagonal,
as $1 : \sqrt {2}$. \textbf{Theodorus of Cyrene} added to this the fact that the
\index{Theodorus}%
sides of squares represented in length by $\sqrt {3}$, $\sqrt {5}$, etc., up to
$\sqrt {17}$, and Theætetus, that the sides of any square, represented
\index{Theætetus!ref.\ to}%
by a surd, are incommensurable with the linear unit. \textbf{Euclid}
\index{Euclid}%
(about 300~\BC), in his \textit{Elements}, X.~9, generalised still further:
Two magnitudes whose squares are (or are not) to one another
as a square number to a square number are commensurable
(or incommensurable), and conversely. In the tenth book, he
treats of incommensurable quantities at length. He investigates
every possible variety of lines which can be represented
by $\sqrt { \sqrt {\vphantom{b}a} ± \sqrt {b}}$, $a$~and~$b$ representing two commensurable lines,
and obtains 25~species. Every individual of every species
is incommensurable with all the individuals of every other
species. ``This book,'' says De~Morgan, ``has a completeness
\index{Demorgan@{De Morgan}!ref.\ to}%
which none of the others (not even the fifth) can boast of;
and we could almost suspect that Euclid, having arranged his
materials in his own mind, and having completely elaborated
the tenth book, wrote the preceding books after it, and did not
live to revise them thoroughly.''\Reference{9} The theory of incommensurables
\index{Incommensurables}%
remained where Euclid left it, till the fifteenth
century.

Euclid devotes the seventh, eighth, and ninth books of his
\textit{Elements} to arithmetic. Exactly how much contained in
\index{Arithmetic!Pythagoreans|)}%
\index{Arithmetic!Euclid}%
these books is Euclid's own invention, and how much is
borrowed from his predecessors, we have no means of knowing.
%% -----File: 085.png---Folio 71-------
Without doubt, much is original with Euclid. The \textit{seventh
\index{Euclid}%
book} begins with twenty-one definitions. All except that for
`prime' numbers are known to have been given by the
\index{Prime numbers}%
Pythagoreans. Next follows a process for finding the
G.C.D.~of two or more numbers. The \textit{eighth book} deals with
numbers in continued proportion, and with the mutual relations
of squares, cubes, and plane numbers. Thus,~XXII., if
three numbers are in continued proportion, and the first is a
square, so is the third. In the \textit{ninth book}, the same subject is
continued. It contains the proposition that the number of
primes is greater than any given number.

After the death of Euclid, the theory of numbers remained
almost stationary for 400~years. Geometry monopolised
the attention of all Greek mathematicians. Only two are
known to have done work in arithmetic worthy of mention.
\textbf{Eratosthenes} (275--194~\BC) invented a `sieve' for finding prime
\index{Eratosthenes!ref.\ to}%
numbers. All composite numbers are `sifted' out in the
following manner: Write down the odd numbers from $3$~up,
in succession. By striking out every third number after the~$3$,
we remove all multiples of~$3$. By striking out every fifth
number after the~$5$, we remove all multiples of~$5$. In this way,
by rejecting multiples of $7$,~$11$, $13$,~etc., we have left prime
numbers only. \textbf{Hypsicles} (between 200 and 100~\BC) worked
\index{Hypsicles!ref.\ to}%
at the subjects of polygonal numbers and arithmetical progressions,
which Euclid entirely neglected. In his work on
`risings of the stars,' he showed (1)~that in an arithmetical
series of $2n$~terms, the sum of the last~$n$ terms exceeds the sum
of the first~$n$ by a multiple of~$n^2$; (2)~that in such a series of
$2n + 1$~terms, the sum of the series is the number of terms
multiplied by the middle term; (3)~that in such a series of
$2n$~terms, the sum is half the number of terms multiplied by
the two middle terms.\Reference{6}

For two centuries after the time of Hypsicles, arithmetic
%% -----File: 086.png---Folio 72-------
\index{Cube numbers}%
disappears from history. It is brought to light again about
100~\AD\ by \textbf{Nicomachus}, a Neo-Pythagorean, who inaugurated
\index{Nicomachus}%
the final era of Greek mathematics. From now on, arithmetic
was a favourite study, while geometry was neglected. Nicomachus
wrote a work entitled \textit{Introductio Arithmetica}, which
was very famous in its day. The great number of commentators
it has received vouch for its popularity. Boethius
\index{Boethius!ref.\ to}%
translated it into Latin. Lucian could pay no higher
compliment to a calculator than this: ``You reckon like
Nicomachus of Gerasa.'' The \textit{Introductio Arithmetica} was
the first exhaustive work in which arithmetic was treated
quite independently of geometry. Instead of drawing lines,
like Euclid, he illustrates things by real numbers. To be
\index{Euclid!ref.\ to}%
sure, in his book the old geometrical nomenclature is retained,
but the method is inductive instead of deductive. ``Its sole
business is classification, and all its classes are derived from,
and exhibited by, actual numbers.'' The work contains few
results that are really original. We mention one important
proposition which is probably the author's own. He states
that cubical numbers are always equal to the sum of successive
odd numbers. Thus, $8 = 2^{3} = 3 + 5$, $27 = 3^{3} = 7 + 9 + 11$, $64
= 4^{3} = 13 + 15 + 17 + 19$, and so on. This theorem was used
later for finding the sum of the cubical numbers themselves.
\textbf{Theon} of Smyrna is the author of a treatise on ``the mathematical
\index{Theon of Smyrna}%
rules necessary for the study of Plato.'' The work is
ill arranged and of little merit. Of interest is the theorem,
that every square number, or that number minus~$1$, is
divisible by $3$ or $4$ or both. A remarkable discovery is a
proposition given by \textbf{Iamblichus} in his treatise on Pythagorean
\index{Iamblichus}%
philosophy. It is founded on the observation that the
Pythagoreans called $1$, $10$, $100$, $1000$, units of the first, second,
third, fourth `course' respectively. The theorem is this:
If we add any three consecutive numbers, of which the highest
%% -----File: 087.png---Folio 73-------
\index{Arabic numerals and notation}%
is divisible by~$3$, then add the digits of that sum, then, again,
the digits of \emph{that} sum, and so on, the final sum will be~$6$.
Thus, $61 + 62 + 63 = 186$, $1 + 8 + 6 = 15$, $1 + 5 = 6$. This
discovery was the more remarkable, because the ordinary
Greek numerical symbolism was much less likely to suggest
any such property of numbers than our ``Arabic'' notation
\index{Notation!Arabic notation}%
would have been.

The works of Nicomachus, Theon of Smyrna, Thymaridas,
\index{Thymaridas}%
and others contain at times investigations of subjects which
are really algebraic in their nature. Thymaridas in one place
uses the Greek word meaning ``unknown quantity'' in a way
which would lead one to believe that algebra was not far
\index{Algebra!early Greek}%
distant. Of interest in tracing the invention of algebra are
the arithmetical epigrams in the \textit{Palatine Anthology}, which contain
\index{Anthology, Palatine}%
\index{Palatine anthology}%
about fifty problems leading to linear equations. Before
the introduction of algebra these problems were propounded
as puzzles. A riddle attributed to Euclid and contained in
\index{Euclid!ref.\ to}%
the \textit{Anthology} is to this effect: A mule and a donkey were
walking along, laden with corn. The mule says to the
donkey, ``If you gave me one measure, I should carry twice
as much as you. If I gave you one, we should both carry
equal burdens. Tell me their burdens, O~most learned master
of geometry.''\Reference{6}

It will be allowed, says Gow, that this problem, if authentic,
was not beyond Euclid, and the appeal to geometry smacks of
antiquity. A far more difficult puzzle was the famous `cattle-problem,'
\index{Cattle-problem}%
which Archimedes propounded to the Alexandrian
\index{Archimedes!ref.\ to}%
mathematicians. The problem is indeterminate, for from only
seven equations, eight unknown quantities in integral numbers
are to be found. It may be stated thus: The sun had a herd
of bulls and cows, of different colours. (1)~Of Bulls, the
white~($W$) were, in number, $(\frac{1}{2} + \frac{1}{3})$ of the blue~($B$) and yellow~($Y$):
the $B$ were $(\frac{1}{4} + \frac{1}{5})$ of the~$Y$ and piebald~($P$): the
%% -----File: 088.png---Folio 74-------
$P$ were $(\frac{1}{6} + \frac{1}{7})$ of the~$W$ and~$Y$. (2)~Of Cows, which had the
same colours ($w$, $b$, $y$, $p$),
\iffalse
\begin{align*}% [** PP: This commented code visually matches the original]
w=(\tfrac{1}{3} + \tfrac{1}{4})(B + b)
  &: b = (\tfrac{1}{4} + \tfrac{1}{5})(P + p) : p = (\tfrac{1}{5} + \tfrac{1}{6})(Y + y) \\
  &: y = (\tfrac{1}{6} + \tfrac{1}{7})(W + w).
\end{align*}
\fi
%[** PP: Re-breaking and aligning on =, but otherwise retaining verbatim]
\begin{align*}
w = (\tfrac{1}{3} + \tfrac{1}{4})(B + b) : b
 &= (\tfrac{1}{4} + \tfrac{1}{5})(P + p) : p \\
 &= (\tfrac{1}{5} + \tfrac{1}{6})(Y + y) : y
  = (\tfrac{1}{6} + \tfrac{1}{7})(W + w).
\end{align*}
Find the number of bulls and cows.\Reference{6} Another problem in the
\textit{Anthology} is quite familiar to school-boys: ``Of four pipes,
one fills the cistern in one day, the next in two days, the third
in three days, the fourth in four days: if all run together, how
soon will they fill the cistern?'' A great many of these
problems, puzzling to an arithmetician, would have been solved
easily by an algebraist. They became very popular about the
time of Diophantus, and doubtless acted as a powerful stimulus
\index{Diophantus|(}%
on his mind.

\textbf{Diophantus} was one of the last and most fertile mathematicians
of the second Alexandrian school. He died about
330~\AD. His age was eighty-four, as is known from an epitaph
to this effect: Diophantus passed $\frac{1}{6}$~of his life in childhood, $\frac{1}{12}$~in
youth, and $\frac{1}{7}$~more as a bachelor; five years after his marriage
was born a son who died four years before his father, at half
his father's age. The place of nativity and parentage of
Diophantus are unknown. If his works were not written in
Greek, no one would think for a moment that they were the
product of Greek mind. There is nothing in his works that
reminds us of the classic period of Greek mathematics. His
were almost entirely new ideas on a new subject. In the
circle of Greek mathematicians he stands alone in his specialty.
Except for him, we should be constrained to say that
among the Greeks \emph{algebra} was always an unknown science.
\index{Algebra!Diophantus|(}%

Of his works we have lost the \textit{Porisms}, but possess a fragment
of \textit{Polygonal Numbers}, and seven books of his great work
on \textit{Arithmetica}, said to have been written in 13~books.

If we except the Ahmes papyrus, which contains the first
\index{Ahmes!ref.\ to}%
%% -----File: 089.png---Folio 75-------
\index{Tentative assumption}%
suggestions of algebraic notation, and of the solution of
\index{Notation!in algebra}%
equations, then his \textit{Arithmetica} is the earliest treatise on
\index{Equations!theory of}%
algebra now extant. In this work is introduced the idea of
an algebraic equation expressed in algebraic symbols. His
treatment is purely analytical and completely divorced from
geometrical methods. He is, as far as we know, the first to
state that ``a negative number multiplied by a negative number
gives a positive number.'' This is applied to the multiplication
of differences, such as $(x - 1)(x - 2)$. It must be
remarked, however, that Diophantus had no notion whatever
of negative numbers standing by themselves. All he knew
were differences, such as $(2x - 10)$, in which $2x$ could not be
smaller than~$10$ without leading to an absurdity. He appears
to be the first who could perform such operations as $(x - 1)
× (x - 2)$ without reference to geometry. Such identities as
$(a + b)^2 = a^2 + 2ab + b^2$, which with Euclid appear in the elevated
rank of geometric theorems, are with Diophantus the
simplest consequences of the algebraic laws of operation. His
sign for subtraction was~\Glyph{subtract}, for equality~$\iota$. For unknown
quantities he had only one symbol,~$\varsigma$. He had no sign for
addition except juxtaposition. Diophantus used but few symbols,
and sometimes ignored even these by describing an operation
in words when the symbol would have answered just
as well.

In the solution of simultaneous equations Diophantus adroitly
managed with only one symbol for the unknown quantities and
arrived at answers, most commonly, by the method of \emph{tentative
assumption}, which consists in assigning to some of the unknown
\index{Assumption, tentative}%
quantities preliminary values, that satisfy only one or two of
the conditions. These values lead to expressions palpably
wrong, but which generally suggest some stratagem by which
values can be secured satisfying all the conditions of the
problem.
%% -----File: 090.png---Folio 76-------
\index{Numbers!theory of numbers}%
\index{Theory of numbers}%

Diophantus also solved determinate equations of the second
degree. We are ignorant of his method, for he nowhere goes
through with the whole process of solution, but merely states
the result. Thus, ``$84 x^2 + 7x = 7$, whence $x$ is found $= \frac{1}{4}$.''
Notice he gives only one root. His failure to observe that a
quadratic equation has two roots, even when both roots are
positive, rather surprises us. It must be remembered, however,
that this same inability to perceive more than one out of
the several solutions to which a problem may point is common
to all Greek mathematicians. Another point to be observed
is that he never accepts as an answer a quantity which is
negative or irrational.

Diophantus devotes only the first book of his \textit{Arithmetica} to
the solution of determinate equations. The remaining books
extant treat mainly of \emph{indeterminate quadratic equations} of the
\index{Quadratic equations}%
form $Ax^2 + Bx + C = y^2$, or of two simultaneous equations of the
same form. He considers several but not all the possible
cases which may arise in these equations. The opinion of
Nesselmann on the method of Diophantus, as stated by Gow,
\index{Nesselmann}%
is as follows: ``(1)~Indeterminate equations of the second
degree are treated completely only when the quadratic or
the absolute term is wanting: his solution of the equations
$Ax^{2} + C = y^{2}$ and $Ax^{2} + Bx + C = y^{2}$ is in many respects cramped.
(2)~For the `double equation' of the second degree he has a
definite rule only when the quadratic term is wanting in both
expressions: even then his solution is not general. More complicated
expressions occur only under specially favourable
circumstances.'' Thus, he solves $Bx + C^{2} = y^{2}$, $B_{1}x + C_1^2 = y_1^2$.

The extraordinary ability of Diophantus lies rather in
another direction, namely, in his wonderful ingenuity to reduce
all sorts of equations to particular forms which he knows
how to solve. Very great is the variety of problems considered.
The 130~problems found in the great work of Diophantus contain
%% -----File: 091.png---Folio 77-------
\index{Arithmetic!Greek|)}%
\index{Greeks|)}%
over 50~different classes of problems, which are strung
together without any attempt at classification. But still more
multifarious than the problems are the solutions. General
methods are unknown to Diophantus. Each problem has its
own distinct method, which is often useless for the most
closely related problems. ``It is, therefore, difficult for a
modern, after studying 100~Diophantine solutions, to solve
the~101st.''\Reference{7}

That which robs his work of much of its scientific value is
the fact that he always feels satisfied with one solution, though
his equation may admit of an indefinite number of values.
Another great defect is the absence of general methods. Modern
mathematicians, such as Euler, \DPtypo{La~Grange}{Lagrange}, Gauss, had to
\index{Euler!ref.\ to}%
\index{Gauss!ref.\ to}%
\index{Lagrange!ref.\ to}%
begin the study of indeterminate analysis anew and received
no direct aid from Diophantus in the formulation of methods.
In spite of these defects we cannot fail to admire the work
for the wonderful ingenuity exhibited therein in the solution
of particular equations.

It is still an open question and one of great difficulty
whether Diophantus derived portions of his algebra from
\index{Algebra!Diophantus|)}%
\index{Diophantus|)}%
Hindoo sources or not.

\Section{The Romans}
\index{Romans|(}%

Nowhere is the contrast between the Greek and Roman
mind shown forth more distinctly than in their attitude toward
the mathematical science. The sway of the Greek was a
flowering time for mathematics, but that of the Roman a
period of sterility. In philosophy, poetry, and art the Roman
was an imitator. But in mathematics he did not even rise to
the desire for imitation. The mathematical fruits of Greek
genius lay before him untasted. In him a science which had
%% -----File: 092.png---Folio 78-------
no direct bearing on practical life could awake no interest.
As a consequence, not only the higher geometry of Archimedes
\index{Archimedes!ref.\ to}%
and Apollonius, but even the \textit{Elements} of Euclid, were entirely
\index{Apollonius!ref.\ to}%
\index{Euclid!ref.\ to}%
neglected. What little mathematics the Romans possessed
did not come from the Greeks, but from more ancient
sources. Exactly where and how it originated is a matter of
doubt. It seems most probable that the ``Roman notation,''
\index{Notation!Roman}%
as well as the practical geometry of the Romans, came from
the old Etruscans, who, at the earliest period to which our
knowledge of them extends, inhabited the district between the
Arno and Tiber.

Livy tells us that the Etruscans were in the habit of representing
the number of years elapsed, by driving yearly a nail
into the sanctuary of Minerva, and that the Romans continued
this practice. A less primitive mode of designating numbers,
presumably of Etruscan origin, was a notation resembling the
present ``Roman notation.'' This system is noteworthy from
the fact that a principle is involved in it which is not met
with in any other; namely, the principle of subtraction. If a
letter be placed before another of greater value, its value is
not to be added to, but subtracted from, that of the greater.
In the designation of large numbers a horizontal bar placed
over a letter was made to increase its value one thousand fold.
In fractions the Romans used the duodecimal system.
\index{Fractions!Roman}%

Of arithmetical calculations, the Romans employed three
different kinds: Reckoning on the fingers, upon the abacus,
and by tables prepared for the purpose.\Reference{3} Finger-symbolism
was known as early as the time of King Numa, for he had
erected, says Pliny, a statue of the double-faced Janus, of
which the fingers indicated $365$ ($355$?), the number of days in
a year. Many other passages from Roman authors point out
the use of the fingers as aids to calculation. In fact, a finger-symbolism
of practically the same form was in use not only in
%% -----File: 093.png---Folio 79-------
\index{Canon paschalis}%
Rome, but also in Greece and throughout the East, certainly
as early as the beginning of the Christian era, and continued
to be used in Europe during the Middle Ages. We possess no
knowledge as to where or when it was invented. The second
mode of calculation, by the abacus, was a subject of elementary
\index{Abacus}%
\index{Calculation, origin of word}%
instruction in Rome. Passages in Roman writers indicate
that the kind of abacus most commonly used was covered with
dust and then divided into columns by drawing straight lines.
Each column was supplied with pebbles (calculi, whence `calculare'
and `calculate') which served for calculation. Additions
and subtractions could be performed on the abacus quite
easily, but in multiplication the abacus could be used only for
adding the particular products, and in division for performing
the subtractions occurring in the process. Doubtless at this
point recourse was made to mental operations and to the multiplication
table. Possibly finger-multiplication may also have
been used. But the multiplication of large numbers must, by
either method, have been beyond the power of the ordinary
arithmetician. To obviate this difficulty, the arithmetical
tables mentioned above were used, from which the desired
products could be copied at once. Tables of this kind were
prepared by \textit{Victorius} of Aquitania. His tables contain a
\index{Victorius}%
peculiar notation for fractions, which continued in use throughout
the Middle Ages. Victorius is best known for his \textit{canon
paschalis}, a rule for finding the correct date for Easter, which
he published in 457~\AD.

Payments of interest and problems in interest were very old
among the Romans. The Roman laws of inheritance gave
rise to numerous arithmetical examples. Especially unique is
the following: A dying man wills that, if his wife, being with
child, gives birth to a son, the son shall receive~$\frac{2}{3}$ and she $\frac{1}{3}$~of
his estates; but if a daughter is born, she shall receive~$\frac{1}{3}$ and
his wife~$\frac{2}{3}$. It happens that twins are born, a boy and a girl.
%% -----File: 094.png---Folio 80-------
How shall the estates be divided so as to satisfy the will?
The celebrated Roman jurist, Salvianus Julianus, decided that
the estates shall be divided into seven equal parts, of which
the son receives four, the wife two, the daughter one.

We next consider Roman geometry. He who expects to
\index{Geometry!Roman}%
find in Rome a science of geometry, with definitions, axioms,
theorems, and proofs arranged in logical order, will be disappointed.
The only geometry known was a \emph{practical} geometry,
which, like the old Egyptian, consisted only of empirical rules.
This practical geometry was employed in surveying. Treatises
thereon have come down to us, compiled by the Roman surveyors,
called \textit{agrimensores} or \textit{gromatici}. One would naturally
\index{Agrimensores}%
\index{Gromatici}%
expect rules to be clearly formulated. But no; they are left
to be abstracted by the reader from a mass of numerical examples.
``The total impression is as though the Roman gromatic
were thousands of years older than Greek geometry, and as
though a deluge were lying between the two.'' Some of their
rules were probably inherited from the Etruscans, but others
are identical with those of Heron. Among the latter is that
\index{Heron the Elder!ref.\ to}%
for finding the area of a triangle from its sides and the approximate
formula,~$\frac{13}{30}a^{2}$, for the area of equilateral triangles ($a$
being one of the sides). But the latter area was also calculated
by the formulas $\frac{1}{2}(a^{2} + a)$ and $\frac{1}{2} a^{2}$, the first of which was
unknown to Heron. Probably the expression $\frac{1}{2}a^{2}$ was derived
from the Egyptian formula $\dfrac{a + b\Fstr}{2} · \dfrac{c + d}{2}$ for the determination
of the surface of a quadrilateral. This Egyptian formula
was used by the Romans for finding the area, not only of rectangles,
but of any quadrilaterals whatever. Indeed, the gromatici
considered it even sufficiently accurate to determine the
areas of cities, laid out irregularly, simply by measuring their
circumferences.\Reference{7} Whatever Egyptian geometry the Romans
possessed was transplanted across the Mediterranean at the
%% -----File: 095.png---Folio 81-------
\index{Julian calendar}%
time of \textit{Julius Cæsar}, who ordered a survey of the whole
\index{Caesar@{Cæsar}, Julius}%
empire to secure an equitable mode of taxation. Cæsar also
reformed the calendar, and, for that purpose, drew from
\index{Calendar}%
Egyptian learning. He secured the services of the Alexandrian
astronomer, \textit{Sosigenes}.
\index{Sosigenes}%

In the fifth century, the Western Roman Empire was fast
falling to pieces. Three great branches---Spain, Gaul, and
the province of Africa---broke off from the decaying trunk.
In~476, the Western Empire passed away, and the Visigothic
chief, Odoacer, became king. Soon after, Italy was conquered
by the Ostrogoths under Theodoric. It is remarkable that
this very period of political humiliation should be the one
during which Greek science was studied in Italy most zealously.
School-books began to be compiled from the elements
of Greek authors. These compilations are very deficient, but
are of absorbing interest, from the fact that, down to the
twelfth century, they were the only sources of mathematical
knowledge in the Occident. Foremost among these writers is
\textbf{Boethius} (died~524). At first he was a great favourite of King
\index{Boethius}%
Theodoric, but later, being charged by envious courtiers with
treason, he was imprisoned, and at last decapitated. While
in prison he wrote \textit{On the Consolations of Philosophy}. As a
mathematician, Boethius was a Brobdingnagian among Roman
scholars, but a Liliputian by the side of Greek masters. He
wrote an \textit{Institutis Arithmetica}, which is essentially a translation
of the arithmetic of Nicomachus, and a \textit{Geometry} in
\index{Nicomachus!ref.\ to}%
several books. Some of the most beautiful results of Nicomachus
are omitted in Boethius' arithmetic. The first book
on geometry is an extract from Euclid's \textit{Elements}, which contains,
\index{Euclid!ref.\ to}%
in addition to definitions, postulates, and axioms, the
theorems in the first three books, without proofs. How can
this omission of proofs be accounted for? It has been argued
by some that Boethius possessed an incomplete Greek copy of
%% -----File: 096.png---Folio 82-------
\index{Apices of Boethius}%
\index{Gobar numerals}%
the \textit{Elements}; by others, that he had Theon's edition before
\index{Theon of Alexandria!ref.\ to}%
him, and believed that only the theorems came from Euclid,
while the proofs were supplied by Theon. The second book,
as also other books on geometry attributed to Boethius,
teaches, from numerical examples, the mensuration of plane
figures after the fashion of the agrimensores.

A celebrated portion in the geometry of Boethius is that
pertaining to an abacus, which he attributes to the Pythagoreans.
\index{Abacus}%
A considerable improvement on the old abacus is
there introduced. Pebbles are discarded, and \textit{apices} (probably
small cones) are used. Upon each of these apices is drawn
a numeral giving it some value below~$10$. The names of
these numerals are pure Arabic, or nearly so, but are added,
apparently, by a later hand. These figures are obviously the
parents of our modern ``Arabic'' numerals. The~$0$~is not
mentioned by Boethius in the text. These numerals bear
striking resemblance to the Gubar-numerals of the West-Arabs,
\index{Gubar-numerals}%
which are admittedly of Indian origin. These facts
have given rise to an endless controversy. Some contended
that Pythagoras was in India, and from there brought the
\index{Pythagoras!ref.\ to}%
nine numerals to Greece, where the Pythagoreans used them
secretly. This hypothesis has been generally abandoned, for
it is not certain that Pythagoras or any disciple of his ever
was in India, nor is there any evidence in any Greek author,
that the apices were known to the Greeks, or that numeral
signs of any sort were used by them with the abacus. It is
improbable, moreover, that the Indian signs, from which the
apices are derived, are so old as the time of Pythagoras.
A second theory is that the \textit{Geometry} attributed to Boethius
is a forgery; that it is not older than the tenth, or possibly
the ninth, century, and that the apices are derived from the
Arabs. This theory is based on contradictions between passages
in the \textit{Arithmetica} and others in the \textit{Geometry}. But
%% -----File: 097.png---Folio 83-------
there is an Encyclopædia written by \textit{Cassiodorius} (died about
\index{Cassiodorius}%
570) in which both the arithmetic and geometry of Boethius
are mentioned. There appears to be no good reason for doubting
the trustworthiness of this passage in the Encyclopædia.
A third theory (Woepcke's) is that the Alexandrians either
\index{Woepcke}%
directly or indirectly obtained the nine numerals from the
Hindoos, about the second century~\AD, and gave them to
the Romans on the one hand, and to the Western Arabs
\index{Romans|)}%
on the other. This explanation is the most plausible.
%% -----File: 098.png---Folio 84-------


\Chapter{Middle Ages}

\Section{The Hindoos}
\index{Hindoos|(}%

\First{The} first people who distinguished themselves in mathematical
research, after the time of the ancient Greeks, belonged,
like them, to the Aryan race. It was, however, not a European,
but an Asiatic nation, and had its seat in far-off India.

Unlike the Greek, Indian society was fixed into castes. The
only castes enjoying the privilege and leisure for advanced
study and thinking were the \emph{Brahmins}, whose prime business
was religion and philosophy, and the \emph{Kshatriyas}, who attended
to war and government.

Of the development of Hindoo mathematics we know but
little. A few manuscripts bear testimony that the Indians
had climbed to a lofty height, but their path of ascent is no
longer traceable. It would seem that Greek mathematics grew
up under more favourable conditions than the Hindoo, for in
Greece it attained an independent existence, and was studied
for its own sake, while Hindoo mathematics always remained
merely a servant to astronomy. Furthermore, in Greece
mathematics was a science of the people, free to be cultivated
by all who had a liking for it; in India, as in Egypt, it was in
the hands chiefly of the priests. Again, the Indians were in
the habit of putting into verse all mathematical results they
obtained, and of clothing them in obscure and mystic language,
%% -----File: 099.png---Folio 85-------
which, though well adapted to aid the memory of him who
already understood the subject, was often unintelligible to the
uninitiated. Although the great Hindoo mathematicians
doubtless reasoned out most or all of their discoveries, yet
they were not in the habit of preserving the proofs, so that
the naked theorems and processes of operation are all that
have come down to our time. Very different in these respects
were the Greeks. Obscurity of language was generally
avoided, and proofs belonged to the stock of knowledge quite
as much as the theorems themselves. Very striking was the
difference in the bent of mind of the Hindoo and Greek; for,
while the Greek mind was pre-eminently \emph{geometrical}, the
Indian was first of all \emph{arithmetical}. The Hindoo dealt with
number, the Greek with form. Numerical symbolism, the
science of numbers, and algebra attained in India far greater
perfection than they had previously reached in Greece. On
the other hand, we believe that there was little or no geometry
in India of which the source may not be traced back to
Greece. Hindoo trigonometry might possibly be mentioned
as an exception, but it rested on arithmetic more than on
geometry.

An interesting but difficult task is the tracing of the relation
between Hindoo and Greek mathematics. It is well
known that more or less trade was carried on between Greece
and India from early times. After Egypt had become a
Roman province, a more lively commercial intercourse sprang
up between Rome and India, by way of Alexandria. \textit{A~priori},
it does not seem improbable, that with the traffic of merchandise
there should also be an interchange of ideas. That
communications of thought from the Hindoos to the Alexandrians
actually did take place, is evident from the fact that
certain philosophic and theologic teachings of the Manicheans,
Neo-Platonists, Gnostics, show unmistakable likeness to
%% -----File: 100.png---Folio 86-------
Indian tenets. Scientific facts passed also from Alexandria
to India. This is shown plainly by the Greek origin of some
of the technical terms used by the Hindoos. Hindoo astronomy
\index{Astronomy!Hindoo}%
was influenced by Greek astronomy. Most of the geometrical
knowledge which they possessed is traceable to
Alexandria, and to the writings of Heron in particular. In
algebra there was, probably, a mutual giving and receiving.
We suspect that Diophantus got the first glimpses of algebraic
\index{Diophantus!ref.\ to}%
knowledge from India. On the other hand, evidences have
been found of Greek algebra among the Brahmins. The
earliest knowledge of algebra in India may possibly have been
of Babylonian origin. When we consider that Hindoo scientists
looked upon arithmetic and algebra merely as tools
useful in astronomical research, there appears deep irony in
the fact that these secondary branches were after all the only
ones in which they won real distinction, while in their pet
science of astronomy they displayed an inaptitude to observe,
to collect facts, and to make inductive investigations.

We shall now proceed to enumerate the names of the
leading Hindoo mathematicians, and then to review briefly
Indian mathematics. We shall consider the science only in
its complete state, for our data are not sufficient to trace the
history of the development of methods. Of the great Indian
mathematicians, or rather, astronomers,---for India had no
mathematicians proper,---\textbf{Aryabhatta} is the earliest. He was
\index{Aryabhatta}%
born 476~\AD, at Pataliputra, on the upper Ganges. His
celebrity rests on a work entitled \textit{Aryabhattiyam}, of which
the third chapter is devoted to mathematics. About one
hundred years later, mathematics in India reached the highest
mark. At that time flourished \textbf{Brahmagupta} (born 598). In
\index{Brahmagupta}%
628 he wrote his \textit{Brahma-sphuta-siddhanta} (``The Revised System
of Brahma''), of which the twelfth and eighteenth chapters
belong to mathematics. To the fourth or fifth century belongs
%% -----File: 101.png---Folio 87-------
\index{Arabic numerals and notation}%
\index{Numerals!Arabic}%
an anonymous astronomical work, called \textit{Surya-siddhanta}
(``Knowledge from the Sun''), which by native authorities
was ranked second only to the \textit{Brahma-siddhanta}, but is of interest
to us merely as furnishing evidence that Greek science
influenced Indian science even before the time of Aryabhatta.
The following centuries produced only two names of importance;
namely, \textbf{Cridhara}, who wrote a \textit{Ganita-sara} (``Quintessence
\index{Cridhara}%
of Calculation''), and \textbf{Padmanabha}, the author of an
\index{Padmanabha}%
algebra. The science seems to have made but little progress
at this time; for a work entitled \textit{Siddhantaciromani} (``Diadem
of an Astronomical System''), written by \textbf{Bhaskara Acarya} in
\index{Bhaskara}%
1150, stands little higher than that of Brahmagupta, written
over 500~years earlier. The two most important mathematical
chapters in this work are the \textit{Lilavati} ($=$~``the beautiful,'' \ie~the
noble science) and \textit{Viga-ganita} ($=$~``root-extraction''), devoted
to arithmetic and algebra. From now on, the Hindoos
in the Brahmin schools seemed to content themselves with
studying the masterpieces of their predecessors. Scientific
intelligence decreases continually, and in modern times a very
deficient Arabic work of the sixteenth century has been held
in great authority.\Reference{7}

The mathematical chapters of the \textit{Brahma-siddhanta} and
\textit{Siddhantaciromani} were translated into English by H.~T.
Colebrooke, London, 1817. The \textit{Surya-siddhanta} was translated
\index{Colebrooke}%
by E.~Burgess, and annotated by W.~D. Whitney, New
\index{Whitney}%
Haven, Conn., 1860.

The grandest achievement of the Hindoos and the one
which, of all mathematical inventions, has contributed most
to the general progress of intelligence, is the invention of
the principle of position in writing numbers. Generally we
speak of our notation as the ``Arabic'' notation, but it should
\index{Notation!Arabic notation}%
be called the ``Hindoo'' notation, for the Arabs borrowed it
from the Hindoos. That the invention of this notation was
%% -----File: 102.png---Folio 88-------
not so easy as we might suppose at first thought, may be
inferred from the fact that, of other nations, not even the
keen-minded Greeks possessed one like it. We inquire, who
invented this ideal symbolism, and when? But we know
neither the inventor nor the time of invention. That our
system of notation is of Indian origin is the only point of
which we are certain. From the evolution of ideas in general
we may safely infer that our notation did not spring into
existence a completely armed Minerva from the head of
Jupiter. The nine figures for writing the units are supposed
to have been introduced earliest, and the sign of zero and the
\index{Zero!(symbol for)}%
principle of position to be of later origin. This view receives
support from the fact that on the island of Ceylon a notation
resembling the Hindoo, but without the zero has been preserved.
We know that Buddhism and Indian culture were
transplanted to Ceylon about the third century after Christ,
and that this culture remained stationary there, while it made
progress on the continent. It seems highly probable, then,
that the numerals of Ceylon are the old, imperfect numerals
of India. In Ceylon, nine figures were used for the units,
nine others for the tens, one for~$100$, and also one for~$1000$.
These 20~characters enabled them to write all the numbers up
to~$9999$. Thus, $8725$ would have been written with six signs,
representing the following numbers: $8$,~$1000$, $7$,~$100$, $20$,~$5$.
These Singhalesian signs, like the old Hindoo numerals, are
supposed originally to have been the initial letters of the corresponding
numeral adjectives. There is a marked resemblance
between the notation of Ceylon and the one used by Aryabhatta
\index{Aryabhatta!ref.\ to}%
in the first chapter of his work, and there only. Although the
zero and the principle of position were unknown to the scholars
of Ceylon, they were probably known to Aryabhatta; for, in
the second chapter, he gives directions for extracting the square
and cube roots, which seem to indicate a knowledge of them.
%% -----File: 103.png---Folio 89-------
It would appear that the zero and the accompanying principle
of position were introduced about the time of Aryabhatta.
These are the inventions which give the Hindoo system its
great superiority, its admirable perfection.

There appear to have been several notations in use in
different parts of India, which differed, not in principle, but
merely in the forms of the signs employed. Of interest is
also a \emph{symbolical system of position}, in which the figures
generally were not expressed by numerical adjectives, but by
objects suggesting the particular numbers in question. Thus,
for~$1$ were used the words \emph{moon}, \emph{Brahma}, \emph{Creator}, or \emph{form};
for~$4$, the words \emph{Veda}, (because it is divided into four parts)
or \emph{ocean}, etc. The following example, taken from the \textit{Surya-siddhanta},
illustrates the idea. The number $1,577,917,828$ is
expressed from right to left as follows: Vasu (a class of $8$~gods)%
${}+{}$ two${}+{}$eight${}+{}$mountains (the $7$~mountain-chains)
${}+{}$form${}+{}$digits (the $9$~digits)${}+{}$seven${}+{}$mountains${}+{}$lunar
days (half of which equal~$15$). The use of such notations
made it possible to represent a number in several different
ways. This greatly facilitated the framing of verses containing
arithmetical rules or scientific constants, which could
thus be more easily remembered.

At an early period the Hindoos exhibited great skill in
calculating, even with large numbers. Thus, they tell us of
an examination to which Buddha, the reformer of the Indian
\index{Buddha}%
religion, had to submit, when a youth, in order to win the
maiden he loved. In arithmetic, after having astonished his
examiners by naming all the periods of numbers up to the~53d,
he was asked whether he could determine the number
of primary atoms which, when placed one against the other,
would form a line one mile in length. Buddha found the
required answer in this way: $7$~primary atoms make a very
minute grain of dust, $7$~of these make a minute grain of dust,
%% -----File: 104.png---Folio 90-------
\index{Arithmetic!Hindoo|(}%
$7$~of \emph{these} a grain of dust whirled up by the wind, and so on.
Thus he proceeded, step by step, until he finally reached the
length of a mile. The multiplication of all the factors gave
for the multitude of primary atoms in a mile a number consisting
of 15~digits. This problem reminds one of the `Sand-Counter'
\index{Sand-counter}%
of Archimedes.
\index{Archimedes!ref.\ to}%

After the numerical symbolism had been perfected, figuring
was made much easier. Many of the Indian modes of
operation differ from ours. The Hindoos were generally
inclined to follow the motion from left to right, as in writing.
Thus, they \emph{added} the left-hand columns first, and made the
necessary corrections as they proceeded. For instance, they
would have added $254$ and $663$ thus: $2 + 6 = 8$, $5 + 6=11$,
which changes $8$ into~$9$, $4 + 3 = 7$. Hence the sum~$917$. In
\emph{subtraction} they had two methods. Thus in $821 - 348$ they
would say, $8\text{ from }11 = 3$, $4\text{ from }11 = 7$, $3\text{ from }7 = 4$. Or
they would say, $8\text{ from }11 = 3$, $5\text{ from }12 = 7$, $4\text{ from }8 = 4$.
In \emph{multiplication} of a number by another of only one digit, say
$569$ by~$5$, they generally said, $5·5 = 25$, $5·6 = 30$, which
changes $25$ into~$28$, $5·9 = 45$, hence the $0$ must be increased by~$4$.
The product is~$2845$. In the multiplication with each
other of many-figured numbers, they first multiplied, in the
manner just indicated, with the left-hand digit of the multiplier,
which was written above the multiplicand, and placed
the product above the multiplier. On multiplying with the
next digit of the multiplier, the product was not placed in
a new row, as with us, but the first product obtained was
corrected, as the process continued, by erasing, whenever
necessary, the old digits, and replacing them by new ones,
until finally the whole product was obtained. We who possess
the modern luxuries of pencil and paper, would not be likely
to fall in love with this Hindoo method. But the Indians
wrote ``with a cane-pen upon a small blackboard with a white,
%% -----File: 105.png---Folio 91-------
\index{Casting out the 9's}%
thinly liquid paint which made marks that could be easily
erased, or upon a white tablet, less than a foot square, strewn
with red flour, on which they wrote the figures with a small
stick, so that the figures appeared white on a red ground.''\Reference{7}
Since the digits had to be quite large to be distinctly legible,
and since the boards were small, it was desirable to have a
method which would not require much space. Such a one
was the above method of multiplication. Figures could be
easily erased and replaced by others without sacrificing neatness.
But the Hindoos had also other
ways of multiplying, of which we mention
the following: The %[** PP: Width-dependent line break]
%[Illustration]
\begin{wrapfigure}{r}{1.6in}
\hfill\Graphic{1.6in}{105a}
\end{wrapfigure}
tablet was
divided into squares like a chess-board.
Diagonals were also drawn, as seen in
the figure. The multiplication of $12 ×
735 = 8820$ is exhibited in the adjoining diagram.\Reference{3}
The manuscripts
extant give no information of how \emph{divisions} were
executed. The correctness of their additions, subtractions,
and multiplications was tested ``by excess of $9$'s.'' In writing
fractions, the numerator was placed above the denominator,
but no line was drawn between them.

We shall now proceed to the consideration of some arithmetical
problems and the Indian modes of solution. A favourite
method was that of \emph{inversion}. With laconic brevity,
Aryabhatta describes it thus: ``Multiplication becomes division,
\index{Aryabhatta!ref.\ to}%
division becomes multiplication; what was gain becomes
loss, what loss, gain; inversion.'' Quite different from this
quotation in style is the following problem from Aryabhatta,
which illustrates the method:\Reference[]{3} ``Beautiful maiden with beaming
eyes, tell me, as thou understandst the right method of
inversion, which is the number which multiplied by~$3$, then
increased by $\frac{3}{4}$ of the product, divided by~$7$, diminished by $\frac{1}{3}$ of
the quotient, multiplied by itself, diminished by~$52$, the square
%% -----File: 106.png---Folio 92-------
\index{Arithmetic!Hindoo|)}%
root extracted, addition of~$8$, and division by~$10$, gives the
number~$2$?'' The process consists in beginning with $2$ and
working backwards. Thus, $(2·10-8)^{2}+52 = 196$, $\sqrt {196}=14$,
and $14·\frac{3}{2}·7·\frac{4}{7} ÷ 3 = 28$, the answer.

Here is another example taken from \textit{Lilavati}, a chapter in
Bhaskara's great work: ``The square root of half the number
\index{Bhaskara!ref.\ to|(}%
of bees in a swarm has flown out upon a jessamine-bush, $\frac{8}{9}$ of
the whole swarm has remained behind; one female bee flies
about a male that is buzzing within a lotus-flower into which
he was allured in the night by its sweet odour, but is now
imprisoned in it. Tell me the number of bees.'' Answer,~$72$.
The pleasing poetic garb in which all arithmetical problems
are clothed is due to the Indian practice of writing all school-books
in verse, and especially to the fact that these problems,
propounded as puzzles, were a favourite social amusement.
Says Brahmagupta: ``These problems are proposed simply for
\index{Brahmagupta!ref.\ to}%
pleasure; the wise man can invent a thousand others, or he
can solve the problems of others by the rules given here. As
the sun eclipses the stars by his brilliancy, so the man of
knowledge will eclipse the fame of others in assemblies of the
people if he proposes algebraic problems, and still more if he
solves them.''

The Hindoos solved problems in interest, discount, partnership,
alligation, summation of arithmetical and geometric
series, devised rules for determining the numbers of combinations
and permutations, and invented magic squares. It may
\index{Magic squares}%
here be added that chess, the profoundest of all games, had its
\index{Chess}%
origin in India.

The Hindoos made frequent use of the ``rule of three,'' and
\index{Rule of three}%
also of the method of ``falsa positio,'' which is almost identical
\index{Falsa positio}%
with that of the ``tentative assumption'' of Diophantus.
\index{Assumption, tentative}%
\index{Tentative assumption}%
These and other rules were applied to a large number of
problems.
%% -----File: 107.png---Folio 93-------
\index{Notation!in algebra}%
\index{Quadratic equations}%

Passing now to \emph{algebra}, we shall first take up the symbols
\index{Algebra!Hindoo|(}%
of operation. Addition was indicated simply by juxtaposition
as in Diophantine algebra; subtraction, by placing a dot over
the subtrahend; multiplication, by putting after the factors,
\emph{bha}, the abbreviation of the word \emph{bhavita}, ``the product'';
division, by placing the divisor beneath the dividend; square-root,
by writing \emph{ka}, from the word \emph{karana} (irrational), before
the quantity. The unknown quantity was called by Brahmagupta
\emph{yâvattâvat} (\emph{quantum tantum}). When several unknown
quantities occurred, he gave, unlike Diophantus, to each a
\index{Diophantus!ref.\ to}%
distinct name and symbol. The first unknown was designated
by the general term ``unknown quantity.'' The rest were
distinguished by names of colours, as the black, blue, yellow,
red, or green unknown. The initial syllable of each word
constituted the symbol for the respective unknown quantity.
Thus \emph{yâ} meant~$x$; \emph{kâ} (from $\text{\emph{kâlaka}} = \text{black}$) meant~$y$; \emph{yâ kâ
bha}, ``$x$~times~$y$''; \emph{ka}~$15$ \emph{ka}~$10$, ``$\sqrt{15} - \sqrt{10}$.''

The Indians were the first to recognise the existence of
absolutely negative quantities. They brought out the difference
\index{Negative quantities}%
between positive and negative quantities by attaching to
the one the idea of `possession,' to the other that of `debts.'
The conception also of opposite directions on a line, as an
interpretation of $+$~and~$-$ quantities, was not foreign to them.
They advanced beyond Diophantus in observing that a quadratic
has always two roots. Thus Bhaskara gives $x = 50$ and
$x = -5$ for the roots of $x^{2} - 45 x = 250$. ``But,'' says he,
``the second value is in this case not to be taken, for it is
inadequate; people do not approve of negative roots.'' Commentators
\index{Negative roots}%
speak of this as if negative roots were seen, but not
admitted.

Another important generalisation, says Hankel, was this,
\index{Hankel!ref.\ to}%
that the Hindoos never confined their arithmetical operations
to rational numbers. For instance, Bhaskara showed how,
%% -----File: 108.png---Folio 94-------
\index{Irrationals}%
by the formula %[** PP: Displaying]
\[
\sqrt{a + \sqrt{b}}
  = \sqrt{\frac{a + \sqrt{a^{2} - b}}{2}}
  + \sqrt{\frac{a - \sqrt{a^{2} - b}}{2}}
\]
the square root of the sum of rational and irrational numbers
\index{Square root}%
could be found. The Hindoos never discerned the dividing
line between numbers and magnitudes, set up by the Greeks,
which, though the product of a scientific spirit, greatly retarded
the progress of mathematics. They passed from magnitudes
to numbers and from numbers to magnitudes without
anticipating that gap which to a sharply discriminating mind
exists between the continuous and discontinuous. Yet by
doing so the Indians greatly aided the general progress of
mathematics. ``Indeed, if one understands by algebra the
application of arithmetical operations to complex magnitudes
of all sorts, whether rational or irrational numbers or space-magnitudes,
then the learned Brahmins of Hindostan are the
real inventors of algebra.''\Reference{7}

Let us now examine more closely the Indian algebra. In
extracting the square and cube roots they used the formulas
$(a + b)^{2} = a^{2} + 2ab + b^{2}$ and $(a + b)^{3}= a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$.
In this connection Aryabhatta speaks of dividing a number
into periods of two and three digits. From this we infer that
the principle of position and the zero in the numeral notation
were already known to him. In figuring with zeros, a statement
of Bhaskara is interesting. A fraction whose denominator
is zero, says he, admits of no alteration, though much be
added or subtracted. Indeed, in the same way, no change
takes place in the infinite and immutable Deity when worlds
are destroyed or created, even though numerous orders of beings
be taken up or brought forth. Though in this he apparently
evinces clear mathematical notions, yet in other places he
makes a complete failure in figuring with fractions of zero
\index{Fractions!Hindoo}%
denominator.

In the Hindoo solutions of determinate equations, Cantor
%% -----File: 109.png---Folio 95-------
\index{Numbers!theory of numbers}%
\index{Theory of numbers}%
thinks he can see traces of Diophantine methods. Some
technical terms betray their Greek origin. Even if it be true
that the Indians borrowed from the Greeks, they deserve great
credit for improving and generalising the solutions of linear
and quadratic equations. Bhaskara advances far beyond the
\index{Bhaskara!ref.\ to|)}%
Greeks and even beyond Brahmagupta when he says that
\index{Brahmagupta!ref.\ to}%
``the square of a positive, as also of a negative number,
is positive; that the square root of a positive number is
twofold, positive and negative. There is no square root
of a negative number, for it is not a square.'' Of equations
of higher degrees, the Indians succeeded in solving
only some special cases in which both sides of the equation
could be made perfect powers by the addition of certain
terms to each.

Incomparably greater progress than in the solution of determinate
equations was made by the Hindoos in the treatment
of \emph{indeterminate equations}. Indeterminate analysis was a
\index{Indeterminate analysis}%
\index{Indeterminate equations}%
subject to which the Hindoo mind showed a happy adaptation.
We have seen that this very subject was a favourite with Diophantus,
\index{Diophantus!ref.\ to}%
and that his ingenuity was almost inexhaustible in
devising solutions for particular cases. But the glory of
having invented \emph{general} methods in this most subtle branch
of mathematics belongs to the Indians. The Hindoo indeterminate
analysis differs from the Greek not only in method,
but also in aim. The object of the former was to find all
possible integral solutions. Greek analysis, on the other hand,
demanded not necessarily integral, but simply rational answers.
Diophantus was content with a single solution; the Hindoos
endeavoured to find all solutions possible. Aryabhatta gives
solutions in integers to linear equations of the form $ax ± by=c$,
where $a$,~$b$,~$c$ are integers. The rule employed is called the
\emph{pulveriser}. For this, as for most other rules, the Indians give
\index{Pulveriser}%
no proof. Their solution is essentially the same as the one of
%% -----File: 110.png---Folio 96-------
\index{Algebra!Hindoo|)}%
Euler. Euler's process of reducing $\dfrac{a}{b}$ to a continued fraction
\index{Euler!ref.\ to}%
amounts to the same as the Hindoo process of finding the
greatest common divisor of $a$~and~$b$ by division. This is frequently
called the Diophantine method. Hankel protests
\index{Hankel!ref.\ to}%
against this name, on the ground that Diophantus not only
\index{Diophantus!ref.\ to}%
never knew the method, but did not even aim at solutions
purely integral.\Reference{7} These equations probably grew out of problems
in astronomy. They were applied, for instance, to
determine the time when a certain constellation of the planets
would occur in the heavens.

Passing by the subject of linear equations with more than
two unknown quantities, we come to indeterminate quadratic
equations. In the solution of $xy = ax + by + c$, they applied
the method re-invented later by Euler, of decomposing $(ab + c)$
into the product of two integers $m·n$ and of placing $x = m + b$
and $y = n + a$.

Remarkable is the Hindoo solution of the quadratic equation
$cy^{2} = ax^{2} + b$. With great keenness of intellect they
recognised in the special case $y^{2} = ax^{2} + 1$ a fundamental
problem in indeterminate quadratics. They solved it by the
\emph{cyclic method}. ``It consists,'' says De~Morgan, ``in a rule for
\index{Cyclic method}%
\index{Demorgan@{De Morgan}!ref.\ to}%
finding an indefinite number of solutions of $y^{2} = ax^{2} +1$ ($a$~being
an integer which is not a square), by means of one solution
given or found, and of feeling for one solution by making a
solution of $y^2 = ax^{2} + b$ give a solution of $y^{2} = ax^{2} + b^{2}$. It
amounts to the following theorem: If $p$ and~$q$ be one set of
values of $x$ and~$y$ in $y^{2} = ax^{2} + b$ and $p'$ and~$q'$ the same or
another set, then $qp + pq'$ and $app' + qq'$ are values of $x$ and~$y$
in $y^{2} = ax^{2} + b^{2}$. From this it is obvious that one solution of
$y^{2} = ax^{2} + 1$ may be made to give any number, and that if,
taking $b$ at pleasure, $y^{2} = ax^{2} + b^{2}$ can be solved so that $x$ and~$y$
are divisible by~$b$, then one preliminary solution of $y^{2} = ax^{2} + 1$
%% -----File: 111.png---Folio 97-------
\index{Pell's problem}%
\index{Pythagoras!ref.\ to}%
can be found. Another mode of trying for solutions is a
combination of the preceding with the \emph{cuttaca} (pulveriser).''
These calculations were used in astronomy.

Doubtless this ``cyclic method'' constitutes the greatest
\index{Cyclic method}%
invention in the theory of numbers before the time of Lagrange.
The perversity of fate has willed it, that the equation
$y^{2} = ax^{2} + 1$ should now be called \emph{Pell's} problem, while in
recognition of Brahmin scholarship it ought to be called the
``Hindoo problem.'' It is a problem that has exercised the
highest faculties of some of our greatest modern analysts.
By them the work of the Hindoos was done over again; for,
unfortunately, the Arabs transmitted to Europe only a small
part of Indian algebra and the original Hindoo manuscripts,
which we now possess, were unknown in the Occident.

Hindoo \emph{geometry} is far inferior to the Greek. In it are
\index{Geometry!Hindoo}%
found no definitions, no postulates, no axioms, no logical
chain of reasoning or rigid form of demonstration, as with
Euclid. Each theorem stands by itself as an independent
\index{Euclid!ref.\ to}%
truth. Like the early Egyptian, it is empirical. Thus, in the
proof of the theorem of the right triangle, Bhaskara draws
\index{Bhaskara!ref.\ to}%
the %[** PP: Width-dependent line break]
% *[Illustration]
\begin{wrapfigure}[5]{r}{2.25in}
\hfill\Graphic{2.25in}{111a}
\end{wrapfigure}
right triangle four times in the square of the hypotenuse,
so that in the middle
there remains a square
whose side equals the
difference between the
two sides of the right
triangle. Arranging this square and the four triangles in a
different way, they are seen, together, to make up the sum
of the square of the two sides. ``Behold!'' says Bhaskara,
without adding another word of explanation. Bretschneider
\index{Bretschneider}%
conjectures that the Pythagorean proof was substantially the
same as this. In another place, Bhaskara gives a second
demonstration of this theorem by drawing from the vertex of
%% -----File: 112.png---Folio 98-------
\index{Ptolemy!ref.\ to}%
the right angle a perpendicular to the hypotenuse, and comparing
the two triangles thus obtained with the given triangle
to which they are similar. This proof was unknown in Europe
till Wallis \Change{rediscovered}{re-discovered} it. The Brahmins never inquired
\index{Wallis!ref.\ to}%
into the properties of figures. They considered only metrical
relations applicable in practical life. In the Greek sense, the
Brahmins never had a science of geometry. Of interest is
\index{Geometry!Hindoo}%
the formula given by Brahmagupta for the area of a triangle
\index{Brahmagupta!ref.\ to}%
in terms of its sides. In the great work attributed to Heron
\index{Heron the Elder!ref.\ to}%
the Elder this formula is first found. Whether the Indians
themselves invented it, or whether they borrowed it from
Heron, is a disputed question. Several theorems are given
by Brahmagupta on quadrilaterals which are true only of
those which can be inscribed on a circle---a limitation which
he omits to state. Among these is the proposition of Ptolemæus,
that the product of the diagonals is equal to the sum
of the products of the opposite sides. The Hindoos were
familiar with the calculation of the areas of circles and their
segments, of the length of chords and perimeters of regular
inscribed polygons. An old Indian tradition makes $\pi = 3$,
also $= \sqrt{10}$; but Aryabhatta gives the value $\frac{31416}{10000}$. Bhaskara
\index{Aryabhatta!ref.\ to}%
gives two values,---the `accurate,' $\frac{3927}{1250}$, and the `inaccurate,'
Archimedean value,~$\frac{22}{7}$. A commentator on \textit{Lilavati} says
that these values were calculated by beginning with a regular
inscribed hexagon, and applying repeatedly the formula
$AD = \sqrt{2 - \sqrt{\vphantom{\rule{0pt}{12pt}}4 - \smash[t]{\overline{AB}^{2}}}}$, wherein~$AB$ is the side of the given
polygon, and $AD$ that of one with double the number of
sides. In this way were obtained the perimeters of the
inscribed polygons of $12$, $24$, $48$, $96$, $192$, $384$~sides. Taking
the radius $= 100$, the perimeter of the last one gives the value
which Aryabhatta used for~$\pi$.
\index{p@{$\pi$}: values for!Hindoo}%

Greater taste than for geometry was shown by the Hindoos
for \emph{trigonometry}. Like the Babylonians and Greeks, they
\index{Trigonometry|(}%
%% -----File: 113.png---Folio 99-------
divided the circle into quadrants, each quadrant into $90$~degrees
and $5400$ minutes. The whole circle was therefore
made up of $21,600$ equal parts. From Bhaskara's `accurate'
value for~$\pi$ it was found that the radius contained $3438$ of
these circular parts. This last step was not Grecian. The
Greeks might have had scruples about taking a part of a
\emph{curve} as the measure of a \emph{straight line}. Each quadrant was
divided into $24$~equal parts, so that each part embraced $225$
units of the whole circumference, and corresponds to $3 \frac{3}{4}$ degrees.
Notable is the fact that the Indians never reckoned, like the
Greeks, with the whole chord of double the arc, but always
with the \emph{sine} (\emph{joa}) and \emph{versed sine}. Their mode of calculating
\index{Sine}%
\index{Versed sine}%
tables was theoretically very simple. The sine of~$90°$ was
equal to the radius, or~$3438$; the sine of~$30°$ was evidently
half that, or~$1719$. Applying the formula $\sin^{2}a + \cos^{2}a = r^{2}$,
they obtained $\sin 45° = \sqrt{\dfrac{r^2}{2}} = 2431$. Substituting for $\cos a$
its equal $\sin(90 - a)$, and making $a = 60°$, they obtained
$\sin 60° = \dfrac{\sqrt{3r^2}}{2} = 2978$. With the sines of $90$, $60$, $45$, and~$30$
as starting-points, they reckoned the sines of half the angles
by the formula $\versin 2a = 2 \sin^{2}a$, thus obtaining the sines
of $22°\ 30'$, $11°\ 15'$, $7°\ 30'$, $3°\ 45'$. They now figured out the
sines of the complements of these angles, namely, the sines
of $86°\ 15'$, $82°\ 30'$, $78°\ 45'$, $75°$, $67°\ 30'$; then they calculated
the sines of half these angles; then of their complements;
then, again, of half their complements; and so on. By this
very simple process they got the sines of angles at intervals of
$3°\ 45'$. In this table they discovered the unique law that if
$a$,~$b$,~$c$ be three successive arcs such that $a - b = b - c = 3°\ 45'$,
then $\sin a - \sin b = (\sin b - \sin c) - \dfrac{\sin b}{225}$. This formula was
afterwards used whenever a re-calculation of tables had to
be made. No Indian trigonometrical treatise on the triangle
%% -----File: 114.png---Folio 100-------
\index{Hindoos|)}%
\index{Trigonometry|)}%
is extant. In astronomy they solved plane and spherical
\index{Astronomy!Arabic}%
right triangles.\Reference{18}

It is remarkable to what extent Indian mathematics enters
into the science of our time. Both the form and the spirit
of the arithmetic and algebra of modern times are essentially
Indian and not Grecian. Think of that most perfect of
mathematical symbolisms---the Hindoo notation, think of the
Indian arithmetical operations nearly as perfect as our own,
think of their elegant algebraical methods, and then judge
whether the Brahmins on the banks of the Ganges are not
entitled to some credit. Unfortunately, some of the most
brilliant of Hindoo discoveries in indeterminate analysis
reached Europe too late to exert the influence they would
have exerted, had they come two or three centuries earlier.

\Section{The Arabs}
\index{Arabs|(}%

After the flight of Mohammed from Mecca to Medina in
622~\AD, an obscure people of Semitic race began to play an
important part in the drama of history. Before the lapse of
ten years, the scattered tribes of the Arabian peninsula were
fused by the furnace blast of religious enthusiasm into a
powerful nation. With sword in hand the united Arabs
subdued Syria and Mesopotamia. Distant Persia and the
lands beyond, even unto India, were added to the dominions
of the Saracens. They conquered Northern Africa, and nearly
the whole Spanish peninsula, but were finally checked from
further progress in Western Europe by the firm hand of
Charles Martel (732~\AD). The Moslem dominion extended
now from India to Spain; but a war of succession to the
caliphate ensued, and in 755 the Mohammedan empire was
divided,---one caliph reigning at Bagdad, the other at Cordova
%% -----File: 115.png---Folio 101-------
\index{Indeterminate analysis}%
\index{Indeterminate equations}%
in Spain. Astounding as was the grand march of conquest by
the Arabs, still more so was the ease with which they put
aside their former nomadic life, adopted a higher civilisation,
and assumed the sovereignty over cultivated peoples. Arabic
was made the written language throughout the conquered
lands. With the rule of the Abbasides in the East began a
new period in the history of learning. The capital, Bagdad,
situated on the Euphrates, lay half-way between two old
centres of scientific thought,---India in the East, and Greece
in the West. The Arabs were destined to be the custodians
of the torch of Greek and Indian science, to keep it ablaze
during the period of confusion and chaos in the Occident, and
afterwards to pass it over to the Europeans. Thus science
passed from Aryan to Semitic races, and then back again
to the Aryan. The Mohammedans have added but little to
the knowledge in mathematics which they received. They
now and then explored a small region to which the path had
been previously pointed out, but they were quite incapable of
discovering new fields. Even the more elevated regions in
which the Hellenes and Hindoos delighted to wander---namely,
the Greek conic sections and the Indian indeterminate
\index{Conic sections!Arabs}%
analysis---were seldom entered upon by the Arabs. They
were less of a speculative, and more of a practical turn of
mind.

The Abbasides at Bagdad encouraged the introduction of
the sciences by inviting able specialists to their court, irrespective
of nationality or religious belief. Medicine and
astronomy were their favourite sciences. Thus Haroun-al-Raschid,
\index{Astronomy!Arabic}%
the most distinguished Saracen ruler, drew Indian
physicians to Bagdad. In the year~772 there came to the
court of Caliph Almansur a Hindoo astronomer with astronomical
tables which were ordered to be translated into Arabic.
These tables, known by the Arabs as the \textit{Sindhind}, and
%% -----File: 116.png---Folio 102-------
\index{Sine}%
probably taken from the \textit{Brahma-sphuta-siddhanta} of Brahmagupta,
\index{Brahmagupta!ref.\ to}%
stood in great authority. They contained the important
Hindoo table of sines.

Doubtless at this time, and along with these astronomical
tables, the Hindoo numerals, with the zero and the principle
\index{Numerals!Arabic}%
of position, were introduced among the Saracens. Before the
time of Mohammed the Arabs had no numerals. Numbers
were written out in words. Later, the numerous computations
connected with the financial administration over the conquered
lands made a short symbolism indispensable. In some localities,
the numerals of the more civilised conquered nations
were used for a time. Thus in Syria, the Greek notation was
\index{Notation!Arabic notation}%
retained; in Egypt, the Coptic. In some cases, the numeral
adjectives may have been abbreviated in writing. The \textit{Diwani-numerals},
\index{Diwani-numerals}%
found in an Arabic-Persian dictionary, are supposed
to be such abbreviations. Gradually it became the practice to
employ the 28~Arabic letters of the alphabet for numerals, in
analogy to the Greek system. This notation was in turn
superseded by the Hindoo notation, which quite early was
adopted by merchants, and also by writers on arithmetic. Its
superiority was so universally recognised, that it had no rival,
except in astronomy, where the alphabetic notation continued
to be used. Here the alphabetic notation offered no great
disadvantage, since in the sexagesimal arithmetic, taken from
the \textit{Almagest}, numbers of generally only one or two places
had to be written.\Reference{7}

As regards the form of the so-called Arabic numerals, the
\index{Arabic numerals and notation}%
statement of the Arabic writer \textit{Albiruni} (died~1039), who
\index{Albiruni!ref.\ to}%
spent many years in India, is of interest. He says that the
shape of the numerals, as also of the letters in India, differed
in different localities, and that the Arabs selected from the
various forms the most suitable. An Arabian astronomer
says there was among people much difference in the use of
%% -----File: 117.png---Folio 103-------
\index{Gobar numerals}%
symbols, especially of those for $5$, $6$, $7$, and~$8$. The symbols
used by the Arabs can be traced back to the tenth century.
We find material differences between those used by the
Saracens in the East and those used in the West. But
most surprising is the fact that the symbols of both the East
and of the West Arabs deviate so extraordinarily from the
Hindoo \textit{Devanagari} numerals ($=$~divine numerals) of to-day,
\index{Devanagari-numerals}%
\index{Numerals!Arabic}%
and that they resemble much more closely the apices of
\index{Apices of Boethius!ref.\ to}%
the Roman writer Boethius. This strange similarity on the
\index{Boethius!ref.\ to}%
one hand, and dissimilarity on the other, is difficult to explain.
The most plausible theory is the one of Woepcke: (1)~that
\index{Woepcke}%
about the second century after Christ, before the zero had
been invented, the Indian numerals were brought to Alexandria,
whence they spread to Rome and also to West Africa;
(2)~that in the eighth century, after the notation in India had
been already much modified and perfected by the invention of
the zero, the Arabs at Bagdad got it from the Hindoos; (3)~that
the Arabs of the West borrowed the Columbus-egg, the zero,
from those in the East, but retained the old forms of the nine
numerals, if for no other reason, simply to be contrary to their
political enemies of the East; (4)~that the old forms were
remembered by the West-Arabs to be of Indian origin, and
were hence called \textit{Gubar-numerals} ($=$~dust-numerals, in memory
\index{Gubar-numerals}%
of the Brahmin practice of reckoning on tablets strewn
with dust or sand; (5)~that, since the eighth century, the
numerals in India underwent further changes, and assumed
the greatly modified forms of the modern Devanagari-numerals.\Reference{3}
This is rather a bold theory, but, whether true or not,
it explains better than any other yet propounded, the relations
between the apices, the Gubar, the East-Arabic, and Devanagari
numerals.

It has been mentioned that in 772 the Indian \textit{Siddhanta} was
brought to Bagdad and there translated into Arabic. There
%% -----File: 118.png---Folio 104-------
\index{Geometry!Arabic}%
\index{Ptolemy!ref.\ to}%
is no evidence that any intercourse existed between Arabic
and Indian astronomers either before or after this time, excepting
the travels of Albiruni. But we should be very slow
\index{Albiruni!ref.\ to}%
to deny the probability that more extended communications
actually did take place.

Better informed are we regarding the way in which Greek
science, in successive waves, dashed upon and penetrated Arabic
soil. In Syria the sciences, especially philosophy and medicine,
were cultivated by Greek Christians. Celebrated were the
schools at Antioch and Emesa, and, first of all, the flourishing
Nestorian school at Edessa. From Syria, Greek physicians
and scholars were called to Bagdad. Translations of works
from the Greek began to be made. A large number of Greek
manuscripts were secured by Caliph \textit{Al Mamun} (813--833) from
the emperor in Constantinople and were turned over to Syria.
The successors of Al Mamun continued the work so auspiciously
begun, until, at the beginning of the tenth century, the
more important philosophic, medical, mathematical, and astronomical
works of the Greeks could all be read in the Arabic
tongue. The translations of mathematical works must have
been very deficient at first, as it was evidently difficult to
secure translators who were masters of both the Greek and
Arabic and at the same time proficient in mathematics. The
translations had to be revised again and again before they
were satisfactory. The first Greek authors made to speak in
Arabic were Euclid and Ptolemæus. This was accomplished
\index{Euclid!ref.\ to}%
during the reign of the famous Haroun-al-Raschid. A revised
\index{Haroun-al-Raschid}%
translation of Euclid's \textit{Elements} was ordered by Al Mamun.
\index{Elements (Euclid's)}%
As this revision still contained numerous errors, a new translation
was made, either by the learned Honein ben Ishak, or
\index{Honein ben Ishak}%
by his son, Ishak ben Honein. To the thirteen books of the
\index{Ishak ben Honein}%
\textit{Elements} were added the fourteenth, written by Hypsicles,
\index{Hypsicles!ref.\ to}%
and the fifteenth by Damascius. But it remained for Tabit
\index{Damascius!ref.\ to}%
%% -----File: 119.png---Folio 105-------
ben Korra to bring forth an Arabic Euclid satisfying every
\index{Tabit ben Korra!ref.\ to}%
need. Still greater difficulty was experienced in securing an
intelligible translation of the \textit{Almagest}. Among other important
\index{Almagest!ref.\ to}%
translations into Arabic were the works of Apollonius,
\index{Apollonius!ref.\ to}%
Archimedes, Heron, and Diophantus. Thus we see that in
\index{Archimedes!ref.\ to}%
\index{Diophantus!ref.\ to}%
\index{Heron the Elder!ref.\ to}%
the course of one century the Arabs gained access to the vast
treasures of Greek science. Having been little accustomed to
abstract thought, we need not marvel if, during the ninth century,
all their energy was exhausted merely in appropriating
the foreign material. No attempts were made at original
work in mathematics until the next century.

In astronomy, on the other hand, great activity in original
\index{Astronomy!Arabic}%
research existed as early as the ninth century. The religious
observances demanded by Mohammedanism presented to astronomers
several practical problems. The Moslem dominions
being of such enormous extent, it remained in some localities
for the astronomer to determine which way the ``Believer''
must turn during prayer that he may be facing Mecca. The
prayers and ablutions had to take place at definite hours during
the day and night. This led to more accurate determinations
of time. To fix the exact date for the Mohammedan
feasts it became necessary to observe more closely the motions
of the moon. In addition to all this, the old Oriental superstition
that extraordinary occurrences in the heavens in some
mysterious way affect the progress of human affairs added
increased interest to the prediction of eclipses.\Reference{7}

For these reasons considerable progress was made. Astronomical
tables and instruments were perfected, observatories
erected, and a connected series of observations instituted. This
intense love for astronomy and astrology continued during the
whole Arabic scientific period. As in India, so here, we hardly
ever find a man exclusively devoted to pure mathematics. Most
of the so-called mathematicians were first of all astronomers.
%% -----File: 120.png---Folio 106-------
\index{Ptolemy!ref.\ to}%

The first notable author of mathematical books was \textbf{Mohammed
ben Musa Al Hovarezmi}, who lived during the reign of Caliph
\index{Hovarezmi}%
\index{Mohammed ben Musa Hovarezmi}%
Al Mamun (813--833). He was engaged by the caliph in making
extracts from the \textit{Sindhind}, in revising the tablets of Ptolemæus,
in taking observations at Bagdad and Damascus, and in
measuring a degree of the earth's meridian. Important to us
is his work on algebra and arithmetic. The portion on arithmetic
\index{Arithmetic!Arabic}%
is not extant in the original, and it was not till 1857
that a Latin translation of it was found. It begins thus:
``Spoken has Algoritmi. Let us give deserved praise to God,
our leader and defender.'' Here the name of the author, \textit{Al
Hovarezmi}, has passed into \textit{Algoritmi}, from which comes our
modern word, \emph{algorithm}, signifying the art of computing in
\index{Algorithm!origin of term}%
any particular way. The arithmetic of Hovarezmi, being
based on the principle of position and the Hindoo method of
calculation, ``excels,'' says an Arabic writer, ``all others in
brevity and easiness, and exhibits the Hindoo intellect and
sagacity in the grandest inventions.'' This book was followed
by a large number of arithmetics by later authors, which differed
from the earlier ones chiefly in the greater variety of
methods. Arabian arithmetics generally contained the four
operations with integers and fractions, modelled after the
Indian processes. They explained the operation of \emph{casting out
\index{Casting out the 9's}%
\index{Nines, casting out the}%
the~9's}, which was sometimes called the ``Hindoo proof.'' They
contained also the \textit{regula falsa} and the \textit{regula duorum falsorum},
\index{Regula duorum falsorum}%
\index{Regula falsa}%
by which algebraical examples could be solved without algebra.
Both these methods were known to the Indians. The \textit{regula
falsa} or \textit{falsa positio} was the assigning of an assumed value to
the unknown quantity, which value, if wrong, was corrected
by some process like the ``rule of three.'' Diophantus used a
\index{Diophantus!ref.\ to}%
\index{Rule of three}%
method almost identical with this. The \textit{regula duorum falsorum}
was as follows:\Reference[]{7} To solve an equation $f(x) = V$, assume,
for the moment, two values for $x$; namely, $x = a$ and $x = b$.
%% -----File: 121.png---Folio 107-------
\index{Irrationals}%
Then form $f(a) = A$ and $f(b) = B$, and determine the errors
$V - A = E_a$ and $V - B = E_b$; then the required
$x = \dfrac{bE_a - aE_b}{E_a - E_b}$
is generally a close approximation, but is absolutely accurate
whenever $f(x)$ is a linear function of~$x$.

We now return to Hovarezmi, and consider the other part
\index{Hovarezmi!ref.\ to}%
\index{Mohammed ben Musa Hovarezmi!ref.\ to}%
of his work,---the \emph{algebra}. This is the first book known to
\index{Algebra!Arabic}%
\index{Algebra!origin of terms}%
contain this word itself as title. Really the title consists of
two words, \textit{aldshebr walmukabala}, the nearest English translation
of which is ``restoration'' and ``reduction.'' By ``restoration''
was meant the transposing of negative terms to the
other side of the equation; by ``reduction,'' the uniting of
similar terms. Thus, $x^2 - 2x = 5x + 6$ passes by aldshebr into
$x^2 = 5x + 2x + 6$; and this, by walmukabala, into $x^2 = 7x + 6$.
The work on algebra, like the arithmetic, by the same author,
contains nothing original. It explains the elementary operations
and the solutions of linear and quadratic equations.
\index{Quadratic equations}%
From whom did the author borrow his knowledge of algebra?
That it came entirely from Indian sources is impossible, for
the Hindoos had no such rules like the ``restoration'' and
``reduction.'' They were, for instance, never in the habit of
making all terms in an equation positive, as is done by the
process of ``restoration.'' Diophantus gives two rules which
\index{Diophantus!ref.\ to}%
resemble somewhat those of our Arabic author, but the probability
that the Arab got all his algebra from Diophantus is
lessened by the considerations that he recognised both roots
of a quadratic, while Diophantus noticed only one; and that
the Greek algebraist, unlike the Arab, habitually rejected irrational
solutions. It would seem, therefore, that the algebra
of Hovarezmi was neither purely Indian nor purely Greek,
but was a hybrid of the two, with the Greek element predominating.

The algebra of Hovarezmi contains also a few meagre
%% -----File: 122.png---Folio 108-------
fragments on \emph{geometry}. He gives the theorem of the right
\index{Geometry!Arabic}%
triangle, but proves it after Hindoo fashion and only for the
simplest case, when the right triangle is isosceles. He then
calculates the areas of the triangle, parallelogram, and circle.
For $\pi$ he uses the value~$3 \frac{1}{7}$, and also the two Indian, $\pi = \sqrt{10}$
\index{p@{$\pi$}: values for!Arabic}%
and $\pi = \frac{62832}{20000}$. Strange to say, the last value was afterwards
forgotten by the Arabs, and replaced by others less
accurate. This bit of geometry doubtless came from India.
Later Arabic writers got their geometry almost entirely from
Greece.

Next to be noticed are the three sons of \textbf{Musa ben Sakir},
\index{Musa ben Sakir}%
who lived in Bagdad at the court of the Caliph Al Mamun.
They wrote several works, of which we mention a geometry
in which is also contained the well-known formula for the
area of a triangle expressed in terms of its sides. We are
told that one of the sons travelled to Greece, probably to
collect astronomical and mathematical manuscripts, and that
on his way back he made acquaintance with Tabit ben Korra.
\index{Tabit ben Korra}%
Recognising in him a talented and learned astronomer, Mohammed
procured for him a place among the astronomers at
the court in Bagdad. \textbf{Tabit ben Korra} (836--901) was born
at Harran in Mesopotamia. He was proficient not only in
astronomy and mathematics, but also in the Greek, Arabic,
and Syrian languages. His translations of Apollonius, Archimedes,
\index{Apollonius!ref.\ to}%
\index{Archimedes!ref.\ to}%
Euclid, Ptolemy, Theodosius, rank among the best.
\index{Euclid!ref.\ to}%
\index{Ptolemy!ref.\ to}%
\index{Theodosius!ref.\ to}%
His dissertation on \emph{amicable numbers} (of which each is the
\index{Numbers!amicable}%
\index{Numbers!theory of numbers}%
sum of the factors of the other) is the first known specimen
of original work in mathematics on Arabic soil. It shows
that he was familiar with the Pythagorean theory of numbers.
\index{Theory of numbers}%
Tabit invented the following rule for finding amicable numbers:
If $p = 3 · 2^{n} - 1$, $q = 3 · 2^{n-1} - 1$, $r = 9 · 2^{2n-1} - 1$ ($n$~being
a whole number) are three primes, then $a = 2^{n} pq$, $b = 2^{n} r$ are
a pair of amicable numbers. Thus, if $n = 2$, then $p = 11$,
%% -----File: 123.png---Folio 109-------
$q = 5$, $r = 71$, and $a = 220$, $b = 284$. Tabit also trisected an
angle.

Foremost among the astronomers of the ninth century
ranked \textbf{Al Battani}, called \textit{Albategnius} by the Latins. Battan
\index{Albattani@{Al Battani}}%
in Syria was his birthplace. His observations were celebrated
for great precision. His work, \textit{De scientia stellarum}, was translated
into Latin by Plato Tiburtinus, in the twelfth century.
\index{Plato of Tivoli}%
Out of this translation sprang the word `sinus,' as the name
of a trigonometric function. The Arabic word for ``sine,''
\index{Sine}%
\index{Sine!origin of term}%
\textit{dschiba}, was derived from the Sanscrit \textit{jiva}, and resembled
the Arabic word \textit{dschaib}, meaning an indentation or gulf.
Hence the Latin ``sinus.''\Reference{3} Al Battani was a close student
of Ptolemy, but did not follow him altogether. He took an
\index{Ptolemy!ref.\ to}%
important step for the better, when he introduced the Indian
``sine'' or \emph{half} the chord, in place of the \emph{whole} chord of
Ptolemy. Another improvement on Greek trigonometry made
\index{Trigonometry}%
by the Arabs points likewise to Indian influences. Propositions
and operations which were treated by the Greeks geometrically
are expressed by the Arabs algebraically. Thus,
\textit{Al Battani} at once gets from an equation
$\smash[b]{\dfrac{\sin \theta}{\cos \theta}} = D$, the value
of $\theta$ by means of
$\sin \theta = \dfrac{D\Fstr}{\sqrt{1 + D^2}}$,---a process unknown to
the ancients. He knows, of course, all the formulas for
spherical triangles given in the \textit{Almagest}, but goes further,
\index{Almagest!ref.\ to}%
and adds an important one of his own for oblique-angled
triangles; namely, $\cos a = \cos b \cos c + \sin b \sin c \cos A$.

At the beginning of the tenth century political troubles
arose in the East, and as a result the house of the Abbasides
lost power. One province after another was taken, till, in~945,
all possessions were wrested from them. Fortunately,
the new rulers at Bagdad, the Persian Buyides, were as much
interested in astronomy as their predecessors. The progress
of the sciences was not only unchecked, but the conditions
%% -----File: 124.png---Folio 110-------
\index{Geometry!Arabic}%
\index{Regular solids}%
\index{Tycho Brahe}%
for it became even more favourable. The Emir \textit{Adud-ed-daula}
(978--983) gloried in having studied astronomy himself. His
son \textit{Saraf-ed-daula} erected an observatory in the garden of his
palace, and called thither a whole group of scholars.\Reference{7} Among
them were \textit{Abul Wefa}, \textit{Al Kuhi}, \textit{Al Sagani}.
\index{Abul Wefa}%

\textbf{Abul Wefa} (940--998) was born at Buzshan in Chorassan, a
region among the Persian mountains, which has brought forth
many Arabic astronomers. He forms an important exception
to the unprogressive spirit of Arabian scientists by his brilliant
discovery of the \emph{variation} of the moon, an inequality
usually supposed to have been first discovered by Tycho
Brahe.\Reference{11} Abul Wefa translated Diophantus. He is one of the
\index{Brahe, Tycho}%
\index{Diophantus!ref.\ to}%
last Arabic translators and commentators of Greek authors.
The fact that he esteemed the algebra of Mohammed ben
Musa Hovarezmi worthy of his commentary indicates that
\index{Hovarezmi!ref.\ to}%
\index{Mohammed ben Musa Hovarezmi!ref.\ to}%
thus far algebra had made little or no progress on Arabic soil.
Abul Wefa invented a method for computing tables of sines
which gives the sine of half a degree correct to nine decimal
places. He did himself credit by introducing the \emph{tangent} into
trigonometry and by calculating a table of tangents. The
\index{Tangents!in trigonometry}%
\index{Trigonometry}%
first step toward this had been taken by Al Battani. Unfortunately,
\index{Albattani@{Al Battani}!ref.\ to}%
this innovation and the discovery of the moon's
variation excited apparently no notice among his contemporaries
and followers. ``We can hardly help looking upon this
circumstance as an evidence of a servility of intellect belonging
to the Arabian period.'' A treatise by Abul Wefa on
``geometric constructions'' indicates that efforts were being
made at that time to improve draughting. It contains a neat
construction of the corners of the regular \DPtypo{polyedrons}{polyhedrons} on the
circumscribed sphere. Here, for the first time, appears the
condition which afterwards became very famous in the Occident,
that the construction be effected with a single opening
of the compass.
%% -----File: 125.png---Folio 111-------
\index{Albiruni}%
\index{Alhogendi@{Al Hogendi}}%
\index{Cube numbers}%
\index{Indeterminate equations}%
\index{Quadratic equations}%

\textbf{Al Kuhi}, the second astronomer at the observatory of the
\index{Alkuhi@{Al Kuhi}}%
emir at Bagdad, was a close student of Archimedes and
Apollonius. He solved the problem, to construct a segment
of a sphere equal in volume to a given segment and having
a curved surface equal in area to that of another given segment.
He, \textbf{Al Sagani}, and \textbf{Al Biruni} made a study of the
\index{Alsagani@{Al Sagani}}%
trisection of angles. \textbf{Abul Gud}, an able geometer, solved the
\index{Abul Gud}%
problem by the intersection of a parabola with an equilateral
hyperbola.

The Arabs had already discovered the theorem that the
sum of two cubes can never be a cube. \textbf{Abu Mohammed Al
Hogendi} of Chorassan thought he had proved this, but we are
told that the demonstration was defective. Creditable work
in theory of numbers and algebra was done by \textbf{Al Karhi} of
\index{Algebra!Arabic}%
\index{Alkarhi@{Al Karhi}}%
\index{Fahri des Al Karhi}%
Bagdad, who lived at the beginning of the eleventh century.
His treatise on algebra is the greatest algebraic work of the
Arabs. In it he appears as a disciple of Diophantus. He
\index{Diophantus!ref.\ to}%
was the first to operate with higher roots and to solve equations
of the form $x^{2n} + ax^{n} = b$. For the solution of quadratic
equations he gives both arithmetical and geometric proofs.
He was the first Arabic author to give and prove the theorems
on the summation of the series:---
\index{Series}%
\begin{align*}
1^2 + 2^2 + 3^2 + \dotsb + n^2
  &= (1 + 2 + \dotsb + n)\DPtypo{^}{\,}\frac{2n+1}{3}, \\
1^3 + 2^3 + 3^3 + \dotsb + n^3
  &= (1 + 2 + \dotsb + n)^2.
\end{align*}

Al Karhi also busied himself with indeterminate analysis.
\index{Indeterminate analysis}%
He showed skill in handling the methods of Diophantus, but
added nothing whatever to the stock of knowledge already
on hand. As a subject for original research, indeterminate
analysis was too subtle for even the most gifted of Arabian
minds. Rather surprising is the fact that Al Karhi's algebra
shows no traces whatever of Hindoo indeterminate analysis.
%% -----File: 126.png---Folio 112-------
\index{Alhayyami@{Al Hayyami}}%
\index{Arabic numerals and notation}%
\index{Cubic equations}%
\index{Notation!Arabic notation}%
But most astonishing it is, that an arithmetic by the same
author completely excludes the Hindoo numerals. It is constructed
\index{Numerals!Arabic}%
wholly after Greek pattern. Abul Wefa also, in the
\index{Abul Wefa!ref.\ to}%
second half of the tenth century, wrote an arithmetic in which
Hindoo numerals find no place. This practice is the very
opposite to that of other Arabian authors. The question,
why the Hindoo numerals were ignored by so eminent authors,
is certainly a puzzle. Cantor suggests that at one time there
\index{Cantor, M.}%
may have been rival schools, of which one followed almost
exclusively Greek mathematics, the other Indian.

The Arabs were familiar with geometric solutions of quadratic
equations. Attempts were now made to solve cubic
equations geometrically. They were led to such solutions by
the study of questions like the Archimedean problem, demanding
the section of a sphere by a plane so that the two segments
shall be in a prescribed ratio. The first to state this
problem in form of a cubic equation was \textbf{Al Mahani} of Bagdad,
\index{Almahani@{Al Mahani}}%
while \textbf{Abu Gafar Al Hazin} was the first Arab to solve the
\index{Alhazin@{Al Hazin}}%
equation by conic sections. Solutions were given also by
\index{Conic sections!Arabs}%
Al Kuhi, Al Hasan ben Al Haitam, and others.\Reference{20} Another
\index{Alhaitam@{Al Haitam}!ref.\ to}%
\index{Alkuhi@{Al Kuhi}!ref.\ to}%
difficult problem, to determine the side of a regular heptagon,
required the construction of the side from the equation
$x^3 - x^2 - 2x + 1 = 0$. It was attempted by many and at last
solved by Abul Gud.

The one who did most to elevate to a \emph{method} the solution
of algebraic equations by intersecting conics, was \textbf{Omar al
Hayyami} of Chorassan, about 1079~\AD\ He divides cubics into
two classes, the trinomial and quadrinomial, and each class
into families and species. Each species is treated separately
but according to a general plan. He believed that cubics
could not be solved by calculation, nor \Change{bi-quadratics}{biquadratics} by geometry.
He rejected negative roots and often failed to discover
\index{Negative roots}%
all the positive ones. Attempts at \Change{bi-quadratic}{biquadratic} equations
\index{Biquadratic equation}%
\index{Quadratic equations}%
%% -----File: 127.png---Folio 113-------
\index{Geometry!Arabic}%
were made by Abul Wefa,\Reference[]{20} who solved geometrically
\index{Abul Wefa!ref.\ to}%
$x^4 = a$ and $x^4 + ax^3 = b$.

The solution of cubic equations by intersecting conics was
\index{Cubic equations}%
the greatest achievement of the Arabs in algebra. The foundation
to this work had been laid by the Greeks, for it was
Menæchmus who first constructed the roots of $x^3 - a = 0$ or
\index{Menæchmus!ref.\ to}%
$x^3 - 2 a^3 = 0$. It was not his aim to find the number corresponding
to~$x$, but simply to determine the side~$x$ of a cube
double another cube of side~$a$. The Arabs, on the other
hand, had another object in view: to find the roots of given
numerical equations. In the Occident, the Arabic solutions
of cubics remained unknown until quite recently. Descartes
\index{Descartes!ref.\ to}%
and Thomas Baker invented these constructions anew. The
\index{Baker, Th}%
works of Al Hayyami, Al Karhi, Abul Gud, show how the
\index{Abul Gud!ref.\ to}%
\index{Alhayyami@{Al Hayyami}!ref.\ to}%
\index{Alkarhi@{Al Karhi}}%
Arabs departed further and further from the Indian methods,
and placed themselves more immediately under Greek influences.
In this way they barred the road of progress against
themselves. The Greeks had advanced to a point where
material progress became difficult with their methods; but the
Hindoos furnished new ideas, many of which the Arabs now
rejected.

With Al Karhi and Omar Al Hayyami, mathematics among
the Arabs of the East reached flood-mark, and now it begins to
ebb. Between 1100 and 1300~\AD\ come the crusades with
war and bloodshed, during which European Christians profited
much by their contact with Arabian culture, then far superior
to their own; but the Arabs got no science from the Christians
in return. The crusaders were not the only adversaries of the
Arabs. During the first half of the thirteenth century, they
had to encounter the wild Mongolian hordes, and, in~1256, were
conquered by them under the leadership of \textit{Hulagu}. The
caliphate at Bagdad now ceased to exist. At the close of the
fourteenth century still another empire was formed by Timur
%% -----File: 128.png---Folio 114-------
or \textit{Tamerlane}, the Tartar. During such sweeping turmoil, it
is not surprising that science declined. Indeed, it is a marvel
that it existed at all. During the supremacy of Hulagu, lived
\textbf{Nasir Eddin} (1201--1274), a man of broad culture and an able
\index{Nasir Eddin}%
\Addendlabel{114}%
astronomer. He persuaded Hulagu to build him and his associates
a large observatory at Maraga. Treatises on algebra,
geometry, arithmetic, and a translation of Euclid's \textit{Elements},
\index{Elements (Euclid's)}%
\index{Euclid!ref.\ to}%
\index{Geometry!Arabic}%
were prepared by him. Even at the court of Tamerlane in
Samarkand, the sciences were by no means neglected. A
group of astronomers was drawn to this court. \textbf{Ulug Beg}
\index{Ulug Beg}%
(1393--1449), a grandson of Tamerlane, was himself an
astronomer. Most prominent at this time was \textbf{Al Kaschi}, the
\index{Alkaschi@{Al Kaschi}}%
author of an arithmetic. Thus, during intervals of peace,
science continued to be cultivated in the East for several
centuries. The last Oriental writer was \textit{Beha Eddin} (1547--1622).
\index{Beha Eddin}%
His \textit{Essence of Arithmetic} stands on about the same
level as the work of Mohammed ben Musa Hovarezmi, written
\index{Hovarezmi!ref.\ to}%
\index{Mohammed ben Musa Hovarezmi!ref.\ to}%
nearly 800 years before.

``Wonderful is the expansive power of Oriental peoples,
with which upon the wings of the wind they conquer half
the world, but more wonderful the energy with which, in
less than two generations, they raise themselves from the
lowest stages of cultivation to scientific efforts.'' During
all these centuries, astronomy and mathematics in the Orient
greatly excel these sciences in the Occident.

Thus far we have spoken only of the Arabs in the East.
Between the Arabs of the East and of the West, which were
under separate governments, there generally existed considerable
political animosity. In consequence of this, and of the
enormous distance between the two great centres of learning,
Bagdad and Cordova, there was less scientific intercourse
among them than might be expected to exist between peoples
having the same religion and written language. Thus the
%% -----File: 129.png---Folio 115-------
\index{Moors}%
course of science in Spain was quite independent of that in
Persia. While wending our way westward to Cordova, we
must stop in Egypt long enough to observe that there, too,
scientific activity was rekindled. Not Alexandria, but Cairo
with its library and observatory, was now the home of learning.
Foremost among her scientists ranked \textbf{Ben Junus} (died
\index{Ben Junus}%
1008), a contemporary of Abul Wefa. He solved some difficult
problems in spherical trigonometry. Another Egyptian astronomer
\index{Spherical trigonometry}%
\index{Trigonometry}%
\index{Trigonometry!spherical}%
was \textbf{Ibn Al Haitam} (died 1038), who wrote on geometric
\index{Alhaitam@{Al Haitam}}%
loci. Travelling westward, we meet in Morocco \textbf{Abul Hasan
\index{Abul Hasan}%
Ali}, whose treatise `on astronomical instruments' discloses a
thorough knowledge of the \textit{Conics} of Apollonius. Arriving
\index{Apollonius!ref.\ to}%
finally in Spain at the capital, Cordova, we are struck by the
magnificent splendour of her architecture. At this renowned
seat of learning, schools and libraries were founded during the
tenth century.

Little is known of the progress of mathematics in Spain.
The earliest name that has come down to us is \textbf{Al Madshriti}
\index{Almadshriti@{Al Madshriti}}%
(died 1007), the author of a mystic paper on `amicable numbers.'
\index{Numbers!amicable}%
His pupils founded schools at Cordova, Dania, and
Granada. But the only great astronomer among the Saracens
in Spain is \textbf{Gabir ben Aflah} of Sevilla, frequently called \textit{Geber}.
\index{Gabir ben Aflah}%
He lived in the second half of the eleventh century. It was
formerly believed that he was the inventor of algebra, and that
\index{Algebra!Arabic}%
\index{Algebra!origin of terms}%
the word \emph{algebra} came from `Gabir' or `Geber.' He ranks
among the most eminent astronomers of this time, but, like so
many of his contemporaries, his writings contain a great deal
of mysticism. His chief work is an astronomy in nine books, of
\index{Astronomy!Arabic}%
which the first is devoted to trigonometry. In his treatment
of spherical trigonometry, he exercises great independence of
thought. He makes war against the time-honoured procedure
adopted by Ptolemy of applying ``the rule of six quantities,''
\index{Ptolemy!ref.\ to}%
and gives a new way of his own, based on the `rule of four
%% -----File: 130.png---Folio 116-------
\index{Geber's theorem}%
quantities.' This is: If $PP_1$~and~$QQ_1$ be two arcs of great
circles intersecting in~$A$, and if $PQ$~and~$P_1Q_1$ be arcs of great
circles drawn perpendicular to~$QQ_1$, then we have the proportion
\[
\sin AP : \sin PQ = \sin AP_1 : \sin P_1Q_1.
\]
From this he derives the formulas for spherical right triangles.
To the four fundamental formulas already given by Ptolemy,
he added a fifth, discovered by himself. If $a$,~$b$,~$c$, be the sides,
and $A$,~$B$,~$C$, the angles of a spherical triangle, right-angled at~$A$,
then $\cos B = \cos b \sin C$. This is frequently called ``Geber's
Theorem.'' Radical and bold as were his innovations in
\Addendlabel{116}%
spherical trigonometry, in plane trigonometry he followed
slavishly the old beaten path of the Greeks. Not even did he
adopt the Indian `sine' and `cosine,' but still used the Greek
\index{Sine}%
`chord of double the angle.' So painful was the departure
from old ideas, even to an independent Arab! After the time
of Gabir ben Aflah there was no mathematician among the
Spanish Saracens of any reputation. In the year in which
Columbus discovered America, the Moors lost their last foothold
\index{Moors}%
on Spanish soil.

We have witnessed a laudable intellectual activity among
the Arabs. They had the good fortune to possess rulers
who, by their munificence, furthered scientific research. At
the courts of the caliphs, scientists were supplied with libraries
and observatories. A large number of astronomical and
mathematical works were written by Arabic authors. Yet
we fail to find a single important principle in mathematics
brought forth by the Arabic mind. Whatever discoveries
they made, were in fields previously traversed by the Greeks
or the Indians, and consisted of objects which the latter had
overlooked in their rapid march. The Arabic mind did not
possess that penetrative insight and invention by which mathematicians
in Europe afterwards revolutionised the science.
%% -----File: 131.png---Folio 117-------
The Arabs were learned, but not original. Their chief service
\index{Arabs|)}%
to science consists in this, that they adopted the learning of
Greece and India, and kept what they received with scrupulous
care. When the love for science began to grow in the
Occident, they transmitted to the Europeans the valuable
treasures of antiquity. Thus a Semitic race was, during the
Dark Ages, the custodian of the Aryan intellectual possessions.


\Section{Europe During the Middle Ages}
\index{Middle Ages|(}%

With the third century after Christ begins an era of migration
of nations in Europe. The powerful Goths quit their
swamps and forests in the North and sweep onward in steady
southwestern current, dislodging the Vandals, Sueves, and
Burgundians, crossing the Roman territory, and stopping and
recoiling only when reaching the shores of the Mediterranean.
From the Ural Mountains wild hordes sweep down on the
Danube. The Roman Empire falls to pieces, and the Dark
Ages begin. But dark though they seem, they are the germinating
season of the institutions and nations of modern Europe.
The Teutonic element, partly pure, partly intermixed with the
Celtic and Latin, produces that strong and luxuriant growth,
the modern civilisation of Europe. Almost all the various
nations of Europe belong to the Aryan stock. As the Greeks
and the Hindoos---both Aryan races---were the great thinkers
of antiquity, so the nations north of the Alps became the great
intellectual leaders of modern times.

\Subsection{Introduction of Roman Mathematics}
\index{Roman mathematics in Occident|(}%

We shall now consider how these as yet barbaric nations of
the North gradually came in possession of the intellectual
%% -----File: 132.png---Folio 118-------
treasures of antiquity. With the spread of Christianity the
Latin language was introduced not only in ecclesiastical but
also in scientific and all important worldly transactions. Naturally
the science of the Middle Ages was drawn largely from
Latin sources. In fact, during the earlier of these ages Roman
authors were the only ones read in the Occident. Though
Greek was not wholly unknown, yet before the thirteenth
century not a single Greek scientific work had been read or
translated into Latin. Meagre indeed was the science which
could be gotten from Roman writers, and we must wait several
centuries before any substantial progress is made in mathematics.

After the time of Boethius and Cassiodorius mathematical
\index{Boethius!ref.\ to}%
\index{Cassiodorius}%
activity in Italy died out. The first slender blossom of science
among tribes that came from the North was an encyclopædia
entitled \textit{Origines}, written by \textbf{Isidorus} (died 636 as bishop of
\index{Isidorus of Seville}%
Seville). This work is modelled after the Roman encyclopædias
of Martianus Capella of Carthage and of Cassiodorius.
Part of it is devoted to the quadrivium, arithmetic, music,
geometry, and astronomy. He gives definitions and grammatical
explications of technical terms, but does not describe the
modes of computation then in vogue. After Isidorus there
follows a century of darkness which is at last dissipated by
the appearance of \textbf{Bede the Venerable} (672--735), the most
\index{Bede, the Venerable}%
learned man of his time. He was a native of Ireland, then
the home of learning in the Occident. His works contain
treatises on the \textit{Computus}, or the computation of Easter-time,
\index{Computus}%
and on finger-reckoning. It appears that a finger-symbolism
\index{Finger-reckoning}%
was then widely used for calculation. The correct determination
of the time of Easter was a problem which in those days
greatly agitated the Church. It became desirable to have at
least one monk at each monastery who could determine the
day of religious festivals and could compute the calendar.
%% -----File: 133.png---Folio 119-------
Such determinations required some knowledge of arithmetic.
\index{Arithmetic!Middle Ages}%
Hence we find that the art of calculating always found some
little corner in the curriculum for the education of monks.

The year in which Bede died is also the year in which
\textbf{Alcuin} (735--804) was born. Alcuin was educated in Ireland,
\index{Alcuin}%
and was called to the court of Charlemagne to direct the progress
of education in the great Frankish Empire. Charlemagne
was a great patron of learning and of learned men. In the
great sees and monasteries he founded schools in which were
taught the psalms, writing, singing, computation (\textit{computus}),
\index{Computus}%
and grammar. By \textit{computus} was here meant, probably, not
merely the determination of Easter-time, but the art of computation
in general. Exactly what modes of reckoning were
then employed we have no means of knowing. It is not likely
that Alcuin was familiar with the apices of Boethius or with
\index{Apices of Boethius!ref.\ to}%
the Roman method of reckoning on the abacus. He belongs
\index{Abacus}%
to that long list of scholars who dragged the theory of numbers
\index{Numbers!theory of numbers}%
\index{Theory of numbers}%
into theology. Thus the number of beings created by
God, who created all things well, is~$6$, because $6$~is a perfect
number (the sum of its divisors being $1 + 2 + 3 = 6$); $8$,~on
the other hand, is an imperfect number ($1 + 2 + 4 < 8$); hence
the second origin of mankind emanated from the number~$8$,
which is the number of souls said to have been in Noah's ark.

There is a collection of ``Problems for Quickening the
Mind'' (\textit{propositiones ad acuendos iuvenes}), which are certainly
\index{Propositiones ad acuendos iuvenes}%
as old as 1000~\AD\ and possibly older. Cantor is of the opinion
that they were written much earlier and by Alcuin. The
following is a specimen of these ``Problems'': A dog chasing
a rabbit, which has a start of $150$~feet, jumps $9$~feet every time
the rabbit jumps~$7$. In order to determine in how many leaps
the dog overtakes the rabbit, $150$~is to be divided by~$2$. In
this collection of problems, the areas of triangular and quadrangular
pieces of land are found by the same formulas of
%% -----File: 134.png---Folio 120-------
\index{Anthology, Palatine}%
\index{Palatine anthology}%
\index{Propositiones ad acuendos iuvenes}%
\index{Sylvester II. (Gerbert)|(}%
approximation as those used by the Egyptians and given by
Boethius in his geometry. An old problem is the ``cistern-problem''
(given the time in which several pipes can fill a
cistern singly, to find the time in which they fill it jointly),
which has been found previously in Heron, in the Greek \textit{Anthology},
and in Hindoo works. Many of the problems show that
the collection was compiled chiefly from Roman sources. The
problem which, on account of its uniqueness, gives the most
positive testimony regarding the Roman origin is that on the
interpretation of a will in a case where twins are born. The
problem is identical with the Roman, except that different
ratios are chosen. Of the exercises for recreation, we mention
the one of the wolf, goat, and cabbage, to be rowed across a
river in a boat holding only one besides the ferry-man. Query:
How must he carry them across so that the goat shall not eat
the cabbage, nor the wolf the goat? The solutions of the
``problems for quickening the mind'' require no further knowledge
than the recollection of some few formulas used in
surveying, the ability to solve linear equations and to perform
the four fundamental operations with integers. Extraction
of roots was nowhere demanded; fractions hardly
\index{Fractions!Middle Ages}%
ever occur.\Reference{3}

The great empire of Charlemagne tottered and fell almost
immediately after his death. War and confusion ensued.
Scientific pursuits were abandoned, not to be resumed until
the close of the tenth century, when under Saxon rule in Germany
and Capetian in France, more peaceful times began.
The thick gloom of ignorance commenced to disappear. The
zeal with which the study of mathematics was now taken up
by the monks is due principally to the energy and influence
of one man,---\textbf{Gerbert}. He was born in Aurillac in Auvergne.
\index{Gerbert|(}%
After receiving a monastic education, he engaged in study,
chiefly of mathematics, in Spain. On his return he taught
%% -----File: 135.png---Folio 121-------
school at Rheims for ten years and became distinguished for
his profound scholarship. By King Otto~I. and his successors
Gerbert was held in highest esteem. He was elected bishop
of Rheims, then of Ravenna, and finally was made Pope under
the name of Sylvester~II. by his former pupil Emperor Otho~III.
He died in~1003, after a life intricately involved in many
political and ecclesiastical quarrels. Such was the career of
the greatest mathematician of the tenth century in Europe.
By his contemporaries his mathematical knowledge was considered
wonderful. Many even accused him of criminal intercourse
with evil spirits.

Gerbert enlarged the stock of his knowledge by procuring
copies of rare books. Thus in Mantua he found the geometry
\index{Geometry!Middle Ages}%
of Boethius. Though this is of small scientific value, yet it
\index{Boethius!ref.\ to}%
is of great importance in history. It was at that time the
only book from which European scholars could learn the elements
of geometry. Gerbert studied it with zeal, and is
generally believed himself to be the author of a geometry.
H.~Weissenborn denies his authorship, and claims that the
book in question consists of three parts which cannot come
from one and the same author.\Reference{21} This geometry contains
nothing more than the one of Boethius, but the fact that
occasional errors in the latter are herein corrected shows that
the author had mastered the subject. ``The first mathematical
paper of the Middle Ages which deserves this name,''
says Hankel, ``is a letter of Gerbert to Adalbold, bishop of
Utrecht,'' in which is explained the reason why the area of a
triangle, obtained ``geometrically'' by taking the product of
the base by half its altitude, differs from the area calculated
``arithmetically,'' according to the formula
$\frac{1}{2} a (a + 1)$, used
by surveyors, where $a$ stands for a side of an equilateral triangle.
He gives the correct explanation that in the latter
formula all the small squares, in which the triangle is supposed
%% -----File: 136.png---Folio 122-------
to be divided, are counted in wholly, even though parts
of them project beyond it.

Gerbert made a careful study of the arithmetical works of
Boethius. He himself published two works,---\textit{Rule of Computation
on the Abacus}, and \textit{A Small Book on the Division of
\index{Abacus}%
Numbers}. They give an insight into the methods of calculation
practised in Europe before the introduction of the
Hindoo numerals. Gerbert used the abacus, which was probably
unknown to Alcuin. \textbf{Bernelinus}, a pupil of Gerbert,
\index{Bernelinus}%
describes it as consisting of a smooth board upon which geometricians
were accustomed to strew blue sand, and then to draw
their diagrams. For arithmetical purposes the board was
divided into 30 columns, of which 3 were reserved for fractions,
while the remaining 27 were divided into groups with
3 columns in each. In every group the columns were marked
respectively by the letters C (\textit{centum}), D (\textit{decem}), and
S (\textit{singularis}) or M (\textit{monas}). Bernelinus gives the nine
numerals used, which are the apices of Boethius, and then
remarks that the Greek letters may be used in their place.\Reference{3}
By the use of these columns any number can be written
without introducing a zero, and all operations in arithmetic
\index{Arithmetic!Middle Ages}%
can be performed in the same way as we execute ours without
the columns, but with the symbol for zero. Indeed, the
methods of adding, subtracting, and multiplying in vogue
among the abacists agree substantially with those of to-day.
But in a division there is very great difference. The early rules
for division appear to have been framed to satisfy the following
three conditions: (1)~The use of the multiplication table shall
be restricted as far as possible; at least, it shall never be
required to multiply mentally a figure of two digits by another
of one digit. (2)~Subtractions shall be avoided as much as
possible and replaced by additions. (3)~The operation shall
proceed in a purely mechanical way, without requiring trials.\Reference{7}
%% -----File: 137.png---Folio 123-------
\index{Arithmetic!Middle Ages}%
That it should be necessary to make such conditions seems
strange to us; but it must be remembered that the monks of
the Middle Ages did not attend school during childhood and
learn the multiplication table while the memory was fresh.
Gerbert's rules for division are the oldest extant. They are
so brief as to be very obscure to the uninitiated. They were
probably intended simply to aid the memory by calling to
mind the successive steps in the work. In later manuscripts
they are stated more fully. In dividing any number by another
of one digit, say $668$~by~$6$, the divisor was first increased to~10
by adding~4. The process is exhibited in the adjoining figure.\Reference{3}
As it continues, we must imagine the digits
which are crossed out, to be erased and then
replaced by the ones beneath. It is as follows:
$600 ÷ 10 = 60$, but, to rectify the error, $4 × 60$,
or~$240$, must be added; $200 ÷ 10 = 20$, but $4 × 20$,
or~$80$, must be added. We now write for
$60 + 40 + 80$, %[** PP: Width-dependent line break]
\begin{wrapfigure}[21]{r}{0.875in}
\setlength{\TmpLen}{1.75em}%[** PP: Sets width of \cancel{digit}]
\hfill\begin{tabular}{|*{3}{@{}c@{}|}}
\multicolumn{3}{@{}c@{}}{\smash[b]{\makebox[0pt][c]{\Graphic{3\TmpLen}{arc}}}} \\
\multicolumn{2}{|@{}c@{}}{\smash{\Graphic{2\TmpLen}{arc}}} & \Graphic{\TmpLen}{arc} \\
 \sffamily C & \sffamily D & \sffamily S \\
    &   & $6$ \\
    &   & $4$ \\
\hline
  $6$ & $6$ & $8$ \\
\hline
  \cancel{6} & \cancel{6} & \cancel{8} \\
  \cancel{2} & \cancel{4} & \cancel{4} \\
  \cancel{1} & \cancel{8} & \cancel{8} \\
  \cancel{1} & \cancel{4} & \cancel{8} \\
             & \cancel{2} & $2$ \\
             & \cancel{4} &  \\
             & \cancel{6} &  \\
             & \cancel{2} &  \\
             & \cancel{2} &  \\
\hline
             & \cancel{6} & \cancel{6} \\
             & \cancel{2} & \cancel{2} \\
             & \cancel{1} & \cancel{2} \\
             & $1$         & $1$
\end{tabular}
\end{wrapfigure}
its sum~$180$, and continue thus:
$100 ÷ 10 = 10$; the correction necessary is $4 × 10$,
or~$40$, which, added to~$80$, gives~$120$. Now
$100 ÷ 10 = 10$, and the correction $4 × 10$, together
with the~$20$, gives~$60$. Proceeding as
before, $60 ÷ 10 = 6$; the correction is $4 × 6 = 24$.
Now $20 ÷ 10 = 2$, the correction being $4 × 2 = 8$.
In the column of units we have now $8 + 4 + 8$,
or~$20$. As before, $20 ÷ 10 = 2$; the correction
is $2 × 4 = 8$, which is not divisible by~$10$, but
only by~$6$, giving the quotient~$1$ and the remainder~$2$.
All the partial quotients taken
together give $60 + 20 + 10 + 10 + 6 + 2 + 2 + 1 = 111$, and
the remainder~$2$.

Similar but more complicated, is the process when the
divisor contains two or more digits. Were the divisor~27,
%% -----File: 138.png---Folio 124-------
\index{Roman mathematics in Occident|)}%
\index{Sylvester II. (Gerbert)|)}%
then the next higher multiple of~$10$, or~$30$, would be taken
for the divisor, but corrections would be required for the~$3$.
He who has the patience to carry such a division through
to the end, will understand why it has been said of Gerbert
that ``Regulas dedit, quæ a sudantibus abacistis vix intelliguntur.''
He will also perceive why the Arabic method of
division, when first introduced, was called the \textit{divisio aurea},
but the one on the abacus, the \textit{divisio ferrea}.

In his book on the abacus, Bernelinus devotes a chapter to
fractions. These are, of course, the \textit{duodecimals}, first used
\index{Duodecimals}%
\index{Fractions!Middle Ages}%
\index{Fractions!duodecimal}%
by the Romans. For want of a suitable notation, calculation
with them was exceedingly difficult. It would be so even to
us, were we accustomed, like the early abacists, to express
them, not by a numerator or denominator, but by the application
of names, such as \textit{uncia} for~$\frac{1}{12}$, \textit{quincunx} for~$\frac{5}{12}$, \textit{dodrans}
for~$\frac{9}{12}$.

In the tenth century, Gerbert was the central figure among
the learned. In his time the Occident came into secure possession
of all mathematical knowledge of the Romans. During
the eleventh century it was studied assiduously. Though
numerous works were written on arithmetic and geometry,
mathematical knowledge in the Occident was still very insignificant.
Scanty indeed were the mathematical treasures
obtained from Roman sources.

\Subsection{Translation of Arabic Manuscripts}
\index{Arabic manuscripts|(}%

By his great erudition and phenomenal activity, Gerbert
\index{Gerbert|)}%
infused new life into the study not only of mathematics, but
also of philosophy. Pupils from France, Germany, and Italy
gathered at Rheims to enjoy his instruction. When they
themselves became teachers, they taught of course not only
the use of the abacus and geometry, but also what they had
%% -----File: 139.png---Folio 125-------
\index{Geometry!Middle Ages}%
\index{Moors}%
\index{Sine}%
learned of the philosophy of Aristotle. His philosophy was
\index{Aristotle!ref.\ to}%
known, at first, only through the writings of Boethius. But
the growing enthusiasm for it created a demand for his complete
works. Greek texts were wanting. But the Latins
heard that the Arabs, too, were great admirers of Peripatetism,
and that they possessed translations of Aristotle's works and
commentaries thereon. This led them finally to search for
and translate Arabic manuscripts. During this search, mathematical
works also came to their notice, and were translated
into Latin. Though some few unimportant works may have
been translated earlier, yet the period of greatest activity
began about~1100. The zeal displayed in acquiring the
Mohammedan treasures of knowledge excelled even that of
the Arabs themselves, when, in the eighth century, they
plundered the rich coffers of Greek and Hindoo science.

Among the earliest scholars engaged in translating manuscripts
into Latin was \textbf{Athelard of Bath}. The period of his
\index{Athelard of Bath}%
\Addendlabel{125}%
activity is the first quarter of the twelfth century. He
travelled extensively in Asia Minor, Egypt, and Spain, and
braved a thousand perils, that he might acquire the language
and science of the Mohammedans. He made the earliest
translations, from the Arabic, of Euclid's \textit{Elements} and of
\index{Elements (Euclid's)}%
\index{Euclid!ref.\ to}%
the astronomical tables of Mohammed ben Musa Hovarezmi.
\index{Hovarezmi!ref.\ to}%
\index{Mohammed ben Musa Hovarezmi!ref.\ to}%
In~1857, a manuscript was found in the library at Cambridge,
which proved to be the arithmetic by Mohammed ben Musa
in Latin. This translation also is very probably due to
Athelard.

At about the same time flourished \textit{Plato of Tivoli} or \textit{Plato
\index{Plato of Tivoli}%
Tiburtinus}. He effected a translation of the astronomy of
Al Battani and of the \textit{Sphærica} of Theodosius. Through the
\index{Albattani@{Al Battani}!ref.\ to}%
\index{Theodosius!ref.\ to}%
former, the term \textit{sinus} was introduced into trigonometry.

About the middle of the twelfth century there was a group
of Christian scholars busily at work at Toledo, under the
%% -----File: 140.png---Folio 126-------
\index{Algorithm!Middle Ages}%
\index{Arithmetic!Middle Ages}%
\index{Gerard of Cremona}%
\index{Sexagesimal system}%
leadership of Raymond, then archbishop of Toledo. Among
those who worked under his direction, \textbf{John of Seville} was
\index{John of Seville}%
most prominent. He translated works chiefly on Aristotelian
philosophy. Of importance to us is a \textit{liber algorismi}, compiled
by him from Arabic authors. On comparing works like
this with those of the abacists, we notice at once the most
\index{Abacists}%
striking difference, which shows that the two parties drew
from independent sources. It is argued by some that Gerbert
got his apices and his arithmetical knowledge, not from
\index{Apices of Boethius!ref.\ to}%
Boethius, but from the Arabs in Spain, and that part or the
whole of the geometry of Boethius is a forgery, dating from
the time of Gerbert. If this were the case, then the writings
of Gerbert would betray Arabic sources, as do those of John
of Seville. But no points of resemblance are found. Gerbert
could not have learned from the Arabs the use of the abacus,
\index{Abacus}%
because all evidence we have goes to show that they did not
employ it. Nor is it probable that he borrowed from the
Arabs the apices, because they were never used in Europe
except on the abacus. In illustrating an example in division,
mathematicians of the tenth and eleventh centuries state an
example in Roman numerals, then draw an abacus and insert
in it the necessary numbers with the apices. Hence it seems
probable that the abacus and apices were borrowed from the
same source. The contrast between authors like John of
Seville, drawing from Arabic works, and the abacists, consists
in this, that, unlike the latter, the former mention the Hindoos,
use the term \emph{algorism}, calculate with the zero, and do
not employ the abacus. The former teach the extraction of
roots, the abacists do not; they teach the sexagesimal fractions
\index{Fractions!sexagesimal}%
\index{Fractions!duodecimal}%
used by the Arabs, while the abacists employ the duodecimals
\index{Duodecimals}%
of the Romans.\Reference{3}

A little later than John of Seville flourished \textbf{Gerard of
Cremona} in Lombardy. Being desirous to gain possession of
%% -----File: 141.png---Folio 127-------
\index{Alfonso's tables}%
\index{Alphonso's tables}%
\index{Arabic numerals and notation|(}%
\index{Elements (Euclid's)}%
\index{Geometry!Middle Ages}%
\index{Girard!ref.\ to}%
the \textit{Almagest}, he went to Toledo, and there, in~1175, translated
\index{Almagest!ref.\ to}%
this great work of Ptolemy. Inspired by the richness of
Mohammedan literature, he gave himself up to its study. He
translated into Latin over 70~Arabic works. Of mathematical
treatises, there were among these, besides the \textit{Almagest}, the
15~books of Euclid, the \textit{Sphærica} of Theodosius, a work of
\index{Euclid!ref.\ to}%
\index{Theodosius!ref.\ to}%
Menelaus, the algebra of Mohammed ben Musa Hovarezmi, the
\index{Hovarezmi!ref.\ to}%
\index{Mohammed ben Musa Hovarezmi!ref.\ to}%
astronomy of Dshabir ben Aflah, and others less important.
\index{Astronomy!Middle Ages}%
\index{Gabir ben Aflah!ref.\ to}%

In the thirteenth century, the zeal for the acquisition of
Arabic learning continued. Foremost among the patrons of
science at this time ranked Emperor Frederick~II. of Hohenstaufen
(died 1250). Through frequent contact with Mohammedan
scholars, he became familiar with Arabic science.
He employed a number of scholars in translating Arabic
manuscripts, and it was through him that we came in possession
of a new translation of the \textit{Almagest}. Another royal
head deserving mention as a zealous promoter of Arabic
science was Alfonso~X. of Castile (died 1284). He gathered
around him a number of Jewish and Christian scholars, who
translated and compiled astronomical works from Arabic
sources. \textbf{Rabbi Zag} and \textbf{Iehuda ben Mose Cohen} were the
\index{Iehuda ben Mose Cohen}%
\index{Zag}%
most prominent among them. Astronomical tables prepared
by these two Jews spread rapidly in the Occident, and constituted
the basis of all astronomical calculation till the
sixteenth century.\Reference{7} The number of scholars who aided in
transplanting Arabic science upon Christian soil was large.
But we mention only one more. \textbf{Giovanni Campano} of Novara
\index{Giovanni Campano}%
(about 1260) brought out a new translation of Euclid, which
drove the earlier ones from the field, and which formed the
basis of the printed editions.\Reference{7}

At the close of the twelfth century, the Occident was in
possession of the so-called Arabic notation. The Hindoo
\index{Notation!Arabic notation|(}%
methods of calculation began to supersede the cumbrous methods
%% -----File: 142.png---Folio 128-------
\index{Arabic manuscripts|)}%
\index{Elements (Euclid's)}%
inherited from Rome. Algebra, with its rules for solving
linear and quadratic equations, had been made accessible to
the Latins. The geometry of Euclid, the \textit{Sphærica} of Theodosius,
\index{Geometry!Middle Ages}%
the astronomy of Ptolemy, and other works were now
accessible in the Latin tongue. Thus a great amount of new
scientific material had come into the hands of the Christians.
The talent necessary to digest this heterogeneous mass of
knowledge was not wanting. The figure of Leonardo of Pisa
\index{Leonardo of Pisa}%
adorns the vestibule of the thirteenth century.

It is important to notice that no work either on mathematics
or astronomy was translated directly from the Greek previous
to the fifteenth century.

\Subsection{The First Awakening and its Sequel}

Thus far, France and the British Isles have been the headquarters
of mathematics in Christian Europe. But at the
beginning of the thirteenth century the talent and activity
of one man was sufficient to assign the mathematical science
a new home in Italy. This man was not a monk, like Bede,
Alcuin, or Gerbert, but a merchant, who in the midst of
business pursuits found time for scientific study. \textbf{Leonardo
of Pisa} is the man to whom we owe the first renaissance of
mathematics on Christian soil. He is also called \textit{Fibonacci},
\ie~son of Bonaccio. His father was secretary at one of the
numerous factories erected on the south and east coast of the
Mediterranean by the enterprising merchants of Pisa. He
made Leonardo, when a boy, learn the use of the abacus. The
boy acquired a strong taste for mathematics, and, in later years,
during his extensive business travels in Egypt, Syria, Greece,
and Sicily, collected from the various peoples all the knowledge
he could get on this subject. Of all the methods of
calculation, he found the Hindoo to be unquestionably the
%% -----File: 143.png---Folio 129-------
\index{Algorithm!Middle Ages}%
\index{Arabic numerals and notation|)}%
best. Returning to Pisa, he published, in~1202, his great
work, the \textit{Liber Abaci}. A revised edition of this appeared in~1228.
This work contains about all the knowledge the Arabs
possessed in arithmetic and algebra, and treats the subject in
a free and independent way. This, together with the other
books of Leonardo, shows that he was not merely a compiler,
or, like other writers of the Middle Ages, a slavish imitator
of the form in which the subject had been previously presented,
but that he was an original worker of exceptional
power.

He was the first great mathematician to advocate the adoption
of the ``Arabic notation.'' The calculation with the zero
\index{Zero!origin of term}%
was the portion of Arabic mathematics earliest adopted by
the Christians. The minds of men had been prepared for the
reception of this by the use of the abacus and the apices.
\index{Abacus}%
\index{Apices of Boethius!ref.\ to}%
The reckoning with columns was gradually abandoned, and
the very word \emph{abacus} changed its meaning and became a
synonym for \emph{algorism}. For the zero, the Latins adopted
the name \emph{zephirum}, from the Arabic \textit{sifr} (\textit{sifra}=empty);
hence our English word \emph{cipher}. The new notation was
\index{Notation!Arabic notation|)}%
accepted readily by the enlightened masses, but, at first,
rejected by the learned circles. The merchants of Italy used
it as early as the thirteenth century, while the monks in the
monasteries adhered to the old forms. In~1299, nearly 100~years
after the publication of Leonardo's \textit{Liber Abaci}, the
Florentine merchants were forbidden the use of the Arabic
numerals in book-keeping, and ordered either to employ the
Roman numerals or to write the numeral adjectives out in
full. In the fifteenth century the abacus with its counters
ceased to be used in Spain and Italy. In France it was used
later, and it did not disappear in England and Germany before
the middle of the seventeenth century.\Reference{22} Thus, in the \textit{Winter's
Tale} (iv.~3), Shakespeare lets the clown be embarrassed by
%% -----File: 144.png---Folio 130-------
a problem which he could not do without counters. Iago
(in \textit{Othello}, i.~1) expresses his contempt for Michael Cassio,
``forsooth a great mathematician,'' by calling him a ``counter-caster.''
So general, indeed, says Peacock, appears to have
\index{Peacock!ref.\ to, X}%
been the practice of this species of arithmetic, that its rules
\index{Arithmetic!Middle Ages}%
and principles form an essential part of the arithmetical
treatises of that day. The real fact seems to be that the old
methods were used long after the Hindoo numerals were in
common and general use. With such dogged persistency does
man cling to the old!

The \textit{Liber Abaci} was, for centuries, the storehouse from
which authors got material for works on arithmetic and
algebra. In it are set forth the most perfect methods of
calculation with integers and fractions, known at that time;
the square and cube root are explained; equations of the first
and second degree leading to problems, either determinate
or indeterminate, are solved by the methods of `single' or
`double position,' and also by real algebra. The book contains
a large number of problems. The following was proposed
to Leonardo of Pisa by a magister in Constantinople,
as a difficult problem: If A gets from B $7$~denare, then A's
sum is five-fold B's; if B gets from A $5$~denare, then B's sum
is seven-fold A's. How much has each? The \textit{Liber Abaci}
contains another problem, which is of historical interest,
because it was given with some variations by Ahmes, 3000~years
\index{Ahmes!ref.\ to}%
earlier: $7$~old women go to Rome; each woman has
$7$~mules, each mule carries $7$~sacks, each sack contains $7$~loaves,
with each loaf are $7$~knives, each knife is put up in $7$~sheaths.
What is the sum total of all named? \textit{Ans.}~$137,256$.\Reference{3}

In~1220, Leonardo of Pisa published his \textit{Practica Geometriæ},
which contains all the knowledge of geometry and trigonometry
\index{Geometry!Middle Ages}%
transmitted to him. The writings of Euclid and of some
other Greek masters were known to him, either from Arabic
%% -----File: 145.png---Folio 131-------
\index{Theory of numbers}%
manuscripts directly or from the translations made by his
countrymen, Gerard of Cremona and Plato of Tivoli. Leonardo's
\textit{Geometry} contains an elegant geometrical demonstration
\index{Geometry!Middle Ages}%
of Heron's formula for the area of a triangle, as a
\index{Heron the Elder!ref.\ to}%
function of its three sides. Leonardo treats the rich material
before him with skill and Euclidean rigour.

Of still greater interest than the preceding works are those
containing Fibonacci's original investigations. We must here
preface that after the publication of the \textit{Liber Abaci}, Leonardo
was presented by the astronomer Dominicus to Emperor
Frederick~II. of Hohenstaufen. On that occasion, John of
Palermo, an imperial notary, proposed several problems, which
Leonardo solved promptly. The first problem was to find a
number $x$, such that $x^2 + 5$ and $x^2 - 5$ are each square numbers.
\index{Numbers!theory of numbers}%
The answer is $x = 3\frac{5}{12}$; for $(3 \frac{5}{12})^2 + 5 = (4\frac{1}{12})^2$, $(3\frac{5}{12})^2 - 5
= (2\frac{7}{12})^2$. His masterly solution of this is given in his \textit{liber
quadratorum}, a copy of which work was sent by him to
Frederick~II\@. The problem was not original with John of
Palermo, since the Arabs had already solved similar ones.
Some parts of Leonardo's solution may have been borrowed
from the Arabs, but the method which he employed of building
squares by the summation of odd numbers is original with him.

The second problem proposed to Leonardo at the famous
scientific tournament which accompanied the presentation of
this celebrated algebraist to that great patron of learning,
Emperor Frederick~II., was the solving of the equation
$x^3 + 2x^2 + 10x = 20$. As yet cubic equations had not been
solved algebraically. Instead of brooding stubbornly over
this knotty problem, and after many failures still entertaining
new hopes of success, he changed his method of inquiry and
showed by clear and rigorous demonstration that the roots
of this equation could not be represented by the Euclidean
irrational quantities, or, in other words, that they could not be
%% -----File: 146.png---Folio 132-------
constructed with the ruler and compass only. He contented
himself with finding a very close approximation to the
required root. His work on this cubic is found in the \textit{Flos},
together with the solution of the following third problem
given him by John of Palermo: Three men possess in common
an unknown sum of money~$t$; the share of the first
is~$\dfrac{t}{2}$; that of the second,~$\dfrac{t}{3\Fstr}$; that of the third,~$\smash[t]{\dfrac{t}{6}}$. Desirous of
depositing the sum at a safer place, each takes at hazard a
certain amount; the first takes~$x$, but deposits only~$\dfrac{x}{2}$; the second
carries~$y$, but deposits only~$\smash{\dfrac{y}{3}}$; the third takes~$z$, and
deposits~$\smash[t]{\dfrac{z}{6}}$. Of the amount deposited each one must receive
exactly~$\frac{1}{3}$, in order to possess his share of the whole sum.
Find $x$, $y$,~$z$. Leonardo shows the problem to be indeterminate.
Assuming 7 for the sum drawn by each from the
deposit, he finds $t = 47$, $x = 33$, $y = 13$, $z = 1$.

One would have thought that after so brilliant a beginning,
the sciences transplanted from Mohammedan to Christian soil
would have enjoyed a steady and vigorous development. But
this was not the case. During the fourteenth and fifteenth
centuries, the mathematical science was almost stationary.
Long wars absorbed the energies of the people and thereby
kept back the growth of the sciences. The death of Frederick~II.
in 1254 was followed by a period of confusion in Germany.
The German emperors and the popes were continually quarrelling,
and Italy was inevitably drawn into the struggles between
the Guelphs and the Ghibellines. France and England were
engaged in the Hundred Years' War (1338--1453). Then
followed in England the Wars of the Roses. The growth of
science was retarded not only by war, but also by the injurious
influence of scholastic philosophy. The intellectual
leaders of those times quarrelled over subtle subjects in metaphysics
%% -----File: 147.png---Folio 133-------
and theology. Frivolous questions, such as ``How
many angels can stand on the point of a needle?'' were discussed
with great interest. Indistinctness and confusion of
ideas characterised the reasoning during this period. Among
the mathematical productions of the Middle Ages, the works
of Leonardo of Pisa appear to us like jewels among quarry-rubbish.
\index{Leonardo of Pisa!ref.\ to}%
The writers on mathematics during this period were
not few in number, but their scientific efforts were vitiated
by the method of scholastic thinking. Though they possessed
the \textit{Elements} of Euclid, yet the true nature of a mathematical
\index{Elements (Euclid's)}%
proof was so little understood, that Hankel believes it no
exaggeration to say that ``since Fibonacci, not a single proof,
not borrowed from Euclid, can be found in the whole literature
of these ages, which fulfils all necessary conditions.''

The only noticeable advance is a simplification of numerical
operations and a more extended application of them. Among
the Italians are evidences of an early maturity of arithmetic.
\index{Arithmetic!Middle Ages}%
Peacock\Reference[]{22} says: The Tuscans generally, and the Florentines
\index{Peacock!ref.\ to, X}%
in particular, whose city was the cradle of the literature and
arts of the thirteenth and fourteenth centuries, were celebrated
for their knowledge of arithmetic and book-keeping, which
were so necessary for their extensive commerce; the Italians
were in familiar possession of commercial arithmetic long
before the other nations of Europe; to them we are indebted
for the formal introduction into books of arithmetic, under
distinct heads, of questions in the single and double rule of
three, loss and gain, fellowship, exchange, simple and compound
interest, discount, and so on.

There was also a slow improvement in the algebraic notation.
\index{Notation!in algebra}%
The Hindoo algebra possessed a tolerable symbolic
\index{Algebra!Middle Ages}%
notation, which was, however, completely ignored by the Mohammedans.
In this respect, Arabic algebra approached
much more closely to that of Diophantus, which can scarcely
%% -----File: 148.png---Folio 134-------
\index{Albertus Magnus}%
\index{Exponents}%
\index{Stevin!ref.\ to}%
be said to employ symbols in a systematic way. Leonardo of
Pisa possessed no algebraic symbolism. Like the Arabs, he
expressed the relations of magnitudes to each other by lines
or in words. But in the mathematical writings of the monk
\textit{Luca Pacioli} (also called Lucas de Burgo sepulchri) symbols
\index{Pacioli!ref.\ to}%
began to appear. They consisted merely in abbreviations of
Italian words, such as \textit{p} for \textit{piu} (more), \textit{m} for \textit{meno} (less), \textit{co}
for \textit{cosa} (the thing or unknown quantity). ``Our present
notation has arisen by almost insensible degrees as convenience
\index{Notation!in algebra}%
suggested different marks of abbreviation to different
authors; and that perfect symbolic language which addresses
itself solely to the eye, and enables us to take in at a glance
the most complicated relations of quantity, is the result of a
large series of small improvements.''\Reference{23}

We shall now mention a few authors who lived during the
thirteenth and fourteenth and the first half of the fifteenth
centuries. About the time of Leonardo of Pisa (1200~\AD),
lived the German monk \textbf{Jordanus Nemorarius}, who wrote a once
\index{Jordanus Nemorarius}%
famous work on the properties of numbers (1496), modelled
after the arithmetic of Boethius. The most trifling numeral
\index{Arithmetic!Middle Ages}%
\index{Boethius!ref.\ to}%
properties are treated with nauseating pedantry and prolixity.
A practical arithmetic based on the Hindoo notation was
also written by him. \textbf{John Halifax} (Sacro Bosco, died 1256)
\index{Halifax}%
taught in Paris and made an extract from the \textit{Almagest} containing
\index{Almagest!ref.\ to}%
only the most elementary parts of that work. This
extract was for nearly 400~years a work of great popularity
and standard authority. Other prominent writers are \textbf{Albertus
Magnus} and \textbf{George Purbach} in Germany, and \textbf{Roger Bacon} in
\index{Bacon, R.}%
\index{Purbach}%
England. It appears that here and there some of our modern
ideas were anticipated by writers of the Middle Ages. Thus,
\textbf{Nicole Oresme}, a bishop in Normandy (died 1382), first conceived
\index{Oresme}%
a notation of fractional powers, afterwards re-discovered
by Stevinus, and gave rules for operating with them.
%% -----File: 149.png---Folio 135-------
\index{Elements (Euclid's)}%
\index{Infinitesimals}%
\index{Infinity}%
His notation was totally different from ours. \textbf{Thomas Bradwardine},
\index{Bradwardine}%
archbishop of Canterbury, studied star-polygons,---a
\index{Star-polygons}%
subject which has recently received renewed attention. The
first appearance of such polygons was with Pythagoras and
\index{Pythagoras!ref.\ to}%
his school. We next meet with such polygons in the geometry
of Boethius and also in the translation of Euclid from
\index{Boethius!ref.\ to}%
the Arabic by Athelard of Bath. Bradwardine's philosophic
\index{Athelard of Bath!ref.\ to}%
writings contain discussions on the infinite and the infinitesimal---subjects
never since lost sight of. To England
falls the honour of having produced the earliest European
writers on trigonometry. The writings of Bradwardine, of
\index{Trigonometry}%
Richard of Wallingford, and John Maudith, both professors
\index{Maudith}%
\index{Richard of Wallingford}%
at Oxford, and of Simon Bredon of Winchecombe, contain
\index{Bredon}%
trigonometry drawn from Arabic sources.

The works of the Greek monk \textbf{Maximus Planudes}, who lived
\index{Planudes, M.}%
in the first half of the fourteenth century, are of interest only
as showing that the Hindoo numerals were then known in
Greece. A writer belonging, like Planudes, to the Byzantine
school, was \textbf{Moschopulus}, who lived in Constantinople in the
\index{Moschopulus}%
early part of the fifteenth century. To him appears to be
due the introduction into Europe of magic squares. He wrote
\index{Magic squares}%
a treatise on this subject. Magic squares were known to the
Arabs, and perhaps to the Hindoos. Mediæval astrologers
and physicians believed them to possess mystical properties
and to be a charm against plague, when engraved on silver
plate.

In 1494 was \emph{printed} the \textit{Summa de Arithmetica, Geometria,
Proportione et Proportionalita}, written by the Tuscan monk
\textbf{Lucas Pacioli}, who, as we remarked, first introduced symbols
\index{Pacioli}%
in algebra. This contains all the knowledge of his day on
\index{Algebra!Middle Ages}%
arithmetic, algebra, and trigonometry, and is the first comprehensive
work which appeared after the \textit{Liber Abaci} of
Fibonacci. It contains little of importance which cannot be
%% -----File: 150.png---Folio 136-------
\index{Elements (Euclid's)}%
\index{Halifax!ref.\ to}%
found in Fibonacci's great work, published three centuries
earlier.\Reference{1}

Perhaps the greatest result of the influx of Arabic learning
was the establishment of universities. What was their
\index{Universities of Cologne, Leipzig, Oxford, Paris, and Prague}%
attitude toward mathematics? The \textit{University of Paris}, so
famous at the beginning of the twelfth century under the
teachings of Abelard, paid but little attention to this science
during the Middle Ages. Geometry was neglected, and Aristotle's
logic was the favourite study. In~1336, a rule was
introduced that no student should take a degree without
attending lectures on mathematics, and from a commentary
on the first six books of Euclid, dated~1536, it appears that
\index{Euclid!ref.\ to}%
candidates for the degree of A.M. had to give an oath that
they had attended lectures on these books.\Reference{7} Examinations,
when held at all, probably did not extend beyond the first
book, as is shown by the nickname ``magister matheseos,''
\index{Magister matheseos}%
applied to the Theorem of Pythagoras, the last in the first
book. More attention was paid to mathematics at the \textit{University
of Prague}, founded 1384. For the Baccalaureate degree,
students were required to take lectures on Sacro Bosco's
famous work on astronomy. Of candidates for the A.M. were
required not only the six books of Euclid, but an additional
knowledge of applied mathematics. Lectures were given on
the \textit{Almagest}. At the \textit{University of Leipzig}, the daughter of
\index{Almagest!ref.\ to}%
Prague, and at \textit{Cologne}, less work was required, and, as late
as the sixteenth century, the same requirements were made at
these as at Prague in the fourteenth. The universities of
Bologna, Padua, Pisa, occupied similar positions to the ones
in Germany, only that purely astrological lectures were given
in place of lectures on the \textit{Almagest}. At Oxford, in the
middle of the fifteenth century, the first two books of Euclid
were read.\Reference{6}

Thus it will be seen that the study of mathematics was
%% -----File: 151.png---Folio 137-------
\index{Middle Ages|)}%
maintained at the universities only in a half-hearted manner.
No great mathematician and teacher appeared, to inspire the
students. The best energies of the schoolmen were expended
upon the stupid subtleties of their philosophy. The genius
of Leonardo of Pisa left no permanent impress upon the age,
\index{Leonardo of Pisa!ref.\ to}%
and another Renaissance of mathematics was wanted.
%% -----File: 152.png---Folio 138-------
\index{Elements (Euclid's)}%
\index{Geometry!Renaissance}%


\Chapter{Modern Europe}
\index{Modern Europe|etseq}%

\First{We} find it convenient to choose the time of the capture of
Constantinople by the Turks as the date at which the Middle
Ages ended and Modern Times began. In~1453, the Turks
battered the walls of this celebrated metropolis with cannon,
and finally captured the city; the Byzantine Empire fell, to
rise no more. Calamitous as was this event to the East, it
acted favourably upon the progress of learning in the West.
A great number of learned Greeks fled into Italy, bringing
with them precious manuscripts of Greek literature. This
contributed vastly to the reviving of classic learning. Up
to this time, Greek masters were known only through the
often very corrupt Arabic manuscripts, but now they began
to be studied from original sources and in their own language.
The first English translation of Euclid was made in 1570 from
\index{Euclid!ref.\ to}%
the Greek by \textit{Sir Henry Billingsley}, assisted by \textit{John Dee}.\Reference{29}
\index{Billingsley}%
\index{Dee}%
About the middle of the fifteenth century, printing was invented;
books became cheap and plentiful; the printing-press
transformed Europe into an audience-room. Near the close of
the fifteenth century, America was discovered, and, soon after,
the earth was circumnavigated. The pulse and pace of the
world began to quicken. Men's minds became less servile;
they became clearer and stronger. The indistinctness of
thought, which was the characteristic feature of mediæval
learning, began to be remedied chiefly by the steady cultivation
%% -----File: 153.png---Folio 139-------
\index{Ptolemy!ref.\ to}%
of Pure Mathematics and Astronomy. Dogmatism was
attacked; there arose a long struggle with the authority of
the Church and the established schools of philosophy. The
Copernican System was set up in opposition to the time-honoured
\index{Copernican System}%
Ptolemaic System. The long and eager contest between
the two culminated in a crisis at the time of Galileo, and
\index{Galileo!ref.\ to}%
resulted in the victory of the new system. Thus, by slow
degrees, the minds of men were cut adrift from their old
scholastic moorings and sent forth on the wide sea of scientific
inquiry, to discover new islands and continents of truth.

\Section{The Renaissance}
\index{Renaissance|(}%

With the sixteenth century began a period of increased
intellectual activity. The human mind made a vast effort to
achieve its freedom. Attempts at its emancipation from
Church authority had been made before, but they were stifled
and rendered abortive. The first great and successful revolt
against ecclesiastical authority was made in Germany. The
new desire for judging freely and independently in matters
of religion was preceded and accompanied by a growing spirit
of scientific inquiry. Thus it was that, for a time, Germany
led the van in science. She produced \textit{Regiomontanus}, \textit{Copernicus,
\index{Copernicus}%
\index{Regiomontanus!ref.\ to}%
Rhæticus}, \textit{Kepler}, and \textit{Tycho Brahe}, at a period when
\index{Brahe, Tycho}%
\index{Kepler!ref.\ to}%
\index{Rhæticus!ref.\ to}%
\index{Tycho Brahe}%
France and England had, as yet, brought forth hardly any
great scientific thinkers. This remarkable scientific productiveness
was no doubt due, to a great extent, to the commercial
prosperity of Germany. Material prosperity is an essential
condition for the progress of knowledge. As long as every
individual is obliged to collect the necessaries for his subsistence,
there can be no leisure for higher pursuits. At this
time, Germany had accumulated considerable wealth. The
%% -----File: 154.png---Folio 140-------
Hanseatic League commanded the trade of the North. Close
commercial relations existed between Germany and Italy.
Italy, too, excelled in commercial activity and enterprise.
We need only mention Venice, whose glory began with the crusades,
and Florence, with her bankers and her manufacturers
of silk and wool. These two cities became great intellectual
centres. Thus, Italy, too, produced men in art, literature, and
science, who shone forth in fullest splendour. In fact, Italy
was the fatherland of what is termed the Renaissance.

For the first great contributions to the mathematical sciences
we must, therefore, look to Italy and Germany. In Italy
brilliant accessions were made to algebra, in Germany to
\index{Algebra!Renaissance}%
astronomy and trigonometry.
\index{Trigonometry}%

On the threshold of this new era we meet in Germany with
the figure of John Mueller, more generally called \textbf{Regiomontanus}
\index{Regiomontanus}%
(1436--1476). Chiefly to him we owe the revival of
trigonometry. He studied astronomy and trigonometry at
Vienna under the celebrated George Purbach. The latter
\index{Purbach!ref.\ to}%
perceived that the existing Latin translations of the \textit{Almagest}
\index{Almagest!ref.\ to}%
were full of errors, and that Arabic authors had not remained
true to the Greek original. Purbach therefore began to make
a translation directly from the Greek. But he did not live to
finish it. His work was continued by Regiomontanus, who
went beyond his master. Regiomontanus learned the Greek
language from Cardinal Bessarion, whom he followed to Italy,
where he remained eight years collecting manuscripts from
Greeks who had fled thither from the Turks. In addition to
the translation of and the commentary on the \textit{Almagest}, he
prepared translations of the \textit{Conics} of Apollonius, of Archimedes,
\index{Apollonius!ref.\ to}%
\index{Archimedes!ref.\ to}%
and of the mechanical works of Heron. Regiomontanus
\index{Heron the Elder!ref.\ to}%
and Purbach adopted the Hindoo \emph{sine} in place of the Greek
\index{Sine}%
\emph{chord of double the arc}. The Greeks and afterwards the Arabs
divided the radius into $60$~equal parts, and each of these again
%% -----File: 155.png---Folio 141-------
\index{Sine}%
into $60$~smaller ones. The Hindoos expressed the length of
the radius by parts of the circumference, saying that of the
$21,600$ equal divisions of the latter, it took $3438$ to measure
the radius. Regiomontanus, to secure greater precision, constructed
\index{Regiomontanus}%
one table of sines on a radius divided into $600,000$
parts, and another on a radius divided decimally into $10,000,000$
divisions. He emphasised the use of the \emph{tangent} in trigonometry.
\index{Trigonometry}%
Following out some ideas of his master, he calculated
a table of tangents. German mathematicians were not the
\index{Tangents!in trigonometry}%
first Europeans to use this function. In England it was known
a century earlier to Bradwardine, who speaks of tangent (\textit{umbra
\index{Bradwardine!ref.\ to}%
recta}) and cotangent (\textit{umbra versa}), and to John Maudith.
\index{Cotangent}%
\index{Maudith!ref.\ to}%
Regiomontanus was the author of an arithmetic and also of
a complete treatise on trigonometry, containing solutions of
both plane and spherical triangles. The form which he gave
to trigonometry has been retained, in its main features, to the
present day.

Regiomontanus ranks among the greatest men that Germany
has ever produced. His complete mastery of astronomy and
mathematics, and his enthusiasm for them, were of far-reaching
influence throughout Germany. So great was his
reputation, that Pope Sixtus~IV. called him to Italy to
improve the calendar. Regiomontanus left his beloved city
\index{Calendar}%
of Nürnberg for Rome, where he died in the following year.

After the time of Purbach and Regiomontanus, trigonometry
and especially the calculation of tables continued to occupy
German scholars. More refined astronomical instruments were
made, which gave observations of greater precision; but these
would have been useless without trigonometrical tables of corresponding
accuracy. Of the several tables calculated, that
by \textit{Georg Joachim} of Feldkirch in Tyrol, generally called
\textbf{Rhæticus}, deserves special mention. He calculated a table of
\index{Rhæticus}%
sines with the radius $= 10,000,000,000$ and from $10''$ to~$10''$;
%% -----File: 156.png---Folio 142-------
\index{Cubic equations|(}%
and, later on, another with the radius $= 1,000,000,000,000,000$,
and proceeding from $10''$ to~$10''$. He began also the construction
of tables of tangents and secants, to be carried to
\index{Secants}%
\index{Tangents!in trigonometry}%
the same degree of accuracy; but he died before finishing them.
For twelve years he had had in continual employment several
calculators. The work was completed by his pupil, \textbf{Valentine
Otho}, in~1596. This was indeed a gigantic work,---a monument
\index{Otho}%
of German diligence and indefatigable perseverance.
The tables were republished in 1613 by \textbf{Pitiscus}, who spared
\index{Pitiscus}%
no pains to free them of errors. Astronomical tables of
so great a degree of accuracy had never been dreamed of
by the Greeks, Hindoos, or Arabs. That Rhæticus was not a
\index{Rhæticus!ref.\ to}%
ready calculator only, is indicated by his views on \DPtypo{trignometrical}{trigonometrical}
lines. Up to his time, the trigonometric functions had
been considered always with relation to the arc; he was the
first to construct the right triangle and to make them depend
directly upon its angles. It was from the right triangle that
Rhæticus got his idea of calculating the hypotenuse; \ie~he
was the first to plan a table of secants. Good work in trigonometry
was done also by Vieta and Romanus.
\index{Romanus!ref.\ to}%
\index{Vieta!ref.\ to}%

We shall now leave the subject of trigonometry to witness
the progress in the solution of algebraical equations. To do
so, we must quit Germany for Italy. The first comprehensive
algebra printed was that of Lucas Pacioli. He closes his
\index{Algebra!Renaissance|(}%
\index{Pacioli!ref.\ to}%
book by saying that the solution of the equations $x^{3} + mx = n$,
$x^{3} + n = mx$ is as impossible at the present state of science as
the quadrature of the circle. This remark doubtless stimulated
thought. The first step in the algebraic solution of
cubics was taken by \textbf{Scipio Ferro} (died 1526), a professor of
\index{Ferro, Scipio}%
mathematics at Bologna, who solved the equation $x^{3} + mx = n$.
Nothing more is known of his discovery than that he imparted
it to his pupil, \textit{Floridas}, in~1505. It was the practice in those
\index{Floridas}%
days and for two centuries afterwards to keep discoveries
%% -----File: 157.png---Folio 143-------
secret, in order to secure by that means an advantage over
rivals by proposing problems beyond their reach. This practice
gave rise to numberless disputes regarding the priority of
inventions. A second solution of cubics was given by \textit{Nicolo}
of Brescia (1506(?)--1557). When a boy of six, Nicolo was
so badly cut by a French soldier that he never again gained
the free use of his tongue. Hence he was called \textbf{Tartaglia},
\index{Tartaglia|(}%
\ie~the stammerer. His widowed mother being too poor to
pay his tuition in school, he learned to read and picked up a
knowledge of Latin, Greek, and mathematics by himself.
Possessing a mind of extraordinary power, he was able to
appear as teacher of mathematics at an early age. In~1530,
one Colla proposed him several problems, one leading to the
\index{Colla}%
equation $x^3 + px^{2} = q$. Tartaglia found an imperfect method
for solving this, but kept it secret. He spoke about his secret
in public and thus caused Ferro's pupil, Floridas, to proclaim
his own knowledge of the form $x^{3} + mx = n$. Tartaglia, believing
him to be a mediocrist and braggart, challenged him to a
public discussion, to take place on the 22d of February, 1535.
Hearing, meanwhile, that his rival had gotten the method
from a deceased master, and fearing that he would be beaten
in the contest, Tartaglia put in all the zeal, industry, and
skill to find the rule for the equations, and he succeeded in it
ten days before the appointed date, as he himself modestly
says.\Reference{7} The most difficult step was, no doubt, the passing from
quadratic irrationals, used in operating from time of old, to
cubic irrationals. Placing $x = \sqrt[3]{t} - \sqrt[3]{u}$, Tartaglia perceived
that the irrationals disappeared from the equation $x^3 + mx = n$,
making $n = t - u$. But this last equality, together with
$(\frac{1}{3}m)^3 = tu$, gives at once
\[
t = \sqrt{\left( \frac{n}{2} \right)^3 + \left( \frac{m}{3} \right)^3} + \frac{n}{2}, \quad
u = \sqrt{\left( \frac{n}{2} \right)^2 + \left( \frac{m}{2} \right)^3} - \frac{n}{2}.
\]
%% -----File: 158.png---Folio 144-------
This is Tartaglia's solution of $x^{3} + mx = n$. On the 13th~of
February, he found a similar solution for $x^{3} = mx + n$. The
contest began on the~22d. Each contestant proposed thirty
problems. The one who could solve the greatest number within
fifty days should be the victor. Tartaglia solved the thirty
problems proposed by Floridas in two hours; Floridas could
\index{Floridas}%
not solve any of Tartaglia's. From now on, Tartaglia studied
cubic equations with a will. In 1541 he discovered a general
solution for the cubic $x^{3} ± px^{2} = ± q$, by transforming it into
the form $x^{3} ± mx = ± n$. The news of Tartaglia's victory
spread all over Italy. Tartaglia was entreated to make known
his method, but he declined to do so, saying that after his
completion of the translation from the Greek of Euclid and
\index{Euclid!ref.\ to}%
Archimedes, he would publish a large algebra containing his
\index{Archimedes!ref.\ to}%
method. But a scholar from Milan, named \textbf{Hieronimo Cardano}
(1501--1576), after many solicitations, and after giving the
most solemn and sacred promises of secrecy, succeeded in
obtaining from Tartaglia a knowledge of his rules.

At this time Cardan was writing his \textit{Ars Magna}, and he
\index{Cardan}%
knew no better way to crown his work than by inserting the
much sought for rules for solving cubics. Thus Cardan broke
his most solemn vows, and published in 1545 in his \textit{Ars Magna}
Tartaglia's solution of cubics. Tartaglia became desperate.
His most cherished hope, of giving to the world an immortal
work which should be the monument of his deep learning and
power for original research, was suddenly destroyed; for the
crown intended for his work had been snatched away. His
first step was to write a history of his invention; but, to completely
annihilate his enemies, he challenged Cardan and his
pupil Lodovico Ferrari to a contest: each party should propose
\index{Ferrari!ref.\ to}%
thirty-one questions to be solved by the other within fifteen
days. Tartaglia solved most questions in seven days, but the
other party did not send in their solution before the expiration
%% -----File: 159.png---Folio 145-------
of the fifth month; moreover, all their solutions except one
were wrong. A replication and a rejoinder followed. Endless
were the problems proposed and solved on both sides. The
dispute produced much chagrin and heart-burnings to the parties,
and to Tartaglia especially, who met with many other
disappointments. After having recovered himself again, Tartaglia
\index{Tartaglia|)}%
began, in 1556, the publication of the work which he
had had in his mind for so long; but he died before he reached
the consideration of cubic equations. Thus the fondest wish
\index{Cubic equations|)}%
of his life remained unfulfilled; the man to whom we owe the
greatest contribution to algebra made in the sixteenth century
was forgotten, and his method came to be regarded as the discovery
of Cardan and to be called Cardan's solution.

Remarkable is the great interest that the solution of cubics
excited throughout Italy. It is but natural that after this
great conquest mathematicians should attack \Change{bi-quadratic}{biquadratic} equations.
As in the case of cubics, so here, the first impulse was
given by Colla, who, in 1540, proposed for solution the equation
\index{Colla}%
$x^{4} + 6x^{2} + 36 = 60x$. To be sure, Cardan had studied
particular cases as early as~1539. Thus he solved the equation
$13x^{2} = x^{4} + 2x^{3} + 2x + 1$ by a process similar to that employed
by Diophantus and the Hindoos; namely, by adding
to both sides $3x^{2}$ and thereby rendering both numbers
complete squares. But Cardan failed to find a general solution;
it remained for his pupil \textbf{Ferrari} to prop the reputation
\index{Ferrari}%
of his master by the brilliant discovery of the general
solution of \Change{bi-quadratic}{biquadratic} equations. Ferrari reduced Colla's
equation to the form $(x^{2}+ 6)^{2} = 60x + 6x^{2}$. In order to
give also the right member the form of a complete square
he added to both members the expression $2 (x^{2} + 6) y + y^{2}$,
containing a new unknown quantity~$y$. This gave him $(x^{2} + 6
+ y)^{2} = (6 + 2y) x^{2} + 60x + (12y + y^{2})$. The condition that
the right member be a complete square is expressed by the
%% -----File: 160.png---Folio 146-------
\index{Biquadratic equation}%
cubic equation $(2y + 6) (12y + y^{2}) = 900$. Extracting the
square root of the \Change{bi-quadratic}{biquadratic}, he got $x^{2} + 6 + y = x \sqrt{2y + 6}
+ \dfrac{900\Fstr}{\sqrt{2y + 6}}$. Solving the cubic for~$y$ and substituting, it remained
only to determine~$x$ from the resulting quadratic.
Ferrari pursued a similar method with other numerical \Change{bi-quadratic}{biquadratic}
equations.\Reference{7} Cardan had the pleasure of publishing
this discovery in his \textit{Ars Magna} in~1545. Ferrari's solution
is sometimes ascribed to \textit{Bombelli}, but he is no more the discoverer
\index{Bombelli}%
of it than Cardan is of the solution called by his
name.

To Cardan algebra is much indebted. In his \textit{Ars Magna}
he takes notice of negative roots of an equation, calling them
\index{Negative roots}%
\emph{fictitious}, while the positive roots are called \emph{real}. Imaginary
\index{Imaginary quantities}%
roots he does not consider; cases where they appear he calls
impossible. Cardan also observed the difficulty in the irreducible
case in the cubics, which, like the quadrature of the
circle, has since ``so much tormented the perverse ingenuity of
mathematicians.'' But he did not understand its nature. It remained
for \textbf{Raphael Bombelli} of Bologna, who published in 1572
an algebra of great merit, to point out the reality of the apparently
imaginary expression which the root assumes, and thus
to lay the foundation of a more intimate knowledge of imaginary
quantities.

After this brilliant success in solving equations of the third
and fourth degrees, there was probably no one who doubted,
that with aid of irrationals of higher degrees, the solution of
equations of any degree whatever could be found. But all
attempts at the algebraic solution of the quintic were fruitless,
and, finally, Abel demonstrated that all hopes of finding algebraic
\index{Abel!ref.\ to}%
solutions to equations of higher than the fourth degree
were purely Utopian.

Since no solution by radicals of equations of higher degrees
%% -----File: 161.png---Folio 147-------
\index{Equations!numerical}%
\index{Falsa positio}%
could be found, there remained nothing else to be done than
the devising of rules by which at least the numerical values of
the roots could be ascertained. Cardan applied the Hindoo
rule of ``false position'' (called by him \textit{regula aurea}) to the
cubic, but this mode of approximating was exceedingly rough.
An incomparably better method was invented by \textbf{Franciscus
Vieta}, a French mathematician, whose transcendent genius
\index{Vieta}%
enriched mathematics with several important innovations.
Taking the equation $f(x) = Q$, wherein $f(x)$ is a polynomial
containing different powers of~$x$, with numerical coefficients,
and $Q$ is a given number, Vieta first substitutes in~$f(x)$ a
known approximate value of the root, and then shows that
another figure of the root can be obtained by division. A repetition
of the same process gives the next figure of the root,
and so on. Thus, in $x^2 + 14x = 7929$, taking $80$ for the approximate
root, and placing $x = 80 + b$, we get
\begin{DPalign*}
&(80 + b)^2 + 14 (80 + b) = 7929,\\
\lintertext{or} &174b + b^2 = 409.
\end{DPalign*}
Since $174b$ is much greater than~$b^2$, we place $174b = 409$, and
obtain thereby $b = 2$. Hence the second approximation is~$82$.
Put $x = 82 + c$, then $(82 + c)^2 + 14(82 + c) = 7929$, or $178 c + c^2
= 57$. As before, place $178c = 57$, then $c = .3$, and the third
approximation gives $82.3$. Assuming $x = 82.3 + d$, and substituting,
gives $178.6 d + d^2 = 3.51$, and $178.6d = 3.51$, $\therefore d = .01$;
giving for the fourth approximation $82.31$. In the same way,
$e = .009$, and the value for the root of the given equation is
$82.319 \dots$. For this process, Vieta was greatly admired by his
contemporaries. It was employed by Harriot, Oughtred, Pell,
\index{Harriot!ref.\ to}%
\index{Oughtred!ref.\ to}%
\index{Pell}%
and others. Its principle is identical with the main principle
involved in the methods of approximation of Newton and
\index{Newton!ref.\ to}%
Horner. The only change lies in the arrangement of the
\index{Horner}%
%% -----File: 162.png---Folio 148-------
work. This alteration was made to afford facility and security
in the process of evolution of the root.

We pause a moment to sketch the life of Vieta, the most
eminent French mathematician of the sixteenth century. He
was born in Poitou in 1540, and died in 1603 at Paris. He
was employed throughout life in the service of the state,
under Henry~III. and Henry~IV\@. He was, therefore, not a
mathematician by profession, but his love for the science was
so great that he remained in his chamber studying, sometimes
several days in succession, without eating and sleeping more
than was necessary to sustain himself. So great devotion to
abstract science is the more remarkable, because he lived at a
time of incessant political and religious turmoil. During the
war against Spain, Vieta rendered service to Henry~IV. by
deciphering intercepted letters written in a species of cipher,
and addressed by the Spanish Court to their governor of Netherlands.
The Spaniards attributed the discovery of the key
to magic.

An ambassador from Netherlands once told Henry~IV. that
France did not possess a single geometer capable of solving a
problem propounded to geometers by a Belgian mathematician,
Adrianus Romanus. It was the solution of the equation of
\index{Romanus!ref.\ to}%
the forty-fifth degree:---
\[
45y - 3795y^3 + 95634y^5 - \dotsb + 945y^{41} - 45y^{43} + y^{45} = C.
\]
Henry~IV. called Vieta, who, having already pursued similar
investigations, saw at once that this awe-inspiring problem
was simply the equation by which $C=2\sin\phi$ was expressed
in terms of $y = 2 \sin \frac{1}{45}\phi$; that, since $45 = 3·3·5$, it was
necessary only to divide an angle once into 5~equal parts, and
then twice into~3,---a division which could be effected by corresponding
equations of the fifth and third degrees. Brilliant
was the discovery by Vieta of 23~roots to this equation, instead
%% -----File: 163.png---Folio 149-------
\index{Biquadratic equation}%
\index{Cubic equations}%
\index{Negative roots}%
\index{Notation!in algebra}%
of only one. The reason why he did not find 45~solutions, is
that the remaining ones involve negative sines, which were
unintelligible to him. Detailed investigations on the famous
old problem of the section of an angle into an odd number of
equal parts, led Vieta to the discovery of a trigonometrical
solution of Cardan's irreducible case in cubics. He applied
\index{Cardan!ref.\ to}%
the equation $\bigl(2 \cos \frac{1}{3} \phi\bigr)^3 - 3 \bigl(2 \cos \frac{1}{3} \phi\bigr) = 2 \cos \phi$ to the solution
of $x^3 - 3a^{2}x = a^{2}b$, when $a > \frac{1}{2} b$, by placing $x = 2a \cos \frac{1}{3}\phi$, and
determining~$\phi$ from $b = 2 a \cos \phi$.

The main principle employed by him in the solution of
equations is that of \emph{reduction}. He solves the quadratic by
\index{Equations!solution of}%
making a suitable substitution which will remove the term
containing $x$ to the first degree. Like Cardan, he reduces the
general expression of the cubic to the form $x^{3}+ mx + n = 0$;
then, assuming $x = \bigl(\frac{1}{3}a - z^2\bigr) ÷ z$ and substituting, he gets
$z^6 - bz^{3} - \frac{1}{27} a^3 = 0$. Putting $z^3 = y$, he has a quadratic. In
the solution of \Change{bi-quadratics}{biquadratics}, Vieta still remains true to his
principle of reduction. This gives him the well-known cubic
resolvent. He thus adheres throughout to his favourite principle,
and thereby introduces into algebra a uniformity of
method which claims our lively admiration. In Vieta's algebra
we discover a partial knowledge of the relations existing
between the coefficients and the roots of an equation. He
shows that if the coefficient of the second term in an equation
of the second degree is minus the sum of two numbers whose
product is the third term, then the two numbers are roots of
the equation. Vieta rejected all except positive roots; hence
it was impossible for him to fully perceive the relations in
question.

The most epoch-making innovation in algebra due to Vieta
is the denoting of general or indefinite quantities by letters
of the alphabet. To be sure, Regiomontanus and Stifel in
\index{Regiomontanus!ref.\ to}%
\index{Stifel!ref.\ to}%
Germany, and Cardan in Italy, used letters before him, but
%% -----File: 164.png---Folio 150-------
\index{Plus and minus, signs for}%
Vieta extended the idea and first made it an essential part of
algebra. The new algebra was called by him \textit{logistica speciosa}
in distinction to the old \textit{logistica numerosa}. Vieta's formalism
differed considerably from that of to-day. The equation
$a^3 + 3 a^2b + 3 ab^2 + b^3 = (a + b)^3$ was written by him ``$a \text{ cubus } + b
\text{ in } a \text{ quadr.\ } 3 + a \text{ in } b \text{ quadr.\ } 3 + b \text{ cubo æqualia } \overline{a + b} \text{ cubo}$.''
In numerical equations the unknown quantity was denoted by~$N$,
its square by~$Q$, and its cube by~$C$. Thus the equation
$x^3 - 8x^2 +16x = 40$ was written $1C - 8Q + 16N \textit{ æqual.\ } 40$.
Observe that exponents and our symbol~($=$) for equality were
not yet in use; but that Vieta employed the Maltese cross~($+$)
as the short-hand symbol for addition, and the~($-$) for
subtraction. These two characters had not been in general
use before the time of Vieta. ``It is very singular,'' says
Hallam, ``that discoveries of the greatest convenience, and,
apparently, not above the ingenuity of a village schoolmaster,
should have been overlooked by men of extraordinary acuteness
like Tartaglia, Cardan, and Ferrari; and hardly less so
that, by dint of that acuteness, they dispensed with the aid of
these contrivances in which we suppose that so much of the
utility of algebraic expression consists.'' Even after improvements
in notation were once proposed, it was with extreme
\index{Notation!in algebra}%
slowness that they were admitted into general use. They
were made oftener by accident than design, and their authors
had little notion of the effect of the change which they were
making. The introduction of the $+$ and $-$ symbols seems
to be due to the Germans, who, although they did not enrich
algebra during the Renaissance with great inventions, as did
\index{Algebra!Renaissance|)}%
the Italians, still cultivated it with great zeal. The arithmetic
\index{Arithmetic!Renaissance}%
of \textbf{John Widmann}, printed \AD~1489 in Leipzig, is the
\index{Widmann}%
earliest book in which the $+$ and $-$ symbols have been found.
There are indications leading us to surmise that they were in
use first among merchants. They occur again in the arithmetic
%% -----File: 165.png---Folio 151-------
\index{Arithmetic!Renaissance}%
\index{Notation!in algebra}%
of Grammateus, a teacher at the University of Vienna.
\index{Grammateus}%
His pupil, Christoff Rudolff, the writer of the first text-book
\index{Rudolff}%
on algebra in the German language (printed in~1525), employs
these symbols also. So did Stifel, who brought out a second
\index{Stifel!ref.\ to}%
edition of Rudolff's \textit{Coss} in~1553. Thus, by slow degrees,
their adoption became universal. There is another short-hand
symbol of which we owe the origin to the Germans. In a
manuscript published sometime in the fifteenth century, a dot
placed before a number is made to signify the extraction of a
root of that number. This dot is the embryo of our present
symbol for the square root. Christoff Rudolff, in his algebra,
remarks that ``the radix quadrata is, for brevity, designated
in his algorithm with the character $\sqrt{\vphantom{4}}$, as~$\sqrt{4}$.'' Here the dot
has grown into a symbol much like our own. This same
symbol was used by \textit{Michael Stifel}. Our sign of equality is due
to \textbf{Robert Recorde} (1510--1558), the author of \textit{The Whetstone of
\index{Recorde}%
Witte} (1557), which is the first English treatise on algebra.
He selected this symbol because no two things could be more
equal than two parallel lines~$=$. The sign~$÷$ for division was
first used by \textit{Johann Heinrich Rahn}, a Swiss, in~1659, and
\index{Rahn}%
was introduced in England by \textit{John Pell} in~1668.
\index{Pell}%

\textbf{Michael Stifel} (1486?--1567), the greatest German algebraist
\index{Stifel}%
of the sixteenth century, was born in Esslingen, and died in
Jena. He was educated in the monastery of his native place,
and afterwards became Protestant minister. The study of the
significance of mystic numbers in Revelation and in Daniel
drew him to mathematics. He studied German and Italian
works, and published in~1544, in Latin, a book entitled
\textit{Arithmetica integra}. Melanchthon wrote a preface to it. Its
three parts treat respectively of rational numbers, irrational
numbers, and algebra. Stifel gives a table containing the numerical
values of the binomial coefficients for powers below the~18th.
He observes an advantage in letting a geometric progression
%% -----File: 166.png---Folio 152-------
\index{Tartaglia!ref.\ to}%
correspond to an arithmetical progression, and arrives at
the designation of integral powers by numbers. Here are the
germs of the theory of exponents. In 1545 Stifel published
\index{Exponents}%
an arithmetic in German. His edition of Rudolff's \textit{Coss} contains
\index{Coss, term for algebra}%
rules for solving cubic equations, derived from the\DPnote{** PP: Removed duplicate `the'.}
\index{Cubic equations}%
writings of Cardan.
\index{Cardan!ref.\ to}%

We remarked above that Vieta discarded negative roots of
\index{Negative roots}%
\index{Vieta!ref.\ to}%
equations. Indeed, we find few algebraists before and during
the Renaissance who understood the significance even of
negative quantities. Fibonacci seldom uses them. Pacioli
\index{Negative quantities}%
\index{Pacioli!ref.\ to}%
states the rule that ``minus times minus gives plus,'' but
applies it really only to the development of the product of
$(a - b) (c - d)$; purely negative quantities do not appear in
his work. The great German ``Cossist'' (algebraist), \textit{Michael
Stifel}, speaks as early as 1544 of numbers which are ``absurd''
or ``fictitious below zero,'' and which arise when ``real numbers
above zero'' are subtracted from zero. Cardan, at last, speaks
of a ``pure minus''; ``but these ideas,'' says Hankel, ``remained
sparsely, and until the beginning of the seventeenth century,
mathematicians dealt exclusively with absolute positive quantities.''
The first algebraist who occasionally places a purely
negative quantity by itself on one side of an equation, is
\textit{Harriot} in England. As regards the recognition of negative
\index{Harriot!ref.\ to}%
roots, Cardan and Bombelli were far in advance of all writers
\index{Bombelli!ref.\ to}%
of the Renaissance, including Vieta. Yet even they mentioned
these so-called false or fictitious roots only in passing, and
without grasping their real significance and importance. On
this subject Cardan and Bombelli had advanced to about the
same point as had the Hindoo Bhaskara, who saw negative roots,
\index{Bhaskara!ref.\ to}%
but did not approve of them. The generalisation of the conception
of quantity so as to include the negative, was an
exceedingly slow and difficult process in the development of
algebra.
\index{Algebra!Renaissance}%
%% -----File: 167.png---Folio 153-------
\index{Cubic equations}%
\index{Equations!solution of}%
\index{Trisection of angles}%

We shall now consider the history of geometry during the
\index{Geometry!Renaissance}%
Renaissance. Unlike algebra, it made hardly any progress.
The greatest gain was a more intimate knowledge of Greek
geometry. No essential progress was made before the time of
Descartes. Regiomontanus, Xylander of Augsburg, Tartaglia,
\index{Regiomontanus!ref.\ to}%
\index{Tartaglia!ref.\ to}%
\index{Xylander}%
Commandinus of Urbino in Italy, Maurolycus, and others,
\index{Commandinus}%
\index{Maurolycus}%
made translations of geometrical works from the Greek. \textbf{John
Werner} of Nürnberg published in 1522 the first work on
\index{Werner}%
conics which appeared in Christian Europe. Unlike the
geometers of old, he studied the sections in relation with the
cone, and derived their properties directly from it. This mode
of studying the conics was followed by \textbf{Maurolycus} of Messina
(1494--1575). The latter is, doubtless, the greatest geometer
of the sixteenth century. From the notes of Pappus, he
\index{Pappus!ref.\ to}%
attempted to restore the missing fifth book of Apollonius
\index{Apollonius!ref.\ to}%
on \textit{maxima} and \textit{minima}. His chief work is his masterly
and original treatment of the conic sections, wherein he discusses
\index{Conic sections!Renaissance}%
tangents and asymptotes more fully than Apollonius
had done, and applies them to various physical and astronomical
problems.

The foremost geometrician of Portugal was \textbf{Nonius}; of
\index{Nonius}%
France, before Vieta, was \textbf{Peter Ramus}, who perished in the
\index{Ramus}%
\index{Vieta!ref.\ to}%
massacre of St.~Bartholomew. \textit{Vieta} possessed great familiarity
with ancient geometry. The new form which he gave
to algebra, by representing general quantities by letters, enabled
him to point out more easily how the construction of
the roots of cubics depended upon the celebrated ancient problems
of the duplication of the cube and the trisection of an
\index{Duplication of the cube}%
angle. He reached the interesting conclusion that the former
problem includes the solutions of all cubics in which the radical
in Tartaglia's formula is real, but that the latter problem
includes only those leading to the irreducible case.

The problem of the quadrature of the circle was revived in
%% -----File: 168.png---Folio 154-------
\index{Apollonius!ref.\ to}%
\index{Gregorian Calendar}%
this age, and was zealously studied even by men of eminence
and mathematical ability. The army of circle-squarers became
\index{Circle}%
most formidable during the seventeenth century. Among the
first to revive this problem was the German Cardinal \textbf{Nicolaus
Cusanus} (died 1464), who had the reputation of being a great
\index{Cusanus}%
logician. His fallacies were exposed to full view by Regiomontanus.
\index{Regiomontanus!ref.\ to}%
As in this case, so in others, every quadrator of
note raised up an opposing mathematician: Orontius was met
\index{Orontius}%
by Buteo and Nonius; Joseph Scaliger by Vieta, Adrianus
\index{Buteo}%
\index{Nonius!ref.\ to}%
\index{Scaliger}%
\index{Vieta!ref.\ to}%
Romanus, and Clavius; A. Quercu by Peter Metius. Two
\index{Clavius!ref.\ to}%
\index{Metius}%
\index{Quercu, a}%
\index{Romanus}%
\index{Romanus!ref.\ to}%
mathematicians of Netherlands, \textbf{Adrianus Romanus} and \textbf{Ludolph
\index{Ludolph}%
van Ceulen}, occupied themselves with approximating to the
ratio between the circumference and the diameter. The former
carried the value~$\pi$ to~$15$, the latter to~$35$, places. The
\index{p@{$\pi$}: values for!Ludolph's}%
value of~$\pi$ is therefore often named ``Ludolph's number.'' His
\index{Ludolph's number}%
performance was considered so extraordinary, that the numbers
were cut on his tomb-stone in St.~Peter's church-yard, at
Leyden. Romanus was the one who propounded for solution
that equation of the forty-fifth degree solved by Vieta. On
receiving Vieta's solution, he at once departed for Paris, to
make his acquaintance with so great a master. Vieta proposed
to him the Apollonian problem, to draw a circle touching
\index{Apollonian Problem}%
three given circles. ``Adrianus Romanus solved the problem
by the intersection of two hyperbolas; but this solution did not
possess the rigour of the ancient geometry. Vieta caused him
\index{Geometry!Renaissance}%
to see this, and then, in his turn, presented a solution which
had all the rigour desirable.''\Reference{25} Romanus did much toward
simplifying spherical trigonometry by reducing, by means of
\index{Trigonometry}%
certain projections, the $28$~cases in triangles then considered
to only six.

Mention must here be made of the improvements of the
Julian calendar. The yearly determination of the movable
\index{Calendar}%
feasts had for a long time been connected with an untold
%% -----File: 169.png---Folio 155-------
amount of confusion. The rapid progress of astronomy led
to the consideration of this subject, and many new calendars
were proposed. Pope Gregory~XIII. convoked a large number
of mathematicians, astronomers, and prelates, who decided
upon the adoption of the calendar proposed by the Jesuit
\textbf{Lilius Clavius}. To rectify the errors of the Julian calendar
\index{Clavius}%
it was agreed to write in the new calendar the 15th~of
October immediately after the 4th~of October of the year
1582. The Gregorian calendar met with a great deal of opposition
both among scientists and among Protestants. Clavius,
who ranked high as a geometer, met the objections of the
former most ably and effectively; the prejudices of the latter
passed away with time.

The passion for the study of mystical properties of numbers
descended from the ancients to the moderns. Much was
written on numerical mysticism even by such eminent men
as Pacioli and Stifel. The \textit{Numerorum Mysteria} of Peter
\index{Pacioli!ref.\ to}%
\index{Stifel!ref.\ to}%
Bungus covered $700$~quarto pages. He worked with great
\index{Bungus}%
industry and satisfaction on~$666$, which is the number of the
beast in Revelation (xiii.~18), the symbol of Antichrist. He
reduced the name of the `impious' Martin Luther to a form
which may express this formidable number. Placing $a = 1$,
$b = 2$,~etc, $k = 10$, $l = 20$,~etc., he finds, after misspelling the
name, that
$\text{M}_{(30)} \text{A}_{(1)} \text{R}_{(80)} \text{T}_{(100)} \text{I}_{(9)} \text{N}_{(40)}$
$\text{L}_{(20)} \text{V}_{(200)} \text{T}_{(100)} \text{E}_{(5)} \text{R}_{(80)} \text{A}_{(1)}$
constitutes the number required. These attacks on the great
reformer were not unprovoked, for his friend, Michael Stifel,
the most acute and original of the early mathematicians of
Germany, exercised an equal ingenuity in showing that the
above number referred to Pope Leo~X.,---a demonstration
which gave Stifel unspeakable comfort.\Reference{22}

Astrology also was still a favourite study. It is well
\index{Astrology}%
known that Cardan, Maurolycus, Regiomontanus, and many
\index{Cardan!ref.\ to}%
\index{Maurolycus!ref.\ to}%
\index{Regiomontanus!ref.\ to}%
other eminent scientists who lived at a period even later than
%% -----File: 170.png---Folio 156-------
\index{Renaissance|)}%
this, engaged in deep astrological study; but it is not so generally
known that besides the occult sciences already named,
men engaged in the mystic study of star-polygons and magic
\index{Star-polygons}%
squares. ``The pentagramma gives you pain,'' says Faust to
Mephistopheles. It is of deep psychological interest to see
scientists, like the great Kepler, demonstrate on one page a
\index{Kepler!ref.\ to}%
theorem on star-polygons, with strict geometric rigour, while
on the next page, perhaps, he explains their use as amulets
or in conjurations.\Reference{1} Playfair, speaking of Cardan as an astrologer,
\index{Cardan!ref.\ to}%
\index{Playfair}%
calls him ``a melancholy proof that there is no folly
or weakness too great to be united to high intellectual attainments.''\Reference{26}
Let our judgment not be too harsh. The period
under consideration is too near the Middle Ages to admit of
complete emancipation from mysticism even among scientists.
Scholars like Kepler, Napier, Albrecht \Change{Duerer}{Dürer}, while in the
\index{Durer@{Dürer, A.}}%
\index{Napier, J.!ref.\ to}%
van of progress and planting one foot upon the firm ground
of truly scientific inquiry, were still resting with the other
foot upon the scholastic ideas of preceding ages.


\Section{Vieta to Descartes}

The ecclesiastical power, which in the ignorant ages was an
unmixed benefit, in more enlightened ages became a serious
evil. Thus, in France, during the reigns preceding that of
Henry~IV., the theological spirit predominated. This is painfully
shown by the massacres of Vassy and of St.~Bartholomew.
Being engaged in religious disputes, people had no
leisure for science and for secular literature. Hence, down
to the time of Henry~IV., the French ``had not put forth a
single work, the destruction of which would now be a loss to
Europe.'' In England, on the other hand, no religious wars
were waged. The people were comparatively indifferent about
%% -----File: 171.png---Folio 157-------
\index{Menelaus!ref.\ to}%
religious strifes; they concentrated their ability upon secular
matters, and acquired, in the sixteenth century, a literature
which is immortalised by the genius of Shakespeare and
Spenser. This great literary age in England was followed
by a great scientific age. At the close of the sixteenth century,
the shackles of ecclesiastical authority were thrown off
by France. The ascension of Henry~IV. to the throne was
followed in 1598 by the Edict of Nantes, granting freedom
of worship to the Huguenots, and thereby terminating religious
wars. The genius of the French nation now began to blossom.
Cardinal Richelieu, during the reign of Louis~XIII., pursued
the broad policy of not favouring the opinions of any sect, but
of promoting the interests of the nation. His age was remarkable
for the progress of knowledge. It produced that
great secular literature, the counterpart of which was found
in England in the sixteenth century. The seventeenth century
was made illustrious also by the great French mathematicians,
Roberval, Descartes, Desargues, Fermat, and Pascal.

More gloomy is the picture in Germany. The great changes
which revolutionised the world in the sixteenth century, and
which led England to national greatness, led Germany to
degradation. The first effects of the Reformation there were
salutary. At the close of the fifteenth and during the sixteenth
century, Germany had been conspicuous for her scientific
pursuits. She had been the leader in astronomy and
trigonometry. Algebra also, excepting for the discoveries in
cubic equations, was, before the time of Vieta, in a more
advanced state there than elsewhere. But at the beginning
of the seventeenth century, when the sun of science began to
rise in France, it set in Germany. Theologic disputes and
religious strife ensued. The Thirty Years' War (1618--1648)
proved ruinous. The German empire was shattered, and
became a mere lax confederation of petty despotisms. Commerce
%% -----File: 172.png---Folio 158-------
was destroyed; national feeling died out. Art disappeared,
and in literature there was only a slavish imitation
of French artificiality. Nor did Germany recover from this
low state for $200$~years; for in 1756 began another struggle,
the Seven Years' War, which turned Prussia into a wasted
land. Thus it followed that at the beginning of the seventeenth
century, the great Kepler was the only German mathematician
\index{Kepler!ref.\ to}%
of eminence, and that in the interval of $200$~years
between Kepler and Gauss, there arose no great mathematician
\index{Gauss!ref.\ to}%
in Germany excepting Leibniz.
\index{Leibniz!ref.\ to}%

Up to the seventeenth century, mathematics was cultivated
but little in Great Britain. During the sixteenth century, she
brought forth no mathematician comparable with Vieta, Stifel,
or Tartaglia. But with the time of Recorde, the English
\index{Recorde!ref.\ to}%
became conspicuous for numerical skill. The first important
arithmetical work of English authorship was published in
Latin in 1522 by \textbf{Cuthbert Tonstall} (1474--1559). He had
\index{Tonstall}%
studied at Oxford, Cambridge, and Padua, and drew freely
from the works of Pacioli and Regiomontanus. Reprints of
\index{Pacioli!ref.\ to}%
\index{Regiomontanus!ref.\ to}%
his arithmetic appeared in England and France. After
\index{Arithmetic!Renaissance|(}%
Recorde the higher branches of mathematics began to be
studied. Later, Scotland brought forth Napier, the inventor
of logarithms. The instantaneous appreciation of their value
\index{Logarithms}%
is doubtless the result of superiority in calculation. In Italy,
and especially in France, geometry, which for a long time had
\index{Geometry!Renaissance}%
been an almost stationary science, began to be studied with
success. Galileo, Torricelli, Roberval, Fermat, Desargues,
Pascal, Descartes, and the English Wallis are the great revolutioners
of this science. Theoretical mechanics began to be
\index{Mechanics!Stevin and Galileo}%
studied. The foundations were laid by Fermat and Pascal
for the theory of numbers and the theory of probability.
\index{Probability}%

We shall first consider the improvements made in the art
of calculating. The nations of antiquity experimented thousands
%% -----File: 173.png---Folio 159-------
\index{Arabic numerals and notation}%
of years upon numeral notations before they happened
to strike upon the so-called ``Arabic notation.'' In the simple
\index{Notation!Arabic notation}%
expedient of the cipher, which was introduced by the Hindoos
about the fifth or sixth century after Christ, mathematics received
one of the most powerful impulses. It would seem that
after the ``Arabic notation'' was once thoroughly understood,
decimal fractions would occur at once as an obvious extension
\index{Decimal fractions|(}%
\index{Fractions!decimal}%
\index{Fractions!continued}%
of it. But ``it is curious to think how much science had
attempted in physical research and how deeply numbers had
been pondered, before it was perceived that the all-powerful
simplicity of the `Arabic notation' was as valuable and as
manageable in an infinitely descending as in an infinitely
ascending progression.''\Reference{28} Simple as decimal fractions appear
to us, the invention of them is not the result of one mind or
even of one age. They came into use by almost imperceptible
degrees. The first mathematicians identified with their history
did not perceive their true nature and importance, and
failed to invent a suitable notation. The idea of decimal
fractions makes its first appearance in methods for approximating
to the square roots of numbers. Thus John of Seville,
\index{John of Seville}%
presumably in imitation of Hindoo rules, adds $2~n$~ciphers
to the number, then finds the square root, and takes this
\index{Square root}%
as the numerator of a fraction whose denominator is 1 followed
by $n$~ciphers. The same method was followed by
\textit{Cardan}, but it failed to be generally adopted even by his
\index{Cardan!ref.\ to}%
Italian contemporaries; for otherwise it would certainly have
been at least mentioned by \textit{Cataldi} (died 1626) in a work
\index{Cataldi}%
devoted exclusively to the extraction of roots. Cataldi finds
the square root by means of continued fractions---a method
\index{Continued fractions}%
ingenious and novel, but for practical purposes inferior to
Cardan's. \textbf{Orontius Finaeus} (died 1555) in France, and \textbf{William
\index{Finæus}%
Buckley} (died about 1550) in England extracted the
\index{Buckley}%
square root in the same way as Cardan and John of Seville.
%% -----File: 174.png---Folio 160-------
The invention of decimals is frequently attributed to Regiomontanus,
\index{Regiomontanus!ref.\ to}%
on the ground that instead of placing the sinus
totus, in trigonometry, equal to a multiple of~$60$, like the
\index{Trigonometry}%
Greeks, he put it $= 100,000$. But here the trigonometrical
lines were expressed in \emph{integers}, and not in fractions. Though
\index{Fractions!decimal}%
he adopted a decimal division of the radius, he and his successors
did not apply the idea outside of trigonometry and,
indeed, had no notion whatever of decimal \emph{fractions}. To
\textbf{Simon Stevin} of Bruges in Belgium (1548--1620), a man who
\index{Stevin}%
did a great deal of work in most diverse fields of science, we
owe the first systematic treatment of decimal fractions. In
his \textit{La Disme} (1585) he describes in very express terms the
advantages, not only of decimal fractions, but also of the
decimal division in systems of weights and measures. Stevin
applied the new fractions ``to all the operations of ordinary
arithmetic.''\Reference{25} What he lacked was a suitable notation. In
\index{Notation!in algebra}%
\index{Notation!decimal fractions}%
place of our decimal point, he used a cipher; to each place in
the fraction was attached the corresponding index. Thus, in
his notation, the number $5.912$ would be
$\overset{0}{5}\overset{1}{9}\overset{2}{1}\overset{3}{2}$ or
$5\CircleDigit{0}9\CircleDigit{1}1\CircleDigit{2}2\CircleDigit{3}$.
These indices, though cumbrous in practice, are of interest,
because they are the germ of an important innovation. To
Stevin belongs the honour of inventing our present mode of
designating powers and also of introducing fractional exponents
\index{Exponents}%
into algebra. Strictly speaking, this had been done much
earlier by \textit{Oresme}, but it remained wholly unnoticed. Not even
\index{Oresme!ref.\ to}%
Stevin's innovations were immediately appreciated or at once
accepted, but, unlike Oresme's, they remained a secure possession.
No improvement was made in the notation of decimals
till the beginning of the seventeenth century. After Stevin,
decimals were used by \textbf{Joost Bürgi}, a Swiss by birth, who prepared
\index{Burgi@{Bürgi}}%
a manuscript on arithmetic soon after 1592, and by
\textbf{Johann Hartmann Beyer}, who assumes the invention as his own.
\index{Beyer}%
In~1603, he published at Frankfurt on the Main a \textit{Logistica}
%% -----File: 175.png---Folio 161-------
\index{Arithmetic!Renaissance|)}%
\index{Trigonometry}%
\textit{Decimalis}. With Bürgi, a zero placed underneath the digit
in unit's place answers as sign of separation. Beyer's notation
resembles Stevin's. The decimal point, says Peacock, is
\index{Decimal point}%
\index{Peacock!ref.\ to, X}%
due to Napier, who in 1617 published his \textit{Rabdologia}, containing
\index{Napier, J.!ref.\ to}%
a treatise on decimals, wherein the decimal point is used
in one or two instances. In the English translation of Napier's
\textit{Mirifici logarithmorum canonis descriptio}, executed by Edward
Wright in 1616, and corrected by the author, the decimal
point occurs in the tables. There is no mention of decimals
in English arithmetics between 1619 and~1631. \textit{Oughtred} in
\index{Oughtred!ref.\ to}%
1631 designates the fraction $.56$ thus, $\olddecimal{0}{56}$. \textit{Albert Girard},
\index{Girard!ref.\ to}%
a pupil of Stevin, in 1629 uses the point on one occasion.
John Wallis in 1657 writes \olddecimal{12}{345}, but afterwards in his
\index{Wallis!ref.\ to}%
algebra adopts the usual point. De Morgan says that ``to the
\index{Demorgan@{De Morgan}!ref.\ to}%
first quarter of the eighteenth century we must refer not only
the complete and final victory of the decimal point, but also
that of the now universal method of performing the operations
of division and extraction of the square root.\Reference{27} We have
dwelt at some length on the progress of the decimal notation,
because ``the history of language \ldots is of the highest order
of interest, as well as utility: its suggestions are the best
lesson for the future which a reflecting mind can have.''\Reference{27}

The miraculous powers of modern calculation are due to
three inventions: the Arabic Notation, Decimal Fractions, and
\index{Decimal fractions|)}%
Logarithms. The invention of logarithms in the first quarter
\index{Logarithms|(}%
of the seventeenth century was admirably timed, for Kepler
\index{Kepler!ref.\ to}%
was then examining planetary orbits, and Galileo had just
\index{Galileo!ref.\ to}%
turned the telescope to the stars. During the Renaissance
German mathematicians had constructed trigonometrical
tables of great accuracy, but this greater precision enormously
increased the work of the calculator. It is no exaggeration
to say that the invention of logarithms ``by shortening the
labours doubled the life of the astronomer.'' Logarithms were
%% -----File: 176.png---Folio 162-------
\index{Euclid!ref.\ to}%
invented by \textbf{John Napier}, Baron of Merchiston, in Scotland
\index{Napier, J.}%
(1550--1617). It is one of the greatest curiosities of the
history of science that Napier constructed logarithms before
exponents were used. To be sure, Stifel and Stevin made
\index{Exponents}%
\index{Stevin!ref.\ to}%
\index{Stifel!ref.\ to}%
some attempts to denote powers by indices, but this notation
was not generally known,---not even to \textit{Harriot}, whose algebra
\index{Harriot!ref.\ to}%
appeared long after Napier's death. That logarithms flow
naturally from the exponential symbol was not observed until
much later. It was Euler who first considered logarithms as
being indices of powers. What, then, was Napier's line of
thought?

Let $AB$ be a definite line, $DE$ a line extending from $D$ indefinitely.
Imagine two points starting at the same %[** PP: Width-dependent line break]
%[**Illustration: ]
\begin{wrapfigure}{l}{2.5in}
\Graphic{2.5in}{176a}
\end{wrapfigure}
moment;
the one moving from
$A$ toward~$B$, the other
from $D$ toward~$E$. Let
the velocity during the
first moment be the
same for both: let that of the point on line~$DE$ be uniform;
but the velocity of the point on $AB$ decreasing in such a way
that when it arrives at any point~$C$, its velocity is proportional
to the remaining distance~$BC$. While the first point moves
over a distance~$AC$, the second one moves over a distance~$DF$.
Napier calls $DF$ the logarithm of~$BC$.

Napier's process is so unique and so different from all other
modes of presenting the subject that there cannot be the
shadow of a doubt that this invention is entirely his own; it
is the result of unaided, isolated speculation. He first sought
the logarithms only of sines; the line~$AB$ was the sine of~$90°$
and was taken $= 10^7$; $BC$ was the sine of the arc, and $DF$ its
logarithm. We notice that as the motion proceeds, $BC$ decreases
in geometrical progression, while $DF$ increases in
arithmetical progression. Let $AB = a = 10^7$, let $x = DF$,
%% -----File: 177.png---Folio 163-------
$y = BC$, then $AC = a - y$. The velocity of the point~$C$ is
$\dfrac{d(a - y)}{dt} = y$; this gives $-\natlog y = t + c$. When $t = 0$,
then $y = a$ and $c = -\natlog a$. Again, let $\smash[t]{\dfrac{dx}{dt}} = a$ be the
velocity of the point~$F$, then $x = at$. Substituting for $t$~and~$c$
their values and remembering that $a = 10^7$ and that by definition
$x = \Naplog y$, we get
\[
\Naplog y = 10^7 \natlog \frac{10^7}{y}.
\]

It is evident from this formula that Napier's logarithms are
\index{Napier, J.}%
not the same as the natural logarithms. Napier's logarithms
increase as the number itself decreases. He took the logarithm
of $\sin 90 = 0$; \ie~the logarithm of $10^7 = 0$. The logarithm
of $\sin \alpha$ increased from zero as $\alpha$ decreased from~$90°$.
Napier's genesis of logarithms from the conception of two
flowing points reminds us of Newton's doctrine of fluxions.
The relation between geometric and arithmetical progressions,
so skilfully utilised by Napier, had been observed by Archimedes,
Stifel, and others. Napier did not determine the base
to his system of logarithms. The notion of a ``base'' in fact
never suggested itself to him. The one demanded by his
reasoning is the reciprocal of that of the natural system, but
such a base would not reproduce accurately all of Napier's
figures, owing to slight inaccuracies in the calculation of the
tables. Napier's great invention was given to the world in
1614 in a work entitled \textit{Mirifici logarithmorum canonis descriptio}.
In it he explained the nature of his logarithms, and
gave a logarithmic table of the natural sines of a quadrant
from minute to minute.

\textbf{Henry Briggs} (1556--1631), in Napier's time professor of
\index{Briggs}%
geometry at Gresham College, London, and afterwards
professor at Oxford, was so struck with admiration of
Napier's book, that he left his studies in London to do
%% -----File: 178.png---Folio 164-------
homage to the Scottish philosopher. Briggs was delayed in
his journey, and Napier complained to a common friend, ``Ah,
\index{Napier, J.!ref.\ to}%
John, Mr. Briggs will not come.'' At that very moment
knocks were heard at the gate, and Briggs was brought into
the lord's chamber. Almost one-quarter of an hour was
spent, each beholding the other without speaking a word. At
last Briggs began: ``My lord, I have undertaken this long
journey purposely to see your person, and to know by what
engine of wit or ingenuity you came first to think of this most
excellent help in astronomy, viz.~the logarithms; but, my
lord, being by you found out, I wonder nobody found it out
before, when now known it is so easy.''\Reference{28} Briggs suggested
to Napier the advantage that would result from retaining zero
for the logarithm of the whole sine, but choosing $10,000,000,000$
for the logarithm of the $10$th part of that same sine, \ie~of
$5°\ 44' 22''$. Napier said that he had already thought of the
change, and he pointed out a slight improvement on Briggs'
idea; viz.~that zero should be the logarithm of~$1$, and
$10,000,000,000$ that of the whole sine, thereby making the
characteristic of numbers greater than unity positive and
not negative, as suggested by Briggs. Briggs admitted this
to be more convenient. The invention of ``Briggian logarithms''
occurred, therefore, to Briggs and Napier independently.
The great practical advantage of the new system
was that its fundamental progression was accommodated to
the base, $10$, of our numerical scale. Briggs devoted all
his energies to the construction of tables upon the new plan.
Napier died in 1617, with the satisfaction of having found
in Briggs an able friend to bring to completion his unfinished
plans. In~1624 Briggs published his \textit{Arithmetica logarithmica},
containing the logarithms to $14$~places of numbers,
from $1$ to~$20,000$ and from $90,000$ to~$100,000$. The gap from
$20,000$ to~$90,000$ was filled up by that illustrious successor of
%% -----File: 179.png---Folio 165-------
\index{Trigonometry}%
Napier and Briggs, \textbf{Adrian Vlacq} of Gouda in Holland. He
\index{Napier, J.!ref.\ to}%
\index{Vlacq}%
published in 1628 a table of logarithms from 1 to~$100,000$, of
which $70,000$ were calculated by himself. The first publication
of Briggian logarithms of trigonometric functions was made
in 1620 by \textbf{Gunter}, a colleague of Briggs, who found the logarithmic
\index{Gunter, E.}%
sines and tangents for every minute to seven places.
Gunter was the inventor of the words \emph{cosine} and \emph{cotangent}.
\index{Cosine}%
\index{Cotangent}%
Briggs devoted the last years of his life to calculating more
extensive Briggian logarithms of trigonometric functions, but
he died in~1631, leaving his work unfinished. It was carried
on by the English \textbf{Henry Gellibrand}, and then published by
\index{Gellibrand}%
Vlacq at his own expense. Briggs divided a degree into $100$~parts,
but owing to the publication by Vlacq of trigonometrical
tables constructed on the old sexagesimal division, Briggs'
innovation remained unrecognised. Briggs and Vlacq published
four fundamental works, the results of which ``have never been
superseded by any subsequent calculations.''

The first logarithms upon the \emph{natural} base $e$ were published
by \textbf{John Speidell} in his \textit{New Logarithmes} (London, 1619), which
\index{Speidell}%
contains the natural logarithms of sines, tangents, and secants.

The only possible rival of John Napier in the invention of
logarithms was the Swiss \textbf{Justus Byrgius} (Joost Bürgi). He
\index{Burgi@{Bürgi}!ref.\ to}%
published a rude table of logarithms six years after the
appearance of the \textit{Canon Mirificus}, but it appears that he
conceived the idea and constructed that table as early, if not
earlier, than Napier did his. But he neglected to have the
results published until Napier's logarithms were known and
\index{Logarithms|)}%
admired throughout Europe.

Among the various inventions of Napier to assist the
memory of the student or calculator, is ``Napier's rule of
circular parts'' for the solution of spherical right triangles.
\index{Napier's rule of circular parts}%
It is, perhaps, ``the happiest example of artificial memory
that is known.''
%% -----File: 180.png---Folio 166-------

The most brilliant conquest in algebra during the sixteenth
\index{Algebra!seventeenth century}%
century had been the solution of cubic and \Change{bi-quadratic}{biquadratic} equations.
\index{Equations!theory of}%
All attempts at solving algebraically equations of higher
degrees remaining fruitless, a new line of inquiry---the properties
of equations and their roots---was gradually opened up.
We have seen that Vieta had attained a partial knowledge of
the relations between roots and coefficients. \textbf{Peletarius}, a
\index{Peletarius}%
Frenchman, had observed as early as~1558, that the root of an
equation is a divisor of the last term. One who extended the
theory of equations somewhat further than Vieta, was \textbf{Albert
Girard} (1590--1634), a Flemish mathematician. Like Vieta,
\index{Girard}%
this ingenious author applied algebra to geometry, and was
the first who understood the use of negative roots in the
\index{Negative roots}%
solution of geometric problems. He spoke of imaginary quantities;
\index{Imaginary quantities}%
inferred by induction that every equation has as many
roots as there are units in the number expressing its degree;
and first showed how to express the sums of their powers in
terms of the coefficients. Another algebraist of considerable
power was the English \textbf{Thomas Harriot} (1560--1621). He
\index{Harriot}%
accompanied the first colony sent out by Sir Walter Raleigh
to Virginia. After having surveyed that country he returned
to England. As a mathematician, he was the boast of his
country. He brought the theory of equations under one
comprehensive point of view by grasping that truth in its
full extent to which Vieta and Girard only approximated; viz.~that
in an equation in its simplest form, the coefficient of
the second term with its sign changed is equal to the sum of
the roots; the coefficient of the third is equal to the sum
of the products of every two of the roots; etc. He was the
first to decompose equations into their simple factors; but,
since he failed to recognise imaginary and even negative roots,
he failed also to prove that every equation could be thus
decomposed. Harriot made some changes in algebraic notation,
%% -----File: 181.png---Folio 167-------
\index{Notation!in algebra}%
adopting small letters of the alphabet in place of the
capitals used by Vieta. The symbols of inequality $>$ and $<$
\index{Vieta!ref.\ to}%
were introduced by him. Harriot's work, \textit{Artis Analyticæ
praxis}, was published in 1631, ten years after his death.
\textbf{William Oughtred} (1574--1660) contributed vastly to the propagation
\index{Oughtred}%
of mathematical knowledge in England by his treatises,
which were long used in the universities. He introduced $×$
as symbol of multiplication, and $::$ as that of proportion. By
him ratio was expressed by only one dot. In the eighteenth
century \textit{Christian Wolf} secured the general adoption of the
\index{Wolf, C.!ref.\ to}%
dot as a symbol of multiplication, and the sign for ratio was
thereupon changed to two dots. Oughtred's ministerial duties
left him but little time for the pursuit of mathematics during
daytime, and evenings his economical wife denied him the
use of a light.

Algebra was now in a state of sufficient perfection to enable
Descartes to take that important step which forms one of the
\index{Descartes!ref.\ to}%
grand epochs in the history of mathematics,---the application
of algebraic analysis to define the nature and investigate the
properties of algebraic curves.

In geometry, the determination of the areas of curvilinear
\index{Geometry!Renaissance}%
figures was diligently studied at this period. \textbf{Paul Guldin}
\index{Guldin}%
(1577--1643), a Swiss mathematician of considerable note,
\Change{rediscovered}{re-discovered} the following theorem, published in his \textit{Centrobaryca},
which has been named after him, though first found
in the \textit{Mathematical Collections} of Pappus: The volume of a
solid of revolution is equal to the area of the generating
figure, multiplied by the circumference described by the centre
of gravity. We shall see that this method excels that of
Kepler and Cavalieri in following a more exact and natural
\index{Cavalieri!ref.\ to}%
\index{Kepler!ref.\ to}%
course; but it has the disadvantage of necessitating the determination
of the centre of gravity, which in itself may be a
more difficult problem than the original one of finding the
%% -----File: 182.png---Folio 168-------
\index{Stereometry}%
volume. Guldin made some attempts to prove his theorem,
but Cavalieri pointed out the weakness of his demonstration.

\textbf{Johannes Kepler} (1571--1630) was a native of Würtemberg
\index{Kepler|(}%
and imbibed Copernican principles while at the University of
Tübingen. His pursuit of science was repeatedly interrupted
by war, religious persecution, pecuniary embarrassments, frequent
changes of residence, and family troubles. In~1600 he
became for one year assistant to the Danish astronomer, Tycho
Brahe, in the observatory near Prague. The relation between
\index{Brahe, Tycho}%
the two great astronomers was not always of an agreeable
character. Kepler's publications are voluminous. His first
attempt to explain the solar system was made in~1596, when he
thought he had discovered a curious relation between the
five regular solids and the number and distance of the planets.
\index{Regular solids}%
The publication of this pseudo-discovery brought him much
fame. Maturer reflection and intercourse with Tycho Brahe
\index{Tycho Brahe}%
and Galileo led him to investigations and results more worthy of
\index{Galileo!ref.\ to}%
his genius---``Kepler's laws.'' He enriched pure mathematics
\index{Kepler's laws}%
as well as astronomy. It is not strange that he was interested
in the mathematical science which had done him so much
service; for ``if the Greeks had not cultivated conic sections,
\index{Conic sections!Kepler}%
Kepler could not have superseded Ptolemy.''\Reference{11} The Greeks
never dreamed that these curves would ever be of practical
use; Aristæus and Apollonius studied them merely to satisfy
their intellectual cravings after the ideal; yet the conic
sections assisted Kepler in tracing the march of the planets in
their elliptic orbits. Kepler made also extended use of logarithms
\index{Logarithms}%
and decimal fractions, and was enthusiastic in diffusing
a knowledge of them. At one time, while purchasing wine, he
was struck by the inaccuracy of the ordinary modes of determining
the contents of kegs. This led him to the study of
the volumes of solids of revolution and to the publication of
the \textit{Stereometria Doliorum} in~1615. In it he deals first with the
%% -----File: 183.png---Folio 169-------
\index{Definite integrals}%
\index{Infinitesimals}%
\index{Method of exhaustion!ref.\ to}%
\index{Rectification of curves}%
solids known to Archimedes and then takes up others. Kepler
\index{Archimedes!ref.\ to}%
introduced a new idea into geometry; namely, that of infinitely
great and infinitely small quantities. Greek mathematicians
always shunned this notion, but with it modern mathematicians
have completely revolutionised the science. In comparing
rectilinear figures, the method of superposition was employed
by the ancients, but in comparing rectilinear and curvilinear
figures with each other, this method failed because no addition
or subtraction of rectilinear figures could ever produce curvilinear
ones. To meet this case, they devised the Method of
Exhaustion, which was long and difficult; it was purely
\index{Exhaustion, method of}%
synthetical, and in general required that the conclusion should
be known at the outset. The new notion of infinity led
\index{Infinity}%
gradually to the invention of methods immeasurably more
powerful. Kepler conceived the circle to be composed of an
infinite number of triangles having their common vertices at
the centre, and their bases in the circumference; and the
sphere to consist of an infinite number of pyramids. He
applied conceptions of this kind to the determination of the
areas and volumes of figures generated by curves revolving
about any line as axis, but succeeded in solving only a few of
the simplest out of the 84~problems which he proposed for
investigation in his \textit{Stereometria}.

Other points of mathematical interest in Kepler's works are
(1)~the statement of the earliest problem of inverse tangents;
\index{Inverse tangents (problem of)}%
\index{Tangents!inverse problem of}%
(2)~an investigation which amounts to the evaluation of
the definite integral $\displaystyle \int_{0}^{\phi} \sin \phi\, d\phi = 1 - \cos \phi$; (3)~the assertion
that the circumference of an ellipse, whose axes are $2a$ and
$2b$, is nearly $\pi (a + b)$; (4)~a passage from which it has been
inferred that Kepler knew the variation of a function near its
maximum value to disappear; (5)~the assumption of the principle
of continuity (which differentiates modern from ancient
\index{Continuity}%
geometry), when he shows that a parabola has a focus at
\index{Focus}%
%% -----File: 184.png---Folio 170-------
\index{Kepler|)}%
infinity, that lines radiating from this ``cæcus focus'' are
\index{Focus}%
parallel and have no other point at infinity.

The \textit{Stereometria} led Cavalieri, an Italian Jesuit, to the
\index{Cavalieri}%
consideration of infinitely small quantities. \textbf{Bonaventura
Cavalieri} (1598--1647), a pupil of Galileo and professor at
\index{Galileo!ref.\ to}%
Bologna, is celebrated for his \textit{Geometria indivisibilibus continuorum
nova quadam ratione promota}, 1635. This work
expounds his method of Indivisibles, which occupies an intermediate
\index{Indivisibles|(}%
place between the method of exhaustion of the Greeks
and the methods of Newton and Leibniz. He considers lines
as composed of an infinite number of points, surfaces as composed
of an infinite number of lines, and solids of an infinite
number of planes. The relative magnitude of two solids or
surfaces could then be found simply by the summation of
series of planes or lines. For example, he finds the sum of the
squares of all lines making up a triangle equal to one-third
the sum of the squares of all lines of a parallelogram of equal
base and altitude; for if in a triangle, the first line at the apex
be~1, then the second is~2, the third is~3, and so on; and the
sum of their squares is
\[
1^2 + 2^2 + 3^2 + \dotsb + n^2 = n(n + 1)(2n + 1) ÷ 6.
\]
In the parallelogram, each of the lines is $n$ and their number is
$n$; hence the total sum of their squares is $n^3$. The ratio
between the two sums is therefore
\[
n(n + 1)(2n + 1) ÷ 6n^3 = \tfrac{1}{3},
\]
since $n$ is infinite. From this he concludes that the pyramid
or cone is respectively $\frac{1}{3}$ of a prism or cylinder of equal base
and altitude, since the polygons or circles composing the former
decrease from the base to the apex in the same way as the
squares of the lines parallel to the base in a triangle decrease
from base to apex. By the Method of Indivisibles, Cavalieri
%% -----File: 185.png---Folio 171-------
solved the majority of the problems proposed by Kepler.
\index{Kepler!ref.\ to}%
Though expeditious and yielding correct results, Cavalieri's
method lacks a scientific foundation. If a line has absolutely
no width, then no number, however great, of lines can ever
make up an area; if a plane has no thickness whatever, then
even an infinite number of planes cannot form a solid. The
reason why this method led to correct conclusions is that one
area is to another area in the same ratio as the sum of the
series of lines in the one is to the sum of the series of lines in
the other. Though unscientific, Cavalieri's method was used
for fifty years as a sort of integral calculus. It yielded solutions
\index{Integral calculus}%
to some difficult problems. Guldin made a severe attack
\index{Guldin!ref.\ to}%
on Cavalieri and his method. The latter published in~1647,
after the death of Guldin, a treatise entitled \textit{Exercitationes
geometricæ sex}, in which he replied to the objections of his
opponent and attempted to give a clearer explanation of his
method. Guldin had never been able to demonstrate the
theorem named after him, except by metaphysical reasoning,
but Cavalieri proved it by the method of indivisibles. A
revised edition of the \textit{Geometry of Indivisibles} appeared in~1653.

There is an important curve, not known to the ancients,
which now began to be studied with great zeal. Roberval
gave it the name of ``trochoid,'' Pascal the name of ``roulette,''
\index{Roulette}%
\index{Trochoid}%
Galileo the name of ``cycloid.'' The invention of this curve
\index{Cycloid}%
\index{Galileo!ref.\ to}%
seems to be due to Galileo, who valued it for the graceful form
it would give to arches in architecture. He ascertained its
area by weighing paper figures of the cycloid against that of
the generating circle, and found thereby the first area to be
nearly but not exactly thrice the latter. A mathematical
determination was made by his pupil, \textbf{Evangelista Torricelli}
\index{Torricelli}%
(1608--1647), who is more widely known as a physicist than
as a mathematician.
%% -----File: 186.png---Folio 172-------

By the Method of Indivisibles he demonstrated its area to
be triple that of the revolving circle, and published his
solution. This same quadrature had been effected a few years
earlier by Roberval in France, but his solution was not known
\index{Roberval}%
\index{Roberval!ref.\ to}%
to the Italians. Roberval, being a man of irritable and
violent disposition, unjustly accused the mild and amiable
Torricelli of stealing the proof. This accusation of plagiarism
created so much chagrin with Torricelli that it is considered
to have been the cause of his early death. \textbf{Vincenzo Viviani},
\index{Viviani}%
another prominent pupil of Galileo, determined the tangent
to the cycloid. This was accomplished in France by Descartes
and Fermat.
\index{Fermat!ref.\ to}%

In France, where geometry began to be cultivated with
greatest success, Roberval, Fermat, Pascal, employed the
Method of Indivisibles and made new improvements in it.
\textbf{Giles Persone de Roberval} (1602--1675), for forty years professor
of mathematics at the College of France in Paris,
claimed for himself the invention of the Method of Indivisibles.
Since his complete works were not published until
after his death, it is difficult to settle questions of priority.
Montucla and Chasles are of the opinion that he invented the
\index{Chasles!ref.\ to}%
\index{Montucla}%
method independent of and earlier than the Italian geometer,
though the work of the latter was published much earlier
than Roberval's. Marie finds it difficult to believe that the
\index{Marie, M.}%
Frenchman borrowed nothing whatever from the Italian, for
both could not have hit independently upon the word \textit{Indivisibles},
which is applicable to infinitely small quantities, as
conceived by Cavalieri, but not as conceived by Roberval.
Roberval and Pascal improved the rational basis of the
Method of Indivisibles, by considering an area as made up
of an indefinite number of rectangles instead of lines, and
a solid as composed of indefinitely small solids instead of
surfaces. Roberval applied the method to the finding of
%% -----File: 187.png---Folio 173-------
\index{Indivisibles|)}%
areas, volumes, and centres of gravity. He effected the quadrature
of a parabola of any degree $y^m = a^{m-1} x$, and also of a
parabola $y^m = a^{m-n} x^n$. We have already mentioned his quadrature
of the cycloid. Roberval is best known for his method
\index{Cycloid}%
of drawing tangents. He was the first to apply motion to
\index{Tangents!in geometry}%
the resolution of this important problem. His method is
allied to Newton's principle of fluxions. Archimedes conceived
\index{Archimedes!ref.\ to}%
\index{Newton!ref.\ to}%
his spiral to be generated by a double motion. This
idea Roberval extended to all curves. Plane curves, as for
instance the conic sections, may be generated by a point
acted upon by two forces, and are the resultant of two
motions. If at any point of the curve the resultant be
resolved into its components, then the diagonal of the parallelogram
determined by them is the tangent to the curve
at that point. The greatest difficulty connected with this
ingenious method consisted in resolving the resultant into
components having the proper lengths and directions. Roberval
did not always succeed in doing this, yet his new idea was
a great step in advance. He broke off from the ancient
definition of a tangent as a straight line having only one point
in common with a curve,---a definition not valid for curves of
higher degrees, nor apt even in curves of the second degree to
bring out the properties of tangents and the parts they may
be made to play in the generation of the curves. The subject
of tangents received special attention also from Fermat,
\index{Fermat}%
\index{Fermat!ref.\ to}%
Descartes, and Barrow, and reached its highest development
\index{Barrow!ref.\ to}%
\index{Descartes!ref.\ to}%
after the invention of the differential calculus. Fermat and
Descartes defined tangents as secants whose two points of
intersection with the curve coincide; Barrow considered a
curve a polygon, and called one of its sides produced a tangent.

A profound scholar in all branches of learning and a mathematician
of exceptional powers was \textbf{Pierre de Fermat} (1601--1665).
He studied law at Toulouse, and in 1631 was made
%% -----File: 188.png---Folio 174-------
\index{Midorge}%
councillor for the parliament of Toulouse. His leisure time
was mostly devoted to mathematics, which he studied with
irresistible passion. Unlike Descartes and Pascal, he led a
\index{Descartes!ref.\ to}%
quiet and unaggressive life. Fermat has left the impress of
his genius upon all branches of mathematics then known. A
great contribution to geometry was his \textit{De maximis et minimis}.
About twenty years earlier, Kepler had first observed that the
\index{Kepler!ref.\ to}%
increment of a variable, as, for instance, the ordinate of a
curve, is evanescent for values very near a maximum or a
minimum value of the variable. Developing this idea, Fermat
obtained his rule for maxima and minima. He substituted
\index{Maxima and minima}%
$x + e$ for $x$ in the given function of $x$ and then equated to each
other the two consecutive values of the function and divided
the equation by~$e$. If $e$ be taken~0, then the roots of this
equation are the values of~$x$, making the function a maximum
or a minimum. Fermat was in possession of this rule in~1629.
The main difference between it and the rule of the differential
\index{Differential calculus!alleged invention by Pascal}%
calculus is that it introduces the indefinite quantity~$e$ instead
of the infinitely small~$dx$. Fermat made it the basis for his
method of drawing tangents.

Owing to a want of explicitness in statement, Fermat's
method of maxima and minima, and of tangents, was severely
attacked by his great contemporary, Descartes, who could
never be brought to render due justice to his merit. In the
ensuing dispute, Fermat found two zealous defenders in Roberval
and Pascal, the father; while Mydorge, Desargues, and
\index{Desargues!ref.\ to}%
Hardy supported Descartes.
\index{Hardy}%

Since Fermat introduced the conception of infinitely small
differences between consecutive values of a function and arrived
at the principle for finding the maxima and minima,
it was maintained by Lagrange, Laplace, and Fourier, that
\index{Fourier!ref.\ to}%
\index{Lagrange!ref.\ to}%
\index{Laplace!ref.\ to}%
Fermat may be regarded as the first inventor of the differential
calculus. This point is not well taken, as will be seen
%% -----File: 189.png---Folio 175-------
from the words of Poisson, himself a Frenchman, who rightly
\index{Poisson!ref.\ to}%
says that the differential calculus ``consists in a system of
rules proper for finding the differentials of all functions,
rather than in the use which may be made of these infinitely
small variations in the solution of one or two isolated problems.''

A contemporary mathematician, whose genius excelled even
that of the great Fermat, was \textbf{Blaise Pascal} (1623--1662). He
\index{Pascal|(}%
was born at Clermont in Auvergne. In~1626 his father retired
to Paris, where he devoted himself to teaching his son, for he
would not trust his education to others. Blaise Pascal's genius
for geometry showed itself when he was but twelve years
old. His father was well skilled in mathematics, but did not
wish his son to study it until he was perfectly acquainted with
Latin and Greek. All mathematical books were hidden out of
his sight. The boy once asked his father what mathematics
treated of, and was answered, in general, ``that it was the
method of making figures with exactness, and of finding out
what proportions they relatively had to one another.'' He
was at the same time forbidden to talk any more about it,
or ever to think of it. But his genius could not submit to be
confined within these bounds. Starting with the bare fact
that mathematics taught the means of making figures infallibly
exact, he employed his thoughts about it and with a piece
of charcoal drew figures upon the tiles of the pavement, trying
the methods of drawing, for example, an exact circle or equilateral
triangle. He gave names of his own to these figures
and then formed axioms, and, in short, came to make perfect
demonstrations. In this way he arrived unaided at the theorem
that the sum of the three angles of a triangle is equal to
two right angles. His father caught him in the act of studying
this theorem, and was so astonished at the sublimity and
force of his genius as to weep for joy. The father now gave
%% -----File: 190.png---Folio 176-------
\index{Conic sections!more recent researches|(}%
him Euclid's \textit{Elements}, which he, without assistance, mastered
easily. His regular studies being languages, the boy employed
only his hours of amusement on the study of geometry, yet he
had so ready and lively a penetration that, at the age of sixteen,
he wrote a treatise upon conics, which passed for such a
surprising effort of genius, that it was said nothing equal to it
in strength had been produced since the time of Archimedes.
Descartes refused to believe that it was written by one so
young as Pascal. This treatise was never published, and is
now lost. Leibniz saw it in Paris and reported on a portion
\index{Leibniz!ref.\ to}%
of its contents. The precocious youth made vast progress in
all the sciences, but the constant application at so tender an
age greatly impaired his health. Yet he continued working,
and at nineteen invented his famous machine for performing
arithmetical operations mechanically. This continued strain
from overwork resulted in a permanent indisposition, and he
would sometimes say that from the time he was eighteen,
he never passed a day free from pain. At the age of
twenty-four he resolved to lay aside the study of the human
sciences and to consecrate his talents to religion. His Provincial
Letters against the Jesuits are celebrated. But at
times he returned to the favourite study of his youth. Being
kept awake one night by a toothache, some thoughts undesignedly
came into his head concerning the roulette or
cycloid; one idea followed another; and he thus discovered
\index{Cycloid}%
properties of this curve even to demonstration. A correspondence
between him and Fermat on certain problems was
the beginning of the theory of probability. Pascal's illness
increased, and he died at Paris at the early age of thirty-nine
years.\Reference{30} By him the answer to the objection to Cavalieri's
Method of Indivisibles was put in the clearest form. Like
\index{Indivisibles}%
Roberval, he explained ``the sum of right lines'' to mean ``the
sum of infinitely small rectangles.'' Pascal greatly advanced
%% -----File: 191.png---Folio 177-------
\index{Centre!of gravity}%
\index{Curves!quadrature of}%
\index{Involution of points}%
\index{Quadrature of curves}%
the knowledge of the cycloid. He determined the area of a
section produced by any line parallel to the base; the volume
generated by it revolving around its base or around the axis;
and, finally, the centres of gravity of these volumes, and also
of half these volumes cut by planes of symmetry. Before
publishing his results, he sent, in~1658, to all mathematicians
that famous challenge offering prizes for the first two solutions
of these problems. Only Wallis and A.~La~Louère competed
\index{Lalouere@{La Louère}}%
\index{Wallis!ref.\ to}%
for them. The latter was quite unequal to the task;
the former, being pressed for time, made numerous mistakes:
neither got a prize. Pascal then published his own solutions,
which produced a great sensation among scientific men. Wallis,
too, published his, with the errors corrected. Though not
competing for the prizes, Huygens, Wren, and Fermat solved
\index{Fermat!ref.\ to}%
\index{Huygens!ref.\ to}%
\index{Wren}%
some of the questions. The chief discoveries of \textbf{Christopher
Wren} (1632--1723), the celebrated architect of St.~Paul's
Cathedral in London, were the rectification of a cycloidal arc
\index{Rectification of curves}%
and the determination of its centre of gravity. Fermat
found the area generated by an arc of the cycloid. Huygens
invented the cycloidal pendulum.

The beginning of the seventeenth century witnessed also
a revival of synthetic geometry. One who treated conics still
by ancient methods, but who succeeded in greatly simplifying
many prolix proofs of Apollonius, was \textbf{Claude Mydorge} in
\index{Mydorge}%
Paris (1585--1647), a friend of Descartes. But it remained
for \textbf{Girard Desargues} (1593--1662) of Lyons, and for Pascal, to
\index{Desargues}%
leave the beaten track and cut out fresh paths. They introduced
the important method of Perspective. All conics on
\index{Perspective}%
a cone with circular base appear circular to an eye at the apex.
Hence Desargues and Pascal conceived the treatment of the
\index{Pascal|)}%
conic sections as projections of circles. Two important and
beautiful theorems were given by Desargues: The one is on
the ``involution of the six points,'' in which a transversal
%% -----File: 192.png---Folio 178-------
\index{Brianchion}%
\index{Conic sections!more recent researches|)}%
\index{Hexagrammum mysticum}%
\index{Homological figures}%
meets a conic and an inscribed quadrangle; the other is that,
if the vertices of two triangles, situated either in space or in
a plane, lie on three lines meeting in a point, then their sides
meet in three points lying on a line; and conversely. This
last theorem has been employed in recent times by Brianchon,
Sturm, Gergonne, and Poncelet. Poncelet made it the basis
\index{Gergonne!ref.\ to}%
\index{Poncelet!ref.\ to}%
\index{Sturm, J.~C.~F.!ref.\ to}%
of his beautiful theory of homoligical figures. We owe to
Desargues the theory of involution and of transversals; also
the beautiful conception that the two extremities of a straight
line may be considered as meeting at infinity, and that parallels
\index{Infinity}%
differ from other pairs of lines only in having their points
of intersection at infinity. Pascal greatly admired Desargues'
\index{Pascal!ref.\ to}%
results, saying (in his \textit{Essais pour les Coniques}), ``I wish to
acknowledge that I owe the little that I have discovered on
this subject, to his writings.'' Pascal's and Desargues' writings
contained the fundamental ideas of modern synthetic
geometry. In Pascal's wonderful work on conics, written
at the age of sixteen and now lost, were given the theorem
on the anharmonic ratio, first found in Pappus, and also that
\index{Anharmonic ratio}%
\index{Pappus!ref.\ to}%
celebrated proposition on the mystic hexagon, known as
``Pascal's theorem,'' viz.~that the opposite sides of a hexagon
\index{Pascal's theorem}%
inscribed in a conic intersect in three points which are collinear.
This theorem formed the keystone to his theory. He
himself said that from this alone he deduced over 400~corollaries,
embracing the conics of Apollonius and many other
results. Thus the genius of Desargues and Pascal uncovered
several of the rich treasures of modern synthetic geometry;
but owing to the absorbing interest taken in the analytical
geometry of Descartes and later in the differential calculus,
the subject was almost entirely neglected until the present
century.

In the theory of numbers no new results of scientific value
\index{Numbers!theory of numbers|(}%
\index{Theory of numbers|(}%
had been reached for over 1000~years, extending from the
%% -----File: 193.png---Folio 179-------
times of Diophantus and the Hindoos until the beginning of
\index{Diophantus!ref.\ to}%
the seventeenth century. But the illustrious period we are
now considering produced men who rescued this science from
the realm of mysticism and superstition, in which it had been
so long imprisoned; the properties of numbers began again
to be studied scientifically. Not being in possession of the
Hindoo indeterminate analysis, many beautiful results of the
Brahmins had to be re-discovered by the Europeans. Thus
a solution in integers of linear indeterminate equations was
re-discovered by the Frenchman \textbf{Bachet de Méziriac} (1581--1638),
\index{Meziriac@{Méziriac}}%
who was the earliest noteworthy European Diophantist.
In~1612 he published \textit{Problèmes plaisants et délectables qui se
font par les nombres}, and in 1621 a Greek edition of \textit{Diophantus}
with notes. The father of the modern theory of numbers is
\textbf{Fermat}. He was so uncommunicative in disposition, that he
\index{Fermat|(}%
generally concealed his methods and made known his results
only. In some cases later analysts have been greatly puzzled
in the attempt of supplying the proofs. Fermat owned a copy
of Bachet's \textit{Diophantus}, in which he entered numerous marginal
notes. In~1670 these notes were incorporated in a new
edition of \textit{Diophantus}, brought out by his son. Other theorems
on numbers, due to Fermat, were published in his \textit{Opera varia}
(edited by his son) and in Wallis's \textit{Commercium epistolicum}
\index{Wallis!ref.\ to}%
of~1658. Of the following theorems, the first seven are found
in the marginal notes:---

(1)~$x^n + y^n = z^n$ is impossible for integral values of $x$,~$y$,
and~$z$, when $n > 2$. Remark: ``I have found for this a truly
wonderful proof, but the margin is too small to hold it.''
Repeatedly was this theorem made the prize question of
learned societies. It has given rise to investigations of
great interest and difficulty on the part of Euler, Lagrange,
\index{Euler!ref.\ to}%
\index{Lagrange!ref.\ to}%
Dirichlet, and Kummer.
\index{Dirichlet!ref.\ to}%
\index{Kummer!ref.\ to}%

(2)~A prime of the form $4n + 1$ is only once the hypothenuse
\index{Prime numbers}%
%% -----File: 194.png---Folio 180-------
\index{Cube numbers}%
of a right triangle; its square is twice; its cube is three
times, etc. Example: $5^2 = 3^2 + 4^2$; $25^2 = 15^2 + 20^2 = 7^2 + 24^2$;
$125^2 = 75^2 + 100^2 = 35^2 +120^2 = 44^2 + 117^2$.

(3)~A prime of the form $4n + 1$ can be expressed once, and
\index{Prime numbers}%
only once, as the sum of two squares. Proved by Euler.
\index{Euler!ref.\ to}%

(4)~A number composed of two cubes can be resolved into
two other cubes in an infinite multiplicity of ways.

(5)~Every number is either a triangular number or the sum
of two or three triangular numbers; either a square or the
\index{Numbers!triangular}%
sum of two, three, or four squares; either a pentagonal number
or the sum of two, three, four, or five pentagonal numbers;
similarly for polygonal numbers in general. The proof of this
and other theorems is promised by Fermat in a future work
which never appeared. This theorem is also given, with
others, in a letter of 1637(?) addressed to \textit{Pater Mersenne}.
\index{Mersenne}%

(6)~As many numbers as you please may be found, such
that the square of each remains a square on the addition to
or subtraction from it of the sum of all the numbers.

(7)~$x^4 + y^4 = z^2$ is impossible.

(8)~In a letter of 1640 he gives the celebrated theorem
generally known as ``Fermat's theorem,'' which we state in
\index{Fermat's theorem}%
Gauss's notation: If $p$ is prime, and $a$ is prime to~$p$, then $a^{p-1} \equiv 1 \pmod{p}$. It was proved by Euler.

(9)~Fermat died with the belief that he had found a long-sought-for
law of prime numbers in the formula $2^{2^n} + 1 =$ a
prime, but he admitted that he was unable to prove it rigorously.
The law is not true, as was pointed out by Euler in the
example $2^{2^5} + 1 = 4,294,967,297 = 6,700,417 \text{ times } 641$. The
American lightning calculator \textit{Zerah Colburn}, when a boy,
\index{Colburn, Z}%
readily found the factors, but was unable to explain the
method by which he made his marvellous mental computation.

(10)~An odd prime number can be expressed as the difference
of two squares in one, and only one, way. This theorem,
%% -----File: 195.png---Folio 181-------
given in the \textit{Relation}, was used by Fermat for the decomposition
of large numbers into prime factors.

(11)~If the integers $a$, $b$, $c$ represent the sides of a right
triangle, then its area cannot be a square number. This was
proved by Lagrange.

(12)~Fermat's solution of $ax^2 + 1 = y^2$, where $a$ is integral
but not a square, has come down in only the broadest outline,
as given in the \textit{Relation}. He proposed the problem to the
Frenchman, \textit{Bernhard Frenicle de Bessy}, and in 1657 to all
\index{Bessy}%
living mathematicians. In England, Wallis and Lord Brounker
conjointly found a laborious solution, which was published in~1658,
and also in~1668, in an algebraical work brought out by
\textit{John Pell}. Though Pell had no other connection with the
\index{Pell}%
problem, it went by the name of ``Pell's problem.'' The first
\index{Pell's problem}%
solution was given by the Hindoos.

We are not sure that Fermat subjected all his theorems to
rigorous proof. His methods of proof were entirely lost
until~1879, when a document was found buried among the
manuscripts of Huygens in the library of Leyden, entitled
\textit{Relation des découvertes en la science des nombres}. It appears
from it that he used an inductive method, called by him \textit{la
descente infinie ou indefinie}. He says that this was particularly
applicable in proving the impossibility of certain relations, as,
for instance, Theorem~11, given above, but that he succeeded
in using the method also in proving affirmative statements.
Thus he proved Theorem~3 by showing that if we suppose
there be a prime~$4n+1$ which does not possess this property,
then there will be a smaller prime of the form~$4n+1$ not
possessing it; and a third one smaller than the second, not
possessing it; and so on. Thus descending indefinitely, he
arrives at the number~$5$, which is the smallest prime factor
of the form $4n+1$. From the above supposition it would
follow that $5$ is not the sum of two squares---a conclusion
%% -----File: 196.png---Folio 182-------
\index{Mechanics!Stevin and Galileo}%
\index{Numbers!theory of numbers|)}%
\index{Theory of numbers|)}%
contrary to fact. Hence the supposition is false, and the
theorem is established. Fermat applied this method of descent
with success in a large number of theorems. By this
method Euler, Legendre, Dirichlet, proved several of his
enunciations and many other numerical propositions.

A correspondence between \textit{Pascal} and \textit{Fermat} relating to a
\index{Pascal!ref.\ to}%
certain game of chance was the germ of the theory of probabilities,
which has since attained a vast growth. Chevalier de
Méré proposed to Pascal the fundamental problem, to determine
\index{Mere@{Méré}}%
the probability which each player has, at any given
\index{Probability}%
stage of the game, of winning the game. Pascal and Fermat
supposed that the players have equal chances of winning a
single point.

The former communicated this problem to Fermat, who
\index{Fermat|)}%
studied it with lively interest and solved it by the theory of
combinations, a theory which was diligently studied both by
him and Pascal. The calculus of probabilities engaged the
attention also of Huygens. The most important theorem
\index{Huygens!ref.\ to}%
reached by him was that, if A has $p$~chances of winning a sum~$a$,
and $q$~chances of winning a sum~$b$, then he may expect to
win the sum $\dfrac{ap+bq}{p+q}$. The next great work on the theory of
probability was the \textit{Ars conjectandi} of Jakob Bernoulli.
\index{Bernoulli, James (born 1654)!ref.\ to}%

Among the ancients, Archimedes was the only one who
\index{Archimedes!ref.\ to}%
attained clear and correct notions on theoretical statics. He
\index{Statics}%
had acquired firm possession of the idea of pressure, which
lies at the root of mechanical science. But his ideas slept
nearly twenty centuries, until the time of \textbf{Stevin} and \textbf{Galileo}.
\index{Galileo}%
\index{Stevin!ref.\ to}%
Stevin determined accurately the force necessary to sustain
a body on a plane inclined at any angle to the horizon. He
was in possession of a complete doctrine of equilibrium. While
Stevin investigated statics, Galileo pursued principally dynamics.
Galileo was the first to abandon the Aristotelian idea
that bodies descend more quickly in proportion as they are
%% -----File: 197.png---Folio 183-------
\index{Centrifugal force}%
\index{Laws of motion}%
heavier; he established the first law of motion; determined
\index{Motion, laws of}%
the laws of falling bodies; and, having obtained a clear notion
of acceleration and of the independence of different motions,
was able to prove that projectiles move in parabolic curves.
Up to his time it was believed that a cannon-ball moved
forward at first in a straight line and then suddenly fell
vertically to the ground. Galileo had an understanding of
\emph{centrifugal forces}, and gave a correct definition of \emph{momentum}.
Though he formulated the fundamental principle of statics,
known as the \emph{parallelogram of forces}, yet he did not fully
\index{Parallelogram of forces}%
recognise its scope. The principle of virtual velocities was
partly conceived by \textbf{Guido Ubaldo} (died 1607), and afterwards
\index{Ubaldo}%
more fully by Galileo.

Galileo is the founder of the science of dynamics. Among
his contemporaries it was chiefly the novelties he detected in
the sky that made him celebrated, but Lagrange claims that
\index{Lagrange!ref.\ to}%
his astronomical discoveries required only a telescope and
perseverance, while it took an extraordinary genius to discover
laws from phenomena, which we see constantly and of
which the true explanation escaped all earlier philosophers.
The first contributor to the science of mechanics after Galileo
was Descartes.
\index{Descartes|(}%

\Section{Descartes to Newton}

Among the earliest thinkers of the seventeenth and eighteenth
centuries, who employed their mental powers toward the
destruction of old ideas and the up-building of new ones, ranks
\textbf{René Descartes} (1596--1650). Though he professed orthodoxy
in faith all his life, yet in science he was a profound sceptic.
He found that the world's brightest thinkers had been long
exercised in metaphysics, yet they had discovered nothing
%% -----File: 198.png---Folio 184-------
certain; nay, had even flatly contradicted each other. This
led him to the gigantic resolution of taking nothing whatever
on authority, but of subjecting everything to scrutinous examination,
according to new methods of inquiry. The certainty
of the conclusions in geometry and arithmetic brought out in
his mind the contrast between the true and false ways of
seeking the truth. He thereupon attempted to apply mathematical
reasoning to all sciences. ``Comparing the mysteries
of nature with the laws of mathematics, he dared to hope that
the secrets of both could be unlocked with the same key.''
Thus he built up a system of philosophy called Cartesianism.

Great as was Descartes' celebrity as a metaphysician, it
may be fairly questioned whether his claim to be remembered
by posterity as a mathematician is not greater. His philosophy
has long since been superseded by other systems, but the analytical
geometry of Descartes will remain a valuable possession
forever. At the age of twenty-one, Descartes enlisted in the
army of Prince Maurice of Orange. His years of soldiering
were years of leisure, in which he had time to pursue his
studies. At that time mathematics was his favourite science.
But in 1625 he ceased to devote himself to pure mathematics.
Sir William Hamilton is in error when he states that Descartes
\index{Hamilton, W.}%
considered mathematical studies absolutely pernicious as a
means of internal culture. In a letter to Mersenne, Descartes
says: ``M.~Desargues puts me under obligations on account of
\index{Desargues!ref.\ to}%
the pains that it has pleased him to have in me, in that he
shows that he is sorry that I do not wish to study more in geometry,
but I have resolved to quit only abstract geometry, that
is to say, the consideration of questions which \textit{serve only to
exercise the mind}, and this, in order to study another kind of
geometry, which has for its object the explanation of the
phenomena of nature\ldots. You know that all my physics is
nothing else than geometry.'' The years between 1629 and
%% -----File: 199.png---Folio 185-------
\index{Analytic geometry|(}%
\index{Co-ordinates}%
\index{Debaune@{De Baune}!ref.\ to}%
1649 were passed by him in Holland in the study, principally,
of physics and metaphysics. His residence in Holland was
during the most brilliant days of the Dutch state. In 1637 he
published his \textit{Discours de la Méthode}, containing among others
an essay of 106~pages on geometry. His \textit{Geometry} is not easy
reading. An edition appeared subsequently with notes by his
friend \textit{De Beaune}, which were intended to remove the difficulties.

It is frequently stated that Descartes was the first to apply
algebra to geometry. This statement is inaccurate, for Vieta
and others had done this before him. Even the Arabs sometimes
used algebra in connection with geometry. The new
step that Descartes did take was the introduction into geometry
of an analytical method based on the notion of variables
and constants, which enabled him to represent curves by algebraic
equations. In the Greek geometry, the idea of motion
was wanting, but with Descartes it became a very fruitful
conception. By him a point on a plane was determined in
position by its distances from two fixed right lines or axes.
These distances varied with every change of position in
the point. This geometric idea of \emph{co-ordinate representation},
together with the algebraic idea of \emph{two variables in one equation}
having an indefinite number of simultaneous values, furnished
a method for the study of loci, which is admirable for
the generality of its solutions. Thus the entire conic sections
of Apollonius is wrapped up and contained in a single equation
of the second degree.

The Latin term for ``ordinate'' used by Descartes comes
from the expression \textit{lineæ ordinatæ}, employed by Roman surveyors
for parallel lines. The term \emph{abscissa} occurs for the
first time in a Latin work of 1659, written by \textit{Stefano degli
Angeli} (1623--1697), a professor of mathematics in Rome.\Reference{3}
\index{Angeli}%
Descartes' geometry was called ``analytical geometry,'' partly
%% -----File: 200.png---Folio 186-------
because, unlike the synthetic geometry of the ancients, it is
\index{Geometry!analytic|(}%
actually \emph{analytical}, in the sense that the word is used in
logic; and partly because the practice had then already
arisen, of designating by the term \emph{analysis} the calculus with
\index{Analysis!Descartes'}%
general quantities.

The first important example solved by Descartes in his
geometry is the ``problem of Pappus''; viz.~``Given several
\index{Pappus!ref.\ to}%
straight lines in a plane, to find the locus of a point such that
the perpendiculars, or more generally, straight lines at given
angles, drawn from the point to the given lines, shall satisfy
the condition that the product of certain of them shall be in
a given ratio to the product of the rest.'' Of this celebrated
problem, the Greeks solved only the special case when the
number of given lines is four, in which case the locus of the
point turns out to be a conic section. By Descartes it was
solved completely, and it afforded an excellent example of
the use which can be made of his analytical method in the
study of loci. Another solution was given later by Newton
\index{Newton!ref.\ to}%
in the \textit{Principia}.

The methods of drawing tangents invented by Roberval
\index{Tangents!in geometry}%
and Fermat were noticed earlier. Descartes gave a third
method. Of all the problems which he solved by his geometry,
none gave him as great pleasure as his mode of constructing
tangents. It is profound but operose, and, on that account,
inferior to Fermat's. His solution rests on the method of
\emph{Indeterminate Coefficients}, of which he bears the honour of
\index{Indeterminate coefficients}%
invention. Indeterminate coefficients were employed by him
also in solving \Change{bi-quadratic}{biquadratic} equations.
\index{Equations!solution of}%

The essays of Descartes on dioptrics and geometry were
sharply criticised by Fermat, who wrote objections to the
former, and sent his own treatise on ``maxima and minima''
\index{Maxima and minima}%
to show that there were omissions in the geometry. Descartes
thereupon made an attack on Fermat's method of tangents.
%% -----File: 201.png---Folio 187-------
Descartes was in the wrong in this attack, yet he continued
\index{Descartes!rule of signs}%
the controversy with obstinacy. He had a controversy also
with Roberval on the cycloid. This curve has been called
\index{Cycloid}%
\index{Roberval!ref.\ to}%
the ``Helen of geometers,'' on account of its beautiful properties
\index{Helen of geometers}%
and the controversies which their discovery occasioned.
Its quadrature by Roberval was generally considered a brilliant
achievement, but Descartes commented on it by saying
that any one moderately well versed in geometry might have
done this. He then sent a short demonstration of his own.
On Roberval's intimating that he had been assisted by a
knowledge of the solution, Descartes constructed the tangent
to the curve, and challenged Roberval and Fermat to do the
same. Fermat accomplished it, but Roberval never succeeded
in solving this problem, which had cost the genius of Descartes
but a moderate degree of attention.

He studied some new curves, now called ``ovals of Descartes,''
\index{Ovals of Descartes}%
which were intended by him to serve in the construction
of converging lenses, but which yielded no results
of practical value.

The application of algebra to the doctrine of curved lines
\index{Algebra!seventeenth century}%
reacted favourably upon algebra. As an abstract science,
Descartes improved it by the systematic use of exponents and
\index{Exponents}%
by the full interpretation and construction of negative quantities.
\index{Negative quantities}%
Descartes also established some theorems on the theory of
equations. Celebrated is his ``rule of signs'' for determining
\index{Rule of signs}%
\index{Signs, rule of}%
the number of positive and negative roots; viz.~an equation
may have as many $+$~roots as there are variations of signs, and
as many $-$~roots as there are permanencies of signs. Descartes
was charged by Wallis with availing himself, without acknowledgment,
\index{Wallis!ref.\ to}%
of Harriot's theory of equations, particularly his mode
\index{Harriot!ref.\ to}%
of generating equations; but there seems to be no good ground
for the charge. Wallis also claimed that Descartes failed to
observe that the above rule of signs is not true whenever the
%% -----File: 202.png---Folio 188-------
\index{Apollonian Problem}%
equation has imaginary roots; but Descartes does not say that
the equation \emph{always has}, but that it \emph{may have} so many roots.
It is true that Descartes does not consider the case of imaginaries
directly, but further on in his \textit{Geometry} he gives
incontestable evidence of being able to handle this case
also.

In mechanics, Descartes can hardly be said to have advanced
\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton}%
beyond Galileo. The latter had overthrown the ideas of
\index{Galileo!ref.\ to}%
Aristotle on this subject, and Descartes simply ``threw himself
upon the enemy'' that had already been ``put to the rout.''
His statement of the first and second laws of motion was an
\index{Laws of motion}%
\index{Motion, laws of}%
improvement in form, but his third law is false in substance.
The motions of bodies in their direct impact was imperfectly
understood by Galileo, erroneously given by Descartes, and
first correctly stated by Wren, Wallis, and Huygens.
\index{Huygens!ref.\ to}%
\index{Wallis!ref.\ to}%
\index{Wren!ref.\ to}%

One of the most devoted pupils of Descartes was the learned
\textit{Princess Elizabeth}, daughter of Frederick~V\@. She applied the
\index{Elizabeth, Princess}%
\index{Princess Elizabeth}%
new analytical geometry to the solution of the ``Apollonian
problem.'' His second royal follower was \textit{Queen Christina},
the daughter of Gustavus Adolphus. She urged upon Descartes
to come to the Swedish court. After much hesitation
he accepted the invitation in~1649. He died at Stockholm one
year later. His life had been one long warfare against the
prejudices of men.

It is most remarkable that the mathematics and philosophy
of Descartes should at first have been appreciated less by his
countrymen than by foreigners. The indiscreet temper of
Descartes alienated the great contemporary French mathematicians,
Roberval, Fermat, Pascal. They continued in investigations
of their own, and on some points strongly opposed
Descartes. The universities of France were under strict
ecclesiastical control and did nothing to introduce his mathematics
and philosophy. It was in the youthful universities of
%% -----File: 203.png---Folio 189-------
\index{Analytic geometry|)}%
\index{Debaune@{De Baune}}%
\index{Inverse tangents (problem of)}%
\index{Maxima and minima}%
Holland that the effect of Cartesian teachings was most
immediate and strongest.

The only prominent Frenchman who immediately followed
in the footsteps of the great master was \textbf{De Beaune} (1601--1652).
He was one of the first to point out that the properties of a
curve can be deduced from the properties of its tangent. This
mode of inquiry has been called the \emph{inverse method of tangents}.
\index{Tangents!inverse problem of}%
He contributed to the theory of equations by considering for
\index{Equations!theory of}%
the first time the upper and lower limits of the roots of
numerical equations.

In the Netherlands a large number of distinguished mathematicians
were at once struck with admiration for the Cartesian
geometry. Foremost among these are \textit{van Schooten}, \textit{John de
\index{Schooten, van}%
\index{Vanschooten@{Van Schooten}}%
Witt}, \textit{van Heuraet}, \textit{Sluze}, and \textit{Hudde}. \textbf{Van Schooten} (died 1660),
\index{Hudde}%
\index{Sluze}%
professor of mathematics at Leyden, brought out an edition
of Descartes' geometry, together with the notes thereon by
\index{Descartes!ref.\ to}%
De~Beaune. His chief work is his \textit{Exercitationes Mathematicæ},
in which he applies the analytical geometry to the solution of
many interesting and difficult problems. The noble-hearted
\textbf{Johann de Witt}, grand-pensioner of Holland, celebrated as a
statesman and for his tragical end, was an ardent geometrician.
He conceived a new and ingenious way of generating conics,
which is essentially the same as that by projective pencils of
rays in modern synthetic geometry. He treated the subject
not synthetically, but with aid of the Cartesian analysis.
\textbf{René François de Sluze} (1622--1685) and \textbf{Johann Hudde} (1633--1704)
made some improvements on Descartes' and Fermat's
\index{Descartes|)}%
methods of drawing tangents, and on the theory of maxima and
minima. With Hudde, we find the first use of three variables
in analytical geometry. He is the author of an ingenious rule
\index{Geometry!analytic|)}%
for finding equal roots. We illustrate it by the equation
$x^3 - x^2 - 8x + 12 = 0$. Taking an arithmetical progression
3,~2, 1,~0, of which the highest term is equal to the degree of
%% -----File: 204.png---Folio 190-------
\index{Circle-squarers}%
\index{Quadrature of curves}%
the equation, and multiplying each term of the equation respectively
by the corresponding term of the progression, we get
$3x^3-2x^2-8x=0$, or $3x^2-2x-8=0$. This last equation
is by one degree lower than the original one. Find the G.C.D.
of the two equations. This is $x - 2$; hence 2~is one of the two
equal roots. Had there been no common divisor, then the
original equation would not have possessed equal roots. Hudde
gave a demonstration for this rule.\Reference{24}

\textbf{Heinrich van Heuraet} must be mentioned as one of the earliest
\index{Heuraet}%
geometers who occupied themselves with success in the
rectification of curves. He observed in a general way that the
\index{Curves!quadrature of}%
\index{Rectification of curves}%
two problems of quadrature and of rectification are really
identical, and that the one can be reduced to the other. Thus
he carried the rectification of the hyperbola back to the
quadrature of the hyperbola. The semi-cubical parabola
\index{Parabola!semi-cubical}%
\index{Semi-cubical parabola}%
$y^3 = ax^2$ was the first curve that was ever rectified absolutely.
This appears to have been accomplished independently by Van
Heuraet in Holland and by \textbf{William Neil} (1637--1670) in England.
\index{Neil}%
According to Wallis the priority belongs to Neil. Soon
after, the cycloid was rectified by Wren and Fermat.
\index{Cycloid}%

The prince of philosophers in Holland, and one of the
greatest scientists of the seventeenth century, was \textbf{Christian
Huygens} (1629--1695), a native of the\DPnote{** Uncapitalized in original} Hague. Eminent as a
\index{Huygens|(}%
physicist and astronomer, as well as mathematician, he was
a worthy predecessor of Sir Isaac Newton. He studied at
Leyden under the younger \textit{Van Schooten}. The perusal of
\index{Schooten, van!ref.\ to}%
\index{Vanschooten@{Van Schooten}!ref.\ to}%
some of his earliest theorems led Descartes to predict his
future greatness. In 1651 Huygens wrote a treatise in which
he pointed out the fallacies of Gregory St.~Vincent (1584--1667)
\index{Stvincent@{St.\ Vincent, Gregory}}%
\index{Vincent, Gregory St.}%
on the subject of quadratures. He himself gave a remarkably
close and convenient approximation to the length of a circular
arc. In 1660 and 1663 he went to Paris and to London. In
1666 he was appointed by Louis~XIV. member of the French
%% -----File: 205.png---Folio 191-------
\index{Analytic geometry}%
\index{Centre!of gravity}%
\index{Centre!of oscillation}%
\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton}%
Academy of Sciences. He was induced to remain in Paris
from that time until~1681, when he returned to his native
city, partly for consideration of his health and partly on
account of the revocation of the Edict of Nantes.

The majority of his profound discoveries were made with
aid of the ancient geometry, though at times he used the
\index{Geometry!analytic}%
geometry of Descartes or of Cavalieri and Fermat. Thus,
\index{Descartes!ref.\ to}%
like his illustrious friend, Sir Isaac Newton, he always showed
\index{Newton!ref.\ to}%
partiality for the Greek geometry. Newton and Huygens
were kindred minds, and had the greatest admiration for each
other. Newton always speaks of him as the ``Summus
Hugenius.''

To the two curves (cubical parabola and cycloid) previously
\index{Cycloid}%
rectified he added a third,---the cissoid. He solved the
\index{Cissoid}%
problem of the catenary, determined the surface of the
\index{Catenary}%
parabolic and hyperbolic conoid, and discovered the properties
of the logarithmic curve and the solids generated by it.
Huygens' \textit{De horologio oscillatorio} (Paris, 1673) is a work that
ranks second only to the \textit{Principia} of Newton and constitutes
\index{Newton's Principia}%
\index{Principia (Newton's)}%
historically a necessary introduction to it.\Reference{13} The book opens
with a description of pendulum clocks, of which Huygens
\index{Pendulum}%
is the inventor. Then follows a treatment of accelerated
motion of bodies falling free, or sliding on inclined planes, or
on given curves,---culminating in the brilliant discovery that
the cycloid is the tautochronous curve. To the theory of
\index{Tautochronous curve}%
curves he added the important theory of ``evolutes.'' After
\index{Evolutes}%
explaining that the tangent of the evolute is normal to the
involute, he applied the theory to the cycloid, and showed by
simple reasoning that the evolute of this curve is an equal
cycloid. Then comes the complete general discussion of the
centre of oscillation. This subject had been proposed for
\index{Oscillation, centre of}%
investigation by Mersenne and discussed by Descartes and
\index{Mersenne}%
Roberval. In Huygens' assumption that the common centre
\index{Roberval!ref.\ to}%
%% -----File: 206.png---Folio 192-------
\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton}%
\index{Saturn's rings}%
\index{Undulatory theory of light}%
of gravity of a group of bodies, oscillating about a horizontal
axis, rises to its original height, but no higher, is expressed
for the first time one of the most beautiful principles of
dynamics, afterwards called the principle of the conservation
\index{Conservation!of \textit{vis viva}}%
of \textit{vis viva}.\Reference{32} The thirteen theorems at the close of the work
relate to the theory of centrifugal force in circular motion.
\index{Centrifugal force}%
This theory aided Newton in discovering the law of gravitation.
\index{Newton!ref.\ to}%

Huygens wrote the first formal treatise on probability. He
\index{Probability}%
proposed the wave-theory of light and with great skill applied
geometry to its development. This theory was long neglected,
but was revived and successfully worked out by Young and
Fresnel a century later. Huygens and his brother improved
the telescope by devising a better way of grinding and polishing
lenses. With more efficient instruments he determined
the nature of Saturn's appendage and solved other astronomical
questions. Huygens' \textit{Opuscula posthuma} appeared
\index{Huygens|)}%
in~1703.

Passing now from Holland to England, we meet there one
of the most original mathematicians of his day---\textbf{John Wallis}
\index{Wallis|(}%
(1616--1703). He was educated for the Church at Cambridge
and entered Holy Orders. But his genius was employed
chiefly in the study of mathematics. In 1649 he was appointed
Savilian professor of geometry at Oxford. He was one of
the original members of the Royal Society, which was founded
in~1663. Wallis thoroughly grasped the mathematical methods
both of Cavalieri and Descartes. His \textit{Conic Sections} is the
\index{Conic sections!more recent researches}%
\index{Descartes!ref.\ to}%
earliest work in which these curves are no longer considered
as sections of a cone, but as curves of the second degree, and
are treated analytically by the Cartesian method of co-ordinates.
In this work Wallis speaks of Descartes in the
highest terms, but in his \textit{Algebra} he, without good reason,
\index{Algebra!seventeenth century}%
accuses Descartes of plagiarising from Harriot. We have
\index{Harriot!ref.\ to}%
%% -----File: 207.png---Folio 193-------
\index{Analytic geometry}%
\index{Curves!quadrature of}%
\index{Descartes!rule of signs}%
\index{Equations!theory of}%
\index{Geometry!analytic}%
\index{Quadrature of curves}%
\index{Rule of signs}%
\index{Signs, rule of}%
already mentioned elsewhere Wallis's solution of the prize
questions on the cycloid, which were proposed by Pascal.

The \textit{Arithmetic of Infinites}, published in 1655, is his greatest
work. By the application of analysis to the Method of Indivisibles,
\index{Indivisibles}%
he greatly increased the power of this instrument for
effecting quadratures. He advanced beyond Kepler by making
more extended use of the ``law of continuity'' and placing
\index{Continuity}%
full reliance in it. By this law he was led to regard the
denominators of fractions as powers with negative exponents.
Thus, the descending geometrical progression $x^3$,~$x^2$, $x^1$, $x^0$, if
continued, gives $x^{-1}$,~$x^{-2}$, $x^{-3}$, etc.; which is the same thing
as $\dfrac{1\Fstr}{x}$,~$\dfrac{1}{x^2}$,~$\dfrac{1}{x^3}$. The exponents of this geometric series are in
continued arithmetical progression, $3$,~$2$, $1$, $0$, $-1$, $-2$,~$-3$.
He also used fractional exponents, which, like the negative,
had been invented long before, but had failed to be generally
introduced. The symbol~$\infty$ for infinity is due to him.
\index{Infinity}%
\index{Infinity!symbol for}%

Cavalieri and the French geometers had ascertained the
\index{Cavalieri!ref.\ to}%
formula for squaring the parabola of any degree, $y = x^m$, $m$~being
a positive integer. By the summation of the powers
of the terms of infinite arithmetical series, it was found that
the curve $y = x^m$ is to the area of the parallelogram having
the same base and altitude as $1$ is to $m + 1$. Aided by the law
of continuity, Wallis arrived at the result that this formula
holds true not only when $m$ is positive and integral, but also
when it is fractional or negative. Thus, in the parabola
$y = \sqrt{px\mathstrut}$, $m = \frac{1}{2}$; hence the area of the parabolic segment
is to that of the circumscribed rectangle as $1 : 1\frac{1}{2}$, or as $2 : 3$.
Again, suppose that in $y = x^m$, $m = -\frac{1}{2}$; then the curve is a
kind of hyperbola referred to its asymptotes, and the hyperbolic
space between the curve and its asymptotes is to the
corresponding parallelogram as $1 : \frac{1}{2}$. If $m = -1$, as in the
common equilateral hyperbola $y = x^{-1}$ or $xy = 1$, then this
ratio is $1 : -1 + 1$, or $1 : 0$, showing that its asymptotic space
%% -----File: 208.png---Folio 194-------
is infinite. But in the case when $m$~is greater than unity and
negative, Wallis was unable to interpret correctly his results.
For example, if $m = -3$, then the ratio becomes $1 : -2$, or
as unity to a negative number. What is the meaning of this?
Wallis reasoned thus: If the denominator is only zero, then
the area is already infinite; but if it is less than zero, then the
area must be more than infinite. It was pointed out later by
Varignon, that this space, supposed to exceed infinity, is really
finite, but taken negatively; that is, measured in a contrary
direction.\Reference{31} The method of Wallis was easily extended to
cases such as $y = ax^\frac{m}{n} + bx^\frac{p}{q}$ by performing the quadrature
for each term separately, and then adding the results.

The manner in which Wallis studied the quadrature of the
circle and arrived at his expression for the value of $\pi$ is
\index{Circle}%
\index{p@{$\pi$}: values for!Wallis'}%
extraordinary. He found that the areas comprised between
the axes, the ordinate corresponding to $x$, and the curves
represented by the equations $y = (1 - x^2)^0$, $y = (1 - x^2)^1$, $y =
(1 - x^2)^2$, $y = (1 - x^2)^3$, etc., are expressed in functions of the
circumscribed rectangles having $x$ and $y$ for their sides, by the
quantities forming the series
\begin{align*}
&x,\\
&x - \tfrac{1}{3}x^3,\\
&x - \tfrac{2}{3}x^3 + \tfrac{1}{5}x^5,\\
&x - \tfrac{3}{3}x^3 + \tfrac{3}{5}x^5 - \tfrac{1}{7}x^7, etc.
\end{align*}
When $x = 1$, these values become respectively $1$,~$\frac{2}{3}$, $\frac{8}{15}$, $\frac{48}{105}$,
etc. Now since the ordinate of the circle is $y = (1 - x^2)^\frac{1}{2}$, the
exponent of which is $\frac{1}{2}$ or the mean value between $0$ and $1$, the
question of this quadrature reduced itself to this: If $0$,~$1$, $2$,
$3$, etc., operated upon by a certain law, give $1$,~$\frac{2}{3}$, $\frac{8}{15}$, $\frac{48}{105}$,
what will $\frac{1}{2}$ give, when operated upon by the same law? He
attempted to solve this by \emph{interpolation}, a method first brought
\index{Interpolation}%
into prominence by him, and arrived by a highly complicated
%% -----File: 209.png---Folio 195-------
\index{Binomial formula}%
\index{p@{$\pi$}: values for!Wallis'}%
\index{Wallis|)}%
and difficult analysis at the following very remarkable expression:
\[
\frac{\pi}{2} = \frac{2 · 2 · 4 · 4 · 6 · 6 · 8 · 8 \dotsm}
                     {1 · 3 · 3 · 5 · 5 · 7 · 7 · 9 \dotsm}
\]

He did not succeed in making the interpolation itself,
because he did not employ literal or general exponents, and
could not conceive a series with more than one term and less
than two, which it seemed to him the interpolated series must
have. The consideration of this difficulty led Newton to the
\index{Newton!ref.\ to}%
discovery of the Binomial Theorem. This is the best place to
\index{Newton's discovery of binomial theorem}%
speak of that discovery. Newton virtually assumed that the
same conditions which underlie the general expressions for
the areas given above must also hold for the expression to be
interpolated. In the first place, he observed that in each
expression the first term is~$x$, that $x$~increases in odd powers,
that the signs alternate $+$~and~$-$, and that the second terms
$\frac{0}{3}x^3$,~$\frac{1}{3}x^3$, $\frac{2}{3}x^3$, $\frac{3}{3}x^3$, are in arithmetical progression. Hence the
first two terms of the interpolated series must be $x - \dfrac{\frac{1}{2}x^3}{3}$.
He next considered that the denominators $1$,~$3$, $5$, $7$, etc., are in
arithmetical progression, and that the coefficients in the numerators
in each expression are the digits of some power of
the number~$11$; namely, for the first expression, $11^0$ or~$1$; for
the second, $11^1$ or $1$,~$1$; for the third, $11^2$ or $1$, $2$,~$1$; for the
fourth, $11^3$ or $1$, $3$, $3$,~$1$; etc. He then discovered that, having
given the second digit (call it~$m$), the remaining digits can
be found by continual multiplication of the terms of the series
$\dfrac{m - 0}{1} · \dfrac{m - 1}{2} · \dfrac{m - 2}{3} · \dfrac{m - 3}{4} ·$ etc. Thus, if $m = 4$, then
$4 · \smash[b]{\dfrac{m - 1}{2}}$ gives~$6$; $6 · \smash[b]{\dfrac{m - 2}{3}}$ gives~$4$; $4 · \smash[b]{\dfrac{m - 3}{4}}$ gives~$1$. Applying
this rule to the required series, since the second term
is $\dfrac{\frac{1}{2}x^3\Fstr[12pt]}{3}$, we have $m = \frac{1}{2}$, and then get for the succeeding coefficients
%% -----File: 210.png---Folio 196-------
\index{Pacioli!ref.\ to}%
in the numerators respectively $-\frac{1}{8}\Fstr$, $+\frac{1}{16}$, $-\frac{5}{128}$,
etc.; hence the required area for the circular segment is
$x - \dfrac{\frac{1}{2}x^3\Fstr[12pt]}{3} - \dfrac{\frac{1}{8}x^5}{5} - \dfrac{\frac{1}{16}x^7}{7} -{}$ etc. Thus he found the interpolated
expression to be an infinite series, instead of one having more
than one term and less than two, as Wallis believed it must
be. This interpolation suggested to Newton a mode of expanding
$(1 - x^2)^\frac{1}{2}$, or, more generally, $(1-x^2)^m$, into a series.
He observed that he had only to omit from the expression
just found the denominators $1$, $3$, $5$, $7$, etc., and to lower
each power of $x$ by unity, and he had the desired expression.
In a letter to Oldenburg (June~13, 1676), Newton states the
theorem as follows: The extraction of roots is much shortened
by the theorem
\index{Binomial formula}%
\[
(P + PQ)^\frac{m}{n} = P^\frac{m}{n} + \frac{m}{n} AQ + \frac{m-n}{2n} BQ + \frac{m-2n}{3n} CQ + \text{etc.},
\]
where $A$ means the first term, $P^\frac{m}{n}$, $B$ the second term, $C$ the
third term, etc. He verified it by actual multiplication, but
gave no regular proof of it. He gave it for any exponent
whatever, but made no distinction between the case when the
exponent is positive and integral, and the others.

It should here be mentioned that very rude beginnings of
the binomial theorem are found very early. The Hindoos and
\index{Newton's discovery of binomial theorem}%
Arabs used the expansions of $(a + b)^2$ and $(a + b)^3$ for extracting
roots; Vieta knew the expansion of $(a + b)^4$; but these
\index{Vieta!ref.\ to}%
were the results of simple multiplication without the discovery
of any law. The binomial coefficients for positive whole exponents
were known to some Arabic and European mathematicians.
Pascal derived the coefficients from the method of
\index{Pascal!ref.\ to}%
what is called the ``arithmetical triangle.'' Lucas de Burgo,
\index{Arithmetical triangle}%
Stifel, Stevinus, Briggs, and others, all possessed something
from which one would think the binomial theorem could have
%% -----File: 211.png---Folio 197-------
\index{Infinite series}%
been gotten with a little attention, ``if we did not know that
such simple relations were difficult to discover.''

Though Wallis had obtained an entirely new expression for~$\pi$,
\index{p@{$\pi$}: values for!Brouncker's}%
\index{Wallis!ref.\ to}%
he was not satisfied with it; for instead of a finite number
of terms yielding an absolute value, it contained merely an
infinite number, approaching nearer and nearer to that value.
He therefore induced his friend, \textbf{Lord Brouncker} (1620?-1684),
\index{Brouncker}%
the first president of the Royal Society, to investigate this
subject. Of course Lord Brouncker did not find what they
were after, but he obtained the following beautiful equality:---
\[
%[** PP: \cfrac leaves a bit too much vertical space]
\pi = \dfrac{4}{1 +
  \dfrac{1\Fstr[10pt]}{2 +
    \dfrac{9\Fstr[10pt]}{2 +
      \dfrac{25\Fstr[10pt]}{2 +
	\dfrac{49\Fstr[10pt]}{2 + \text{etc.}\Fstr[10pt]}
      }
    }
  }
}
\]
Continued fractions, both ascending and descending, appear to
\index{Continued fractions}%
\index{Fractions!continued}%
have been known already to the Greeks and Hindoos, though
not in our present notation. Brouncker's expression gave
birth to the theory of continued fractions.

Wallis' method of quadratures was diligently studied by
his disciples. Lord Brouncker obtained the first infinite
series for the area of an equilateral hyperbola between its
asymptotes. \textbf{Nicolaus Mercator} of Holstein, who had settled
\index{Mercator, N.}%
in England, gave, in his \textit{Logarithmotechnia} (London, 1668), a
similar series. He started with the grand property of the
equilateral hyperbola, discovered in 1647 by \textit{Gregory St.~Vincent},
\index{Stvincent@{St.\ Vincent, Gregory}}%
\index{Vincent, Gregory St.}%
which connected the hyperbolic space between the
asymptotes with the natural logarithms and led to these
\index{Logarithms}%
logarithms being called hyperbolic. By it Mercator arrived
at the logarithmic series, which Wallis had attempted but
\index{Logarithmic series}%
failed to obtain. He showed how the construction of logarithmic
%% -----File: 212.png---Folio 198-------
\index{Prime and ultimate ratios}%
\index{Rectification of curves}%
tables could be reduced to the quadrature of hyperbolic
spaces. Following up some suggestions of Wallis, \textit{William
Neil} succeeded in rectifying the cubical parabola, and \textit{Wren} in
\index{Neil!ref.\ to}%
\index{Parabola}%
\index{Wren!ref.\ to}%
rectifying any cycloidal arc.

A prominent English mathematician and contemporary of
Wallis was \textbf{Isaac Barrow} (1630-1677). He was professor of
\index{Barrow}%
mathematics in London, and then in Cambridge, but in 1669
he resigned his chair to his illustrious pupil, Isaac Newton,
and renounced the study of mathematics for that of divinity.
As a mathematician, he is most celebrated for his method of
tangents. He simplified the method of Fermat by introducing
\index{Fermat!ref.\ to}%
\index{Tangents!direct problem of}%
two infinitesimals instead of one, and approximated to the
course of reasoning afterwards followed by Newton in his doctrine
on Ultimate Ratios.

He considered the infinitesimal right triangle $ABB'$ having
%[** PP: Width-dependent line break]
%[Illustration]
\begin{wrapfigure}[8]{l}{2.125in}
\Graphic{2.125in}{212a}
\end{wrapfigure}
for its sides the difference between two successive ordinates,
the distance between them, and the portion of the curve intercepted
by them. This triangle is similar to $BPT$, formed by
the ordinate, the tangent, and the sub-tangent. Hence, if we
know the ratio of $B'A$ to $BA$, then we know the ratio of the
ordinate and the sub-tangent, and the tangent can be constructed
at once. For any curve, say $y^2 =px$, the ratio of $B'A$
to $BA$ is determined from its
equation as follows: If $x$ receives
an infinitesimal increment
$PP'= e$, then $y$ receives
an increment $B'A = a$, and the
equation for the ordinate $B'P'$
becomes $y^2+2ay+a^2=px+pe$.
Since $y^2 =px$, we get $2ay + a^2 = pe$; neglecting higher powers
of the infinitesimals, we have $2ay = pe$, which gives
\[
a:e=p:2y=p:2\sqrt{px}.
\]
%% -----File: 213.png---Folio 199-------
But $a : e = \text{the ordinate} : \text{the sub-tangent}$; hence
\[
p : 2\sqrt{px} = \sqrt{px} : \text{sub-tangent},
\]
giving $2x$ for the value of the sub-tangent. This method differs
from that of the differential calculus only in notation.\Reference{31}

\Section{Newton to Euler}

It has been seen that in France prodigious scientific progress
was made during the beginning and middle of the seventeenth
century. The toleration which marked the reign of Henry IV.
and Louis~XIII. was accompanied by intense intellectual
activity. Extraordinary confidence came to be placed in the
power of the human mind. The bold intellectual conquests
of Descartes, Fermat, and Pascal enriched mathematics with
imperishable treasures. During the early part of the reign
of Louis~XIV. we behold the sunset splendour of this glorious
period. Then followed a night of mental effeminacy. This
lack of great scientific thinkers during the reign of Louis~XIV.
may be due to the simple fact that no great minds were born;
but, according to Buckle, it was due to the paternalism, to
the spirit of dependence and subordination, and to the lack
of toleration, which marked the policy of Louis~XIV\@.

In the absence of great French thinkers, Louis~XIV. surrounded
himself by eminent foreigners. Römer from Denmark,
\index{Romer@{Römer}}%
Huygens from Holland, Dominic Cassini from Italy,
were the mathematicians and astronomers adorning his court.
They were in possession of a brilliant reputation before going
to Paris. Simply because they performed scientific work in
Paris, that work belongs no more to France than the discoveries
of Descartes belong to Holland, or those of Lagrange
to Germany, or those of Euler and Poncelet to Russia. We
%% -----File: 214.png---Folio 200-------
\index{Fluxions}%
must look to other countries than France for the great scientific
men of the latter part of the seventeenth century.

About the time when Louis~XIV. assumed the direction
of the French government Charles II. became king of England.
At this time England was extending her commerce
and navigation, and advancing considerably in material prosperity.
A strong intellectual movement took place, which
was unwittingly supported by the king. The age of poetry
was soon followed by an age of science and philosophy. In
two successive centuries England produced Shakespeare and
Newton!
\index{Newton!ref.\ to}%

Germany still continued in a state of national degradation.
The Thirty Years' War had dismembered the empire and
brutalised the people. Yet this darkest period of Germany's
history produced Leibniz, one of the greatest geniuses of
\index{Leibniz!ref.\ to}%
modern times.

There are certain focal points in history toward which
the lines of past progress converge, and from which radiate
the advances of the future. Such was the age of Newton
and Leibniz in the history of mathematics. During fifty
years preceding this era several of the brightest and acutest
mathematicians bent the force of their genius in a direction
which finally led to the discovery of the infinitesimal calculus
by Newton and Leibniz. Cavalieri, Roberval, Fermat, Descartes,
Wallis, and others had each contributed to the new
geometry. So great was the advance made, and so near
was their approach toward the invention of the infinitesimal
analysis, that both Lagrange and Laplace pronounced their
countryman, Fermat, to be the true inventor of it. The differential
\index{Differential calculus}%
calculus, therefore, was not so much an individual
discovery as the grand result of a succession of discoveries
by different minds. Indeed, no great discovery ever flashed
upon the mind at once, and though those of Newton will
%% -----File: 215.png---Folio 201-------
influence mankind to the end of the world, yet it must be
admitted that Pope's lines are only a ``poetic fancy'':---
\begin{center}
\footnotesize%
\settowidth{\TmpLen}{``\,God said, `Let Newton be,' and all was light.''}%
\parbox{\TmpLen}{%
  ``\,Nature and Nature's laws lay hid in night; \\
  \settowidth{\TmpLen}{``\,}%
  \makebox[\TmpLen][c]{}God said, `Let Newton be,' and all was light.''}%
\end{center}

\textbf{Isaac Newton} (1642-1727) was born at Woolsthorpe, in
\index{Newton|(}%
Lincolnshire, the same year in which Galileo died. At his
birth he was so small and weak that his life was despaired of.
His mother sent him at an early age to a village school, and
in his twelfth year to the public school at Grantham. At
first he seems to have been very inattentive to his studies
and very low in the school; but when, one day, the little
Isaac received a severe kick upon his stomach from a boy
who was above him, he laboured hard till he ranked higher
in school than his antagonist. From that time he continued
to rise until he was the head boy.\Reference{33} At Grantham, Isaac
showed a decided taste for mechanical inventions. He constructed
a water-clock, a wind-mill, a carriage moved by the
person who sat in it, and other toys. When he had attained
his fifteenth year his mother took him home to assist her in
the management of the farm, but his great dislike for farm-work %[** PP: Only instance]
and his irresistible passion for study, induced her to
send him back to Grantham, where he remained till his
eighteenth year, when he entered Trinity College, Cambridge
(1660). Cambridge was the real birthplace of Newton's
genius. Some idea of his strong intuitive powers may be
drawn from the fact that he regarded the theorems of ancient
geometry as self-evident truths, and that, without any preliminary
study, he made himself master of Descartes' \textit{Geometry}.
He afterwards regarded this neglect of elementary geometry
a mistake in his mathematical studies, and he expressed to
Dr. Pemberton his regret that ``he had applied himself to the
\index{Pemberton}%
works of Descartes and other algebraic writers before he had
%% -----File: 216.png---Folio 202-------
\index{Binomial formula}%
\index{Exponents}%
considered the \textit{Elements} of Euclid with that attention which
so excellent a writer deserves.'' Besides Descartes' \textit{Geometry},
he studied Oughtred's \textit{Clavis}, Kepler's \textit{Optics}, the works of
\index{Kepler!ref.\ to}%
\index{Oughtred!ref.\ to}%
Vieta, Schooten's \textit{Miscellanies}, Barrow's \textit{Lectures}, and the
\index{Barrow!ref.\ to}%
\index{Schooten, van!ref.\ to}%
\index{Vanschooten@{Van Schooten}!ref.\ to}%
\index{Vieta!ref.\ to}%
works of Wallis. He was particularly delighted with Wallis'
\index{Wallis!ref.\ to}%
\textit{Arithmetic of Infinites}, a treatise fraught with rich and varied
suggestions. Newton had the good fortune of having for
a teacher and fast friend the celebrated Dr. Barrow, who
had been elected professor of Greek in~1660, and was made
Lucasian professor of mathematics in~1663. The mathematics
of Barrow and of Wallis were the starting-points
from which Newton, with a higher power than his masters',
moved onward into wider fields. Wallis had effected the
quadrature of curves whose ordinates are expressed by any
\index{Curves!quadrature of}%
integral and positive power of $(1-x^2)$. We have seen how
Wallis attempted but failed to interpolate between the areas
thus calculated, the areas of other curves, such as that of
the circle; how Newton attacked the problem, effected the
interpolation, and discovered the Binomial Theorem, which
afforded a much easier and direct access to the quadrature
of curves than did the method of interpolation; for even
though the binomial expression for the ordinate be raised
to a fractional or negative power, the binomial could at once
be expanded into a series, and the quadrature of each separate
term of that series could be effected by the method of Wallis.
Newton introduced the system of literal indices.

Newton's study of quadratures soon led him to another
and most profound invention. He himself says that in 1665
and 1666 he conceived the method of fluxions and applied
\index{Fluxions|(}%
them to the quadrature of curves. Newton did not communicate
the invention to any of his friends till 1669, when
he placed in the hands of Barrow a tract, entitled \textit{De Analysi
per Æquationes Numero Terminorum Infinitas}, which was sent
%% -----File: 217.png---Folio 203-------
\index{Hudde!ref.\ to}%
by Barrow to Collins, who greatly admired it. In this treatise
\index{Barrow!ref.\ to}%
\index{Collins}%
the principle of fluxions, though distinctly pointed out, is only
partially developed and explained. Supposing the abscissa
to increase uniformly in proportion to the time, he looked
upon the area of a curve as a nascent quantity increasing
by continued fluxion in the proportion of the length of the
ordinate. The expression which was obtained for the fluxion
he expanded into a finite or infinite series of monomial terms,
\index{Infinite series}%
to which Wallis' rule was applicable. Barrow urged Newton
to publish this treatise; ``but the modesty of the author, of
which the excess, if not culpable, was certainly in the present
instance very unfortunate, prevented his compliance.''\Reference{26} Had
this tract been published then, instead of forty-two years
later, there would probably have been no occasion for that
long and deplorable controversy between Newton and Leibniz.

For a long time Newton's method remained unknown, except
to his friends and their correspondents. In a letter
to Collins, dated December~10th, 1672, Newton states the fact
of his invention with one example, and then says: ``This
is one particular, or rather corollary, of a general method,
which extends itself, without any troublesome calculation, not
only to the drawing of tangents to any curve lines, whether
geometrical or mechanical, or anyhow respecting right lines
or other curves, but also to the resolving other abstruser
kinds of problems about the crookedness, areas, lengths,
centres of gravity of curves, etc.; nor is it (as Hudden's
method of Maximis and Minimis) limited to equations which
are free from surd quantities. This method I have interwoven
with that other of working in equations, by reducing
them to infinite series.''

These last words relate to a treatise he composed in the
year 1671, entitled \textit{Method of Fluxions}, in which he aimed
to represent his method as an independent calculus and as
%% -----File: 218.png---Folio 204-------
a complete system. This tract was intended as an introduction
to an edition of Kinckhuysen's \textit{Algebra}, which he had
\index{Kinckhuysen}%
undertaken to publish. ``But the fear of being involved in
disputes about this new discovery, or perhaps the wish to
render it more complete, or to have the sole advantage of
employing it in his physical researches, induced him to abandon
this design.''\Reference{33}

Excepting two papers on optics, all of his works appear
to have been published only after the most pressing solicitations
of his friends and against his own wishes.\Reference{34} His researches
on light were severely criticised, and he wrote in~1675:
``I was so persecuted with discussions arising out of
my theory of light that I blamed my own imprudence for
parting with so substantial a blessing as my quiet to run
after a shadow.''

The \textit{Method of Fluxions}, translated by J. Colson from Newton's
\index{Colson}%
Latin, was first published in~1736, or sixty-five years
after it was written. In it he explains first the expansion
into series of fractional and irrational quantities,---a subject
which, in his first years of study, received the most careful
attention. He then proceeds to the solution of the two following
mechanical problems, which constitute the pillars, so
to speak, of the abstract calculus:---

``I\@. The length of the space described being continually
(\ie~at all times) given; to find the velocity of the motion at
any time proposed.

``II\@. The velocity of the motion being continually given; to
find the length of the space described at any time proposed.''

Preparatory to the solution, Newton says: ``Thus, in the
equation $y = x^2$, if $y$ represents the length of the space at any
time described, which (time) another space~$x$, by increasing
with an uniform celerity~$\dot{x}$, measures and exhibits as described:
then $2x\dot{x}$ will represent the celerity by which the space~$y$,
%% -----File: 219.png---Folio 205-------
\index{Notation!differential calculus}%
at the same moment of time, proceeds to be described; and
contrarywise.''

``But whereas we need not consider the time here, any farther
than it is expounded and measured by an equable local
motion; and besides, whereas only quantities of the same
kind can be compared together, and also their velocities of
increase and decrease; therefore, in what follows I shall have
no regard to time formally considered, but I shall suppose
some one of the quantities proposed, being of the same kind,
to be increased by an equable fluxion, to which the rest may be
referred, as it were to time; and, therefore, by way of analogy,
it may not improperly receive the name of time.'' In this
statement of Newton there is contained a satisfactory answer
to the objection which has been raised against his method,
that it introduces into analysis the foreign idea of motion. A
quantity thus increasing by uniform fluxion, is what we now
call an independent variable.

Newton continues: ``Now those quantities which I consider
as gradually and indefinitely increasing, I shall hereafter call
\emph{fluents}, or \emph{flowing quantities}, and shall represent them by the
\index{Fluents}%
final letters of the alphabet, $v$,~$x$, $y$, and~$z$; \dots and the velocities
by which every fluent is increased by its generating motion
(which I may call \emph{fluxions}, or simply velocities, or celerities),
I shall represent by the same letters pointed, thus, $\dot{v}$,~$\dot{x}$, $\dot{y}$,~$\dot{z}$.
That is, for the celerity of the quantity~$v$ I shall put~$\dot{v}$, and so
for the celerities of the other quantities $x$,~$y$, and~$z$, I shall put
$\dot{x}$,~$\dot{y}$, and~$\dot{z}$, respectively.'' It must here be observed that Newton
does not take the fluxions themselves infinitely small.
The ``moments of fluxions,'' a term introduced further on, are
\index{Moments in fluxionary calculus}%
infinitely small quantities. These ``moments,'' as defined and
used in the \textit{Method of Fluxions}, are substantially the differentials
of Leibniz. De Morgan points out that no small amount of
\index{Demorgan@{De Morgan}!ref.\ to}%
confusion has arisen from the use of the word \emph{fluxion} and the
%% -----File: 220.png---Folio 206-------
\index{Fluents}%
notation~$\dot{x}$ by all the English writers previous to~1704, excepting
Newton and Cheyne, in the sense of an infinitely small increment.\Reference{35}
\index{Cheyne}%
Strange to say, even in the \textit{Commercium Epistolicum}
\index{Commercium epistolicum}%
the words \emph{moment} and \emph{fluxion} appear to be used as synonymous.

After showing by examples how to solve the first problem,
Newton proceeds to the demonstration of his solution:---

``The moments of flowing quantities (that is, their indefinitely
\index{Moments in fluxionary calculus}%
small parts, by the accession of which, in infinitely
small portions of time, they are continually increased) are as
the velocities of their flowing or increasing.

``Wherefore, if the moment of any one (as~$x$) be represented
by the product of its celerity~$\dot{x}$ into an infinitely small quantity~$0$
(\ie by~$\dot{x}0$), the moments of the others, $v$, $y$, $z$, will be represented
by $\dot{v}0$, $\dot{y}0$,~$\dot{z}0$; because $\dot{v}0$,~$\dot{x}0$, $\dot{y}0$, and~$\dot{z}0$ are to each other
as $\dot{v}$,~$\dot{x}$, $\dot{y}$, and~$\dot{z}$.

``Now since the moments, as $\dot{x}0$ and~$\dot{y}0$, are the indefinitely
little accessions of the flowing quantities $x$~and~$y$, by which
those quantities are increased through the several indefinitely
little intervals of time, it follows that those quantities, $x$~and~$y$,
after any indefinitely small interval of time, become
$x + \dot{x}0$ and $y + \dot{y}0$, and therefore the equation, which at all
times indifferently expresses the relation of the flowing quantities,
will as well express the relation between $x + \dot{x}0$ and
$y + \dot{y}0$, as between $x$~and~$y$; so that $x + \dot{x}0$ and $y + \dot{y}0$ may
be substituted in the same equation for those quantities, instead
of $x$~and~$y$. Thus let any equation $x^3 - ax^2 + axy - y^3 = 0$
be given, and substitute $x + \dot{x}0$ for~$x$, and $y + \dot{y}0$ for~$y$, and
there will arise
\[
\left.
\begin{alignedat}{4}
&\phantom{a}x^3    &&+ 3x^2\dot{x}0 &&+ 3x\dot{x}0\dot{x}0 &&+ \dot{x}^3 0^3 \\
-{}&ax^2 &&- 2ax\dot{x}0  &&- a\dot{x}0\dot{x}0 \\
+{}&axy  &&+ ay\dot{x}0   &&+ a\dot{x}0\dot{y}0 \\
&       &&+ ax\dot{y}0 \\
-{}&y^3  &&- 3y^2\dot{y}0 &&- 3y\dot{y}0\dot{y}0 &&- \dot{y}^3 0^3
\end{alignedat}
\right\} = 0.
\]
%% -----File: 221.png---Folio 207-------

``Now, by supposition, $x^3-ax^2+axy-y^3=0$, which therefore,
being expunged and the remaining terms being divided
by 0, there will remain
\begin{align*}
3x^2\dot{x} &- 2ax\dot{x} + ay\dot{x} + ax\dot{y} - 3y^2\dot{y} + 3x\dot{x}\dot{x}0 - a\dot{x}\dot{x}0 + a\dot{x}\dot{y}0 \\
            &- 3y\dot{y}\dot{y}0 + \dot{x}^3 00 - \dot{y}^300 = 0.
\end{align*}
But whereas zero is supposed to be infinitely little, that it
may represent the moments of quantities, the terms that are
multiplied by it will be nothing in respect of the rest (\textit{termini
in eam ducti pro nihilo possunt haberi cum aliis collati}); therefore
I reject them, and there remains
\[
3x^2\dot{x} - 2ax\dot{x} + ay\dot{x} + ax\dot{y} - 3y^2\dot{y} = 0,
\]
as above in Example~I.'' Newton here uses infinitesimals.
\index{Infinitesimals}%

Much greater than in the first problem were the difficulties
encountered in the solution of the second problem, involving,
as it does, inverse operations which have been taxing the skill
of the best analysts since his time. Newton gives first a
special solution to the second problem in which he resorts
to a rule for which he has given no proof.

In the general solution of his second problem, Newton
assumed homogeneity with respect to the fluxions and then
considered three cases: (1)~when the equation contains two
fluxions of quantities and but one of the fluents; (2)~when
the equation involves both the fluents as well as both the fluxions;
(3)~when the equation contains the fluents and the fluxions
of three or more quantities. The first case is the easiest
since it requires simply the integration of $\dfrac{dy}{dx}=f(x)$, to which
his ``special solution'' is applicable. The second case demanded
nothing less than the general solution of a differential
equation of the first order. Those who know what
efforts were afterwards needed for the complete exploration
of this field in analysis, will not depreciate Newton's work
%% -----File: 222.png---Folio 208-------
\index{Partial differential equations}%
even though he resorted to solutions in form of infinite series.
\index{Infinite series}%
Newton's third case comes now under the solution of partial
differential equations. He took the equation $2\dot{x} - \dot{z} + x\dot{y} = 0$
and succeeded in finding a particular integral of it.

The rest of the treatise is devoted to the determination of
maxima and minima, the radius of curvature of curves, and
\index{Maxima and minima}%
other geometrical applications of his fluxionary calculus. All
this was done previous to the year~1672.

It must be observed that in the \textit{Method of Fluxions} (as well
as in his \textit{De Analysi} and all earlier papers) the method
employed by Newton is strictly infinitesimal, and in substance
like that of Leibniz. Thus, the original conception of the
\index{Leibniz!ref.\ to}%
calculus in England, as well as on the Continent, was based
on infinitesimals. The fundamental principles of the fluxionary
\index{Infinitesimals}%
calculus were first given to the world in the \textit{Principia}; but its
\index{Newton's Principia}%
\index{Principia (Newton's)}%
peculiar notation did not appear until published in the second
volume of Wallis' \textit{Algebra} in 1693. The exposition given in
the \textit{Algebra} was substantially a contribution of Newton; it
rests on infinitesimals. In the first edition of the \textit{Principia}
(1687) the description of fluxions is likewise founded on
infinitesimals, but in the second (1713) the foundation is
somewhat altered. In Book~II. Lemma~II. of the first edition
we read: ``Cave tamen intellexeris particulas finitas. \textit{Momenta
quam primum finitæ sunt magnitudinis, desinunt esse momenta.
Finiri enim repugnat aliquatenus perpetuo eorum incremento vel
decremento.} Intelligenda sunt principia jamjam nascentia
finitorum magnitudinum.'' In the second edition the two
sentences which we print in italics are replaced by the
following: ``Particulæ finitæ non sunt momenta sed quantitates
ipsæ ex momentis genitæ.'' Through the difficulty of
the phrases in both extracts, this much distinctly appears,
that in the first, moments are infinitely small quantities.
What else they are in the second is not clear.\Reference{35} In the
%% -----File: 223.png---Folio 209-------
\index{Fresnel's wave-surface}%
\textit{Quadrature of Curves} of 1704, the infinitely small quantity is
completely abandoned. It has been shown that in the \textit{Method
of Fluxions} Newton rejected terms involving the quantity 0,
because they are infinitely small compared with other terms.
This reasoning is evidently erroneous; for as long as 0 is a
quantity, though ever so small, this rejection cannot be made
without affecting the result. Newton seems to have felt this,
for in the \textit{Quadrature of Curves} he remarked that ``in mathematics
the minutest errors are not to be neglected" (errores
quam minimi in rebus mathematicis non sunt contemnendi).

The early distinction between the system of Newton and
Leibniz lies in this, that Newton, holding to the conception
\index{Leibniz!ref.\ to}%
of velocity or fluxion, used the infinitely small increment as a
means of determining it, while with Leibniz the relation of the
infinitely small increments is itself the object of determination.
The difference between the two rests mainly upon a difference
in the mode of generating quantities.\Reference{35}

We give Newton's statement of the method of fluxions or
rates, as given in the introduction to his \textit{Quadrature of Curves}.
``I consider mathematical quantities in this place not as
consisting of very small parts, but as described by a continued
motion. Lines are described, and thereby generated, not by
the apposition of parts, but by the continued motion of points;
superficies by the motion of lines; solids by the motion of
superficies; angles by the rotation of the sides; portions of
time by continual flux: and so on in other quantities. These
geneses really take place in the nature of things, and are
daily seen in the motion of bodies\dots.

``Fluxions are, as near as we please (\textit{quam proxime}), as the
increments of fluents generated in times, equal and as small as
possible, and to speak accurately, they are in the prime ratio
of nascent increments; yet they can be expressed by any lines
whatever, which are proportional to them."
%% -----File: 224.png---Folio 210-------

Newton exemplifies this last assertion by the problem of
tangency: Let $AB$ be the abscissa, $BC$~the ordinate, $VCH$~the
tangent, $Ec$~the increment of the ordinate, which produced
meets $VH$ at~$T$, and $Cc$~the increment of the curve.
The right line~$Cc$ being produced to~$K$, there are formed
three small triangles, the rectilinear~$CEc$, the mixtilinear~$CEc$,
and the rectilinear~$CET$. Of these, the first is evidently
the smallest, and the last the greatest. Now suppose
the ordinate~$bc$ to move into the place~$BC$, so that the
point~$c$ exactly coincides
with %[** PP: Width-dependent line break]
%[Illustration:]
\begin{wrapfigure}[8]{l}{2.75in}
\Graphic{2.75in}{224a}
\end{wrapfigure}
the
point $C$; $CK$, and
therefore the curve
$Cc$, is coincident
with the tangent~$CH$,
$Ec$ is absolutely
equal to $ET$,
and the mixtilinear
evanescent triangle $CEc$ is, in the last form, similar to the
triangle $CET$; and its evanescent sides $CE$, $Ec$, $Cc$, will be
proportional to $CE$, $ET$, and~$CT$, the sides of the triangle~$CET$.
Hence it follows that the fluxions of the lines $AB$,
$BC$, $AC$, being in the last ratio of their evanescent increments,
are proportional to the sides of the triangle~$CET$, or, which is
all one, of the triangle~$VBC$ similar thereunto. As long as
the points $C$~and~$c$ are distant from each other by an interval,
however small, the line~$CK$ will stand apart by a small angle
from the tangent~$CH$. But when $CK$ coincides with~$CH$, and
the lines $CE$, $Ec$, $cC$ reach their ultimate ratios, then the
points $C$~and~$c$ accurately coincide and are one and the same.
Newton then adds that ``in mathematics the minutest errors
are not to be neglected.'' This is plainly a rejection of the
postulates of Leibniz. The doctrine of infinitely small quantities
\index{Leibniz!ref.\ to}%
%% -----File: 225.png---Folio 211-------
is here renounced in a manner which would lead one
to suppose that Newton had never held it himself. Thus it
appears that Newton's doctrine was different in different
periods. Though, in the above reasoning, the Charybdis of
infinitesimals is safely avoided, the dangers of a Scylla stare
\index{Infinitesimals}%
us in the face. We are required to believe that a point may
be considered a triangle, or that a triangle can be inscribed
in a point; nay, that three dissimilar triangles become similar
and equal when they have reached their ultimate form in one
and the same point.

In the introduction to the \textit{Quadrature of Curves} the fluxion
of $x^n$ is determined as follows:---

``In the same time that~$x$, by flowing, becomes~$x + 0$, the
power~$x^n$ becomes $(x+0)^n$, \ie~by the method of infinite series
\[
x^n + n0x^{n-1} + \frac{n^2-n}{2} 0^{2} x^{n-2} + \text{etc.},
\]
and the increments
\[
0 \text{ and } n0 x^{n-1} + \frac{n^2-n}{2} 0^{2} x^{n-2} + \text{etc.},
\]
are to one another as
\[
1 \text{ to } nx^{n-1} + \frac{n^2-n}{2} 0 x^{n-2} + \text{etc.}
\]

``Let now the increments vanish, and their last proportion
will be $1$~to~$nx^{n-1}$: hence the fluxion of the quantity~$x$ is to
the fluxion of the quantity~$x^n$ as~$1 : nx^{n-1}$.

``The fluxion of lines, straight or curved, in all cases whatever,
as also the fluxions of superficies, angles, and other
quantities, can be obtained in the same manner by the method
of prime and ultimate ratios. But to establish in this way
the analysis of infinite quantities, and to investigate prime
and ultimate ratios of finite quantities, nascent or evanescent,
is in harmony with the geometry of the ancients; and I have
endeavoured to show that, in the method of fluxions, it is not
%% -----File: 226.png---Folio 212-------
\index{Astronomy!Newton|(}%
\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton|(}%
necessary to introduce into geometry infinitely small quantities.''
This mode of differentiating does not remove all the
difficulties connected with the subject. When $0$~becomes
nothing, then we get the ratio $\dfrac{0}{0} = nx^{n-1}$, which needs further
elucidation. Indeed, the method of Newton, as delivered by
himself, is encumbered with difficulties and objections. Among
the ablest admirers of Newton, there have been obstinate disputes
respecting his explanation of his method of ``prime and
\index{Prime and ultimate ratios}%
ultimate ratios.''

The so-called ``method of limits'' is frequently attributed
\index{Limits, method of}%
to Newton, but the pure method of limits was never adopted
by him as his method of constructing the calculus. All he
did was to establish in his \textit{Principia} certain principles which
\index{Newton's Principia|(}%
\index{Principia (Newton's)|(}%
are applicable to that method, but which he used for a different
purpose. The first lemma of the first book has been made the
foundation of the method of limits:---

``Quantities and the ratios of quantities, which in any finite
time converge continually to equality, and before the end of
that time approach nearer the one to the other than by any
given difference, become ultimately equal.''

In this, as well as in the lemmas following this, there are
obscurities and difficulties. Newton appears to teach that a
variable quantity and its limit will ultimately coincide and be
equal. But it is now generally agreed that in the clearest
statements which have been made of the theory of limits, the
variable does not actually reach its limit, though the variable
may approach it as near as we please.

The full title of Newton's \textit{Principia} is \textit{Philosophiæ Naturalis
Principia Mathematica}. It was printed in 1687 under
the direction, and at the expense, of Dr.~Edmund Halley.
A second edition was brought out in 1713 with many alterations
and improvements, and accompanied by a preface from
%% -----File: 227.png---Folio 213-------
\index{Fluxions|)}%
\index{Kepler's laws}%
\index{Laws of motion}%
\index{Motion, laws of}%
Mr.~Cotes. It was sold out in a few months, but a pirated
edition published in Amsterdam supplied the demand.\Reference{34} The
third and last edition which appeared in England during
Newton's lifetime was published in 1726 by Henry Pemberton.
The \textit{Principia} consists of three books, of which the first two,
constituting the great bulk of the work, treat of the mathematical
principles of natural philosophy, namely, the laws and
conditions of motions and forces. In the third book is drawn
up the constitution of the universe as deduced from the foregoing
principles. The great principle underlying this memorable
work is that of universal gravitation. The first book
\index{Gravitation, theory of}%
\index{Newton's discovery of universal gravitation}%
was completed on April~28, 1686. After the remarkably short
period of three months, the second book was finished. The
third book is the result of the next nine or ten months'
labours. It is only a sketch of a much more extended elaboration
of the subject which he had planned, but which was never
brought to completion.

The law of gravitation is enunciated in the first book. Its
discovery envelops the name of Newton in a halo of perpetual
glory. The current version of the discovery is as follows: it
was conjectured by Hooke, Huygens, Halley, Wren, Newton,
\index{Halley}%
\index{Hooke}%
\index{Huygens!ref.\ to}%
\index{Wren!ref.\ to}%
and others, that, if Kepler's third law was true (its absolute
\index{Kepler!ref.\ to}%
accuracy was doubted at that time), then the attraction
between the earth and other members of the solar system
varied inversely as the square of the distance. But the proof
of the truth or falsity of the guess was wanting. In~1666
Newton reasoned, in substance, that if $g$~represent the acceleration
of gravity on the surface of the earth, $r$~be the earth's
radius, $R$~the distance of the moon from the earth, $T$~the time
of lunar revolution, and $a$~a degree at the equator, then, if the
law is true,
\[
g \frac{r^2}{R^2} = 4 \pi^2 \frac{R}{T^2}, \text{ or } g = \frac{4\pi}{T^2}\left(\frac{R}{r}\right)^3 · 180a.
\]
%% -----File: 228.png---Folio 214-------
\index{Centrifugal force}%
The data at Newton's command gave $R = 60.4r$, $T = 2,360,628$
seconds, but $a$ only $60$ instead of $69\frac{1}{2}$ English miles. This
wrong value of~$a$ rendered the calculated value of~$g$ smaller
than its true value, as known from actual measurement. It
looked as though the law of inverse squares were not the true
law, and Newton laid the calculation aside. In~1684 he casually
ascertained at a meeting of the Royal Society that Jean
Picard had measured an arc of the meridian, and obtained a
\index{Picard, J.}%
more accurate value for the earth's radius. Taking the corrected
\index{Earth!size of}%
value for~$a$, he found a figure for~$g$ which corresponded
to the known value. Thus the law of inverse squares was
verified. In a scholium in the \textit{Principia}, Newton acknowledged
his indebtedness to Huygens for the laws on centrifugal
\index{Huygens!ref.\ to}%
force employed in his calculation.

The perusal by the astronomer Adams of a great mass of
\index{Adams!ref.\ to}%
unpublished letters and manuscripts of Newton forming the
Portsmouth collection (which remained private property
until~1872, when its owner placed it in the hands of the
University of Cambridge) seems to indicate that the difficulties
encountered by Newton in the above calculation were of a
different nature. According to Adams, Newton's numerical
verification was fairly complete in~1666, but Newton had not
been able to determine what the attraction of a spherical shell
upon an external point would be. His letters to Halley show
\index{Halley}%
that he did not suppose the earth to attract as though all its
mass were concentrated into a point at the centre. He could
not have asserted, therefore, that the assumed law of gravity
was verified by the figures, though for long distances he might
have claimed that it yielded close approximations. When
Halley visited Newton in~1684, he requested Newton to determine
what the orbit of a planet would be if the law of attraction
were that of inverse squares. Newton had solved a
similar problem for Hooke in~1679, and replied at once that it
%% -----File: 229.png---Folio 215-------
\index{Ellipsoid!(attraction of)}%
\index{Solid of least resistance}%
was an ellipse. After Halley's visit, Newton, with Picard's
\index{Picard, J.}%
new value for the earth's radius, reviewed his early calculation,
\index{Earth!size of}%
and was able to show that if the distances between the
bodies in the solar system were so great that the bodies might
be considered as points, then their motions were in accordance
with the assumed law of gravitation. In 1685 he completed
his discovery by showing that a sphere whose density at any
point depends only on the distance from the centre attracts
an external point as though its whole mass were concentrated
at the centre.\Reference{34}

Newton's unpublished manuscripts in the Portsmouth collection
show that he had worked out, by means of fluxions and
fluents, his lunar calculations to a higher degree of approximation
than that given in the \textit{Principia}, but that he was unable
to interpret his results geometrically. The papers in that collection
throw light upon the mode by which Newton arrived
at some of the results in the \textit{Principia}, as, for instance, the
famous construction in Book~II., Prop.~25, which is unproved
in the \textit{Principia}, but is demonstrated by him twice in a draft
of a letter to David Gregory, of Oxford.\Reference{34}
\index{Gregory, David F}%

It is chiefly upon the \textit{Principia} that the fame of Newton
\index{Newton's Principia|)}%
\index{Principia (Newton's)|)}%
rests. Brewster calls it ``the brightest page in the records of
human reason.'' Let us listen, for a moment, to the comments
of Laplace, the foremost among those followers of Newton who
\index{Laplace!ref.\ to}%
grappled with the subtle problems of the motions of planets
under the influence of gravitation: ``Newton has well established
the existence of the principle which he had the merit
of discovering, but the development of its consequences and
advantages has been the work of the successors of this great
mathematician. The imperfection of the infinitesimal calculus,
when first discovered, did not allow him completely to
resolve the difficult problems which the theory of the universe
offers; and he was oftentimes forced to give mere hints, which
%% -----File: 230.png---Folio 216-------
\index{Astronomy!Newton|)}%
\index{Mechanics!Descartes, Wallis, Wren, Huygens, Newton|)}%
were always uncertain till confirmed by rigorous analysis.
Notwithstanding these unavoidable defects, the importance
and the generality of his discoveries respecting the system of
the universe, and the most interesting points of natural philosophy,
the great number of profound and original views,
which have been the origin of the most brilliant discoveries of
the mathematicians of the last century, which were all presented
with much elegance, will insure to the \textit{Principia} a lasting
pre-eminence over all other productions of the human
mind."

Newton's \textit{Arithmetica Universalis}, consisting of algebraical
lectures delivered by him during the first nine years he was
professor at Cambridge, were published in~1707, or more than
thirty years after they were written. This work was published
by Mr.~Whiston. We are not accurately informed how
\index{Whiston}%
Mr.~Whiston came in possession of it, but according to some
authorities its publication was a breach of confidence on his
part.

The \textit{Arithmetica Universalis} contains new and important
results on the theory of equations. His theorem on the
\index{Equations!theory of}%
sums of powers of roots is well known. Newton showed
that in equations with real coefficients, imaginary roots always
occur in pairs. His inventive genius is grandly displayed
in his rule for determining the inferior limit of the number
of imaginary roots, and the superior limits for the number
of positive and negative roots. Though less expeditious than
Descartes', Newton's rule always gives as close, and generally
\index{Descartes!ref.\ to}%
closer, limits to the number of positive and negative roots.
Newton did not prove his rule. It awaited demonstration
for a century and a half, until, at last, Sylvester established
\index{Sylvester~ref.\ to}%
a remarkable general theorem which includes Newton's rule
as a special case.

The treatise on \textit{Method of Fluxions} contains Newton's method
%% -----File: 231.png---Folio 217-------
\index{Cubic curves}%
of approximating to the roots of numerical equations. This
is simply the method of Vieta improved. The same treatise
\index{Vieta!ref.\ to}%
contains ``Newton's parallelogram," which enabled him, in an
\index{Newton's parallelogram}%
equation, $f(x, y) = 0$, to find a series in powers of~$x$ equal to
the variable~$y$. The great utility of this rule lay in its determining
the \emph{form} of the series; for, as soon as the law was
known by which the exponents in the series vary, then the
expansion could be effected by the method of indeterminate
coefficients. The rule is still used in determining the infinite
branches to curves, or their figure at multiple points. Newton
gave no proof for it, nor any clue as to how he discovered it.
The proof was supplied half a century later, by Kaestner and
\index{Kaestner!ref.\ to}%
Cramer, independently.\Reference{37}
\index{Cramer}%

In 1704 was published, as an appendix to the \textit{Opticks}, the
\textit{Enumeratio linearum tertii ordinis}, which contains theorems
on the theory of curves. Newton divides cubics into seventy-two
species, arranged in larger groups, for which his commentators
have supplied the names ``genera'' and ``classes,''
recognising fourteen of the former and seven (or four) of the
latter. He overlooked six species demanded by his principles
of classification, and afterwards added by Stirling, Murdoch,
and Cramer. He enunciates the remarkable theorem that the
five species which he names ``divergent parabolas'' give by
\index{Divergent parabolas}%
their projection every cubic curve whatever. As a rule, the
tract contains no proofs. It has been the subject of frequent
conjecture how Newton deduced his results. Recently we have
gotten at the facts, since much of the analysis used by Newton
and a few additional theorems have been discovered among the
Portsmouth papers. An account of the four holograph manuscripts
on this subject has been published by W.~W. Rouse
Ball, in the \textit{Transactions of the London Mathematical Society}
\index{Ball, W.~W.~R.}%
(vol.~xx., pp.~104--143). It is interesting to observe how
Newton begins his research on the classification of cubic
%% -----File: 232.png---Folio 218-------
curves by the algebraic method, but, finding it laborious,
attacks the problem geometrically, and afterwards returns
again to analysis.\Reference{36}

Space does not permit us to do more than merely mention
Newton's prolonged researches in other departments of science.
He conducted a long series of experiments in optics and is the
author of the corpuscular theory of light. The last of a
\index{Light, theory of}%
number of papers on optics, which he contributed to the Royal
Society, 1687, elaborates the theory of ``fits.'' He explained
the decomposition of light and the theory of the rainbow. By
him were invented the reflecting telescope and the sextant
\index{Sextant}%
(afterwards re-discovered by Thomas Godfrey of Philadelphia\Reference[]{2}
\index{Godfrey}%
and by John Hadley). He deduced a theoretical expression
\index{Hadley}%
for the velocity of sound in air, engaged in experiments on
chemistry, elasticity, magnetism, and the law of cooling, and
entered upon geological speculations.

During the two years following the close of~1692, Newton
\index{Newton|)}%
suffered from insomnia and nervous irritability. Some thought
that he laboured under temporary mental aberration. Though
he recovered his tranquillity and strength of mind, the time
of great discoveries was over; he would study out questions
propounded to him, but no longer did he by his own accord
enter upon new fields of research. The most noted investigation
after his sickness was the testing of his lunar theory
by the observations of Flamsteed, the astronomer royal. In~1695
\index{Flamsteed}%
he was appointed warden, and in~1699 master, of the
mint, which office he held until his death. His body was
interred in Westminster Abbey, where in~1731 a magnificent
monument was erected, bearing an inscription ending with,
``Sibi gratulentur mortales tale tantumque exstitisse humani
generis decus.'' It is not true that the Binomial Theorem is
also engraved on it.

We pass to Leibniz, the second and independent inventor
%% -----File: 233.png---Folio 219-------
\index{Kuhn, J.}%
of the calculus. \textbf{Gottfried Wilhelm Leibniz} (1646--1716) was
\index{Leibniz|(}%
born in Leipzig. No period in the history of any civilised
nation could have been less favourable for literary and scientific
pursuits than the middle of the seventeenth century in Germany.
Yet circumstances seem to have happily combined
to bestow on the youthful genius an education hardly otherwise
obtainable during this darkest period of German history.
He was brought early in contact with the best of the culture
then existing. In his fifteenth year he entered the University
of Leipzig. Though law was his principal study, he applied
himself with great diligence to every branch of knowledge.
Instruction in German universities was then very low. The
higher mathematics was not taught at all. We are told that
a certain John Kuhn lectured on Euclid's \textit{Elements}, but that
his lectures were so obscure that none except Leibniz could
understand them. Later on, Leibniz attended, for a half-year,
at Jena, the lectures of Erhard Weigel, a philosopher and
\index{Weigel}%
mathematician of local reputation. In~1666 Leibniz published
a treatise, \textit{De Arte Combinatoria}, in which he does not pass
beyond the rudiments of mathematics. Other theses written
by him at this time were metaphysical and juristical in
character. A fortunate circumstance led Leibniz abroad. In~1672
he was sent by Baron Boineburg on a political mission
to Paris. He there formed the acquaintance of the most
distinguished men of the age. Among these was Huygens,
\index{Huygens!ref.\ to}%
who presented a copy of his work on the oscillation of the
pendulum to Leibniz, and first led the gifted young German
to the study of higher mathematics. In~1673 Leibniz went
to London, and remained there from January till March. He
there became incidentally acquainted with the mathematician
Pell, to whom he explained a method he had found on the
\index{Pell}%
summation of series of numbers by their differences. Pell
told him that a similar formula had been published by Mouton
\index{Mouton}%
%% -----File: 234.png---Folio 220-------
\index{Calculating machines}%
\index{Inverse tangents (problem of)}%
\index{p@{$\pi$}: values for!Leibniz's}%
as early as~1670, and then called his attention to Mercator's
\index{Mercator, N.!ref.\ to}%
work on the rectification of the parabola. While in London,
Leibniz exhibited to the Royal Society his arithmetical machine,
\index{Arithmetical machine}%
\index{Machine, arithmetical}%
which was similar to Pascal's, but more efficient and
\index{Pascal!ref.\ to}%
perfect. After his return to Paris, he had the leisure to study
mathematics more systematically. With indomitable energy
he set about removing his ignorance of higher mathematics.
Huygens was his principal master. He studied the geometric
works of Descartes, Honorarius Fabri, Gregory St.~Vincent,
\index{Descartes!ref.\ to}%
and Pascal. A careful study of infinite series led him to the
\index{Infinite series}%
discovery of the following expression for the ratio of the
circumference to the diameter of the circle, previously discovered
by James Gregory:---
\[
\frac{\pi}{4} = 1 - \tfrac{1}{3} + \tfrac{1}{5} - \tfrac{1}{7} + \tfrac{1}{9} - \text{etc.}
\]
This elegant series was found in the same way as Mercator's
on the hyperbola. Huygens was highly pleased with it and
urged him on to new investigations. Leibniz entered into a
detailed study of the quadrature of curves and thereby became
\index{Curves!quadrature of}%
\index{Quadrature of curves}%
intimately acquainted with the higher mathematics. Among
the papers of Leibniz is still found a manuscript on quadratures,
written before he left Paris in~1676, but which was
never printed by him. The more important parts of it were
embodied in articles published later in the \textit{Acta Eruditorum}.

In the study of Cartesian geometry the attention of Leibniz
was drawn early to the direct and inverse problems of tangents.
\index{Tangents!inverse problem of}%
The direct problem had been solved by Descartes for
the simplest curves only; while the inverse had completely
transcended the power of his analysis. Leibniz investigated
both problems for any curve; he constructed what he called
the \textit{triangulum characteristicum}---an infinitely small triangle
\index{Triangulum characteristicum}%
between the infinitely small part of the curve coinciding with
the tangent, and the differences of the ordinates and abscissas.
%% -----File: 235.png---Folio 221-------
\index{Differential calculus|(}%
A curve is here considered to be a polygon. The \textit{triangulum
characteristicum} is similar to the triangle formed by the tangent,
the ordinate of the point of contact, and the \DPtypo{subtangent}{sub-tangent},
as well as to that between the ordinate, normal,
and sub-normal. It was first employed by Barrow in England,
\index{Barrow!ref.\ to}%
but appears to have been \DPtypo{reinvented}{re-invented} by Leibniz. From
it Leibniz observed the connection existing between the direct
and inverse problems of tangents. He saw also that the latter
could be carried back to the quadrature of curves. All these
results are contained in a manuscript of Leibniz, written in~1673.
One mode used by him in effecting quadratures was
as follows: The rectangle formed by a sub-tangent~$p$ and
an element~$a$ (\ie~infinitely small part of the abscissa) is
equal to the rectangle formed by the ordinate~$y$ and the element~$l$
of that ordinate; or in symbols, $pa = yl$. But the
summation of these rectangles from zero on gives a right
triangle equal to half the square of the ordinate. Thus,
using Cavalieri's notation, he gets
\index{Cavalieri!ref.\ to}%
\index{Notation!differential calculus}%
\begin{align*}
&\omn pa = \omn yl = \frac{y^2}{2}
  \quad \text{(\textit{omn.}\ meaning \textit{omnia}, all)}.
\intertext{But $y = \omn l$; hence}
&\overline{\omn \overline{\omn l}\, \frac{l}{a}}
  = \overline{\frac{\omn l^2}{2a}}.
\end{align*}
This equation is especially interesting, since it is here that
Leibniz first introduces a new notation. He says: ``It will
be useful to write~$\int$ for \textit{omn.}, as $\int l$ for $\textit{omn.}\,l$, that is, the
sum of the~$l$'s''; he then writes the equation thus:---
\[
\frac{\int \overline{l^2}}{2a}
  = \int \overline{{\textstyle\int \bar{l}} \frac{l}{a}}.
\]
From this he deduced the simplest integrals, such as
\[
\int x = \frac{x^2}{2},\quad \int (x + y) = \int x + \int y.
\]
%% -----File: 236.png---Folio 222-------
\index{Inverse tangents (problem of)}%
\index{Quadrature of curves}%
Since the symbol of summation~$\int$ raises the dimensions, he
concluded that the opposite calculus, or that of differences~$d$,
would lower them. Thus, if $\int l = ya$, then $l = \dfrac{ya}{d}$. The
symbol~$d$ was at first placed by Leibniz in the denominator,
because the lowering of the power of a term was brought
about in ordinary calculation by division. The manuscript
giving the above is dated October~29th, 1675.\Reference{39} This, then,
was the memorable day on which the notation of the new
\index{Notation!differential calculus}%
calculus came to be,---a notation which contributed enormously
to the rapid growth and perfect development of the
calculus.

Leibniz proceeded to apply his new calculus to the solution
of certain problems then grouped together under the name
of the Inverse Problems of Tangents. He found the cubical
\index{Tangents!inverse problem of}%
parabola to be the solution to the following: To find the
curve in which the sub-normal is reciprocally proportional
to the ordinate. The correctness of his solution was tested
by him by applying to the result Sluze's method of tangents
\index{Sluze!ref.\ to}%
and reasoning backwards to the original supposition. In the
solution of the third problem he changes his notation from
$\dfrac{x}{d}$ to the now usual notation~$dx$. It is worthy of remark
that in these investigations, Leibniz nowhere explains the
significance of $dx$ and~$dy$, except at one place in a marginal
note: ``Idem est $dx$~et~$\dfrac{x}{d}$, id est, differentia inter duas $x$~proximas.''
Nor does he use the term \emph{differential}, but
always \emph{difference}. Not till ten years later, in the \textit{Acta
Eruditorum}, did he give further explanations of these symbols.
What he aimed at principally was to determine the
change an expression undergoes when the symbol $\int$~or~$d$ is
placed before it. It may be a consolation to students wrestling
with the elements of the differential calculus to know
that it required Leibniz considerable thought and attention\Reference[]{39}
%% -----File: 237.png---Folio 223-------
\index{Debaune@{De Baune}!ref.\ to}%
\index{Inverse tangents (problem of)}%
to determine whether $dx\,dy$ is the same as~$d(xy)$, and
$\dfrac{dx}{dy}$ the same as~$d \smash[t]{\dfrac{x}{y}}$. After considering these questions at
the close of one of his manuscripts, he concluded that the
expressions were not the same, though he could not give the
true value for each. Ten days later, in a manuscript dated
November~21, 1675, he found the equation $y\, d\overline{x} = d\,\overline{xy} - x\,d\overline{y}$,
giving an expression for~$d(xy)$, which he observed to be true
for all curves. He succeeded also in eliminating~$dx$ from
a differential equation, so that it contained only~$dy$, and
thereby led to the solution of the problem under consideration.
``Behold, a most elegant way by which the problems
of the inverse methods of tangents are solved, or at least
\index{Tangents!direct problem of}%
\index{Tangents!inverse problem of}%
are reduced to quadratures!'' Thus he saw clearly that the
inverse problems of tangents could be solved by quadratures,
or, in other words, by the integral calculus. In course of a
\index{Integral calculus}%
half-year he discovered that the direct problem of tangents,
too, yielded to the power of his new calculus, and that thereby
a more general solution than that of Descartes could be
\index{Descartes!ref.\ to}%
obtained. He succeeded in solving all the special problems
of this kind, which had been left unsolved by Descartes.
Of these we mention only the celebrated problem proposed
to Descartes by De~Beaune, viz.~to find the curve whose
ordinate is to its sub-tangent as a given line is to that part
of the ordinate which lies between the curve and a line drawn
from the vertex of the curve at a given inclination to the axis.

Such was, in brief, the progress in the evolution of the new
calculus made by Leibniz during his stay in Paris. Before
his departure, in October, 1676, he found himself in possession
of the most elementary rules and formulæ of the infinitesimal
calculus.

From Paris, Leibniz returned to Hanover by way of London
and Amsterdam. In London he met Collins, who showed him
\index{Collins}%
%% -----File: 238.png---Folio 224-------
\index{Tchirnhausen!ref.\ to}%
a part of his scientific correspondence. Of this we shall speak
later. In Amsterdam he discussed mathematics with Sluze,
\index{Sluze!ref.\ to}%
and became satisfied that his own method of constructing
tangents not only accomplished all that Sluze's did, but even
more, since it could be extended to three variables, by which
tangent planes to surfaces could be found; and especially,
since neither irrationals nor fractions prevented the immediate
application of his method.

In a paper of July~11, 1677, Leibniz gave correct rules for
the differentiation of sums, products, quotients, powers, and
roots. He had given the differentials of a few negative and
fractional powers, as early as November, 1676, but had made
some mistakes. For $d\sqrt{x}$ he had given the erroneous value
$\smash[b]{\dfrac{1}{\sqrt{x}}}$, and in another place the value $-\frac{1}{2} x^{-\frac{1}{2}}$; for $d \dfrac{1}{\DPtypo{x^2}{x^3}}$\DPnote{** PP: Typo presumed on mathematical context.} occurs in
one place the wrong value, $-\smash{\dfrac{2}{x^2}}$, while a few lines lower is
given $-\dfrac{3}{x^4}$, its correct value.

In 1682 was founded in Berlin the \textit{Acta Eruditorum}, a
journal usually known by the name of \textit{Leipzig Acts}. It
was a partial imitation of the French \textit{Journal des Savans}
(founded in 1665), and the literary and scientific review
published in Germany. Leibniz was a frequent contributor.
Tschirnhaus, who had studied mathematics in Paris with
Leibniz, and who was familiar with the new analysis of
Leibniz, published in the \textit{Acta \DPtypo{Eroditorum}{Eruditorum}} a paper on quadratures,
which consists principally of subject-matter communicated
by Leibniz to Tschirnhaus during a controversy
which they had had on this subject. Fearing that Tschirnhaus
might claim as his own and publish the notation and rules of
the differential calculus, Leibniz decided, at last, to make
public the fruits of his inventions. In~1684, or nine years
after the new calculus first dawned upon the mind of Leibniz,
and nineteen years after Newton first worked at fluxions,
%% -----File: 239.png---Folio 225-------
\index{Debaune@{De Baune}!ref.\ to}%
and three years before the publication of Newton's \textit{Principia},
Leibniz published, in the \textit{Leipzig Acts}, his first paper on the
differential calculus. He was unwilling to give to the world
all his treasures, but chose those parts of his work which were
most abstruse and least perspicuous. This epoch-making paper
of only six pages bears the title: ``Nova methodus pro maximis
et minimis, itemque tangentibus, \Change{quae}{quæ} nec fractas nec irrationales
quantitates moratur, et singulare pro illis calculi
genus.'' The rules of calculation are briefly stated without
proof, and the meaning of $dx$~and~$dy$ is not made clear. It has
been inferred from this that Leibniz himself had no definite
and settled ideas on this subject. Are $dy$~and~$dx$ finite or
infinitesimal quantities? At first they appear, indeed, to have
been taken as finite, when he says: ``We now call any line
selected at random~$dx$, then we designate the line which is to~$dx$
as $y$~is to the sub-tangent, by~$dy$, which is the difference of~$y$.''
Leibniz then ascertains, by his calculus, in what way a
ray of light passing through two differently refracting media,
can travel easiest from one point to another; and then closes
his article by giving his solution, in a few words, of De
Beaune's problem. Two years later (1686) Leibniz published
in the \textit{Acta Eruditorum} a paper containing the rudiments of
the integral calculus. The quantities $dx$~and~$dy$ are there
treated as infinitely small. He showed that by the use of his
notation, the properties of curves could be fully expressed by
equations. Thus the equation
\[
y = \sqrt{2x - x^2} + \int \frac{dx}{\sqrt{2x - x^2}}
\]
characterises the cycloid.\Reference{38}
\index{Cycloid}%

The great invention of Leibniz, now made public by his
articles in the \textit{Leipzig Acts}, made little impression upon the
mass of mathematicians. In Germany no one comprehended
%% -----File: 240.png---Folio 226-------
\index{Analysis situs}%
\index{Singular solutions}%
\index{Tchirnhausen!ref.\ to}%
the new calculus except Tschirnhaus, who remained indifferent
to it. The author's statements were too short and succinct
to make the calculus generally understood. The first
to recognise its importance and to take up the study of it
were two foreigners,---the Scotchman \textit{John Craig}, and
\index{Craig, J.}%
the Swiss \textit{James Bernoulli}. The latter wrote Leibniz a
\index{Bernoulli, James (born 1654)!ref.\ to}%
\index{Bernoulli, John (born 1667)!ref.\ to}%
letter in~1687, wishing to be initiated into the mysteries
of the new analysis. Leibniz was then travelling abroad,
so that this letter remained unanswered till~\DPtypo{1790}{1690}. James
Bernoulli succeeded, meanwhile, by close application, in uncovering
the secrets of the differential calculus without assistance.
He and his brother John proved to be mathematicians
of exceptional power. They applied themselves to the new
science with a success and to an extent which made Leibniz
declare that it was as much theirs as his. Leibniz carried on
an extensive correspondence with them, as well as with other
mathematicians. In a letter to John Bernoulli he suggests,
among other things, that the integral calculus be improved by
reducing integrals back to certain fundamental irreducible
forms. The integration of logarithmic expressions was then
studied. The writings of Leibniz contain many innovations,
and anticipations of since prominent methods. Thus he made
use of variable parameters, laid the foundation of \textit{analysis in
situ}, introduced the first notion of determinants in his effort
\index{Determinants}%
to simplify the expression arising in the elimination of the
unknown quantities from a set of linear equations. He
resorted to the device of breaking up certain fractions into
the sum of other fractions for the purpose of easier integration;
he explicitly assumed the principle of continuity; he gave the
\index{Continuity}%
first instance of a ``singular solution,'' and laid the foundation
to the theory of envelopes in two papers, one of which contains
for the first time the terms \emph{co-ordinate} and \emph{axes of co-ordinates}.
\index{Co-ordinates!first use of term}%
He wrote on osculating curves, but his paper contained the
\index{Curves!osculating}%
\index{Curves!theory of}%
%% -----File: 241.png---Folio 227-------
\index{Fluxional controversy|(}%
error (pointed out by John Bernoulli, but not admitted by
him) that an osculating circle will necessarily cut a curve in
four consecutive points. Well known is his theorem on the
$n$th differential coefficient of the product of two functions of a
\index{Differential calculus|)}%
\index{Differential calculus!controversy between Newton and Leibniz|(}%
variable. Of his many papers on mechanics, some are valuable,
\index{Mechanics!Leibniz}%
while others contain grave errors.

Before tracing the further development of the calculus we
shall sketch the history of that long and bitter controversy
between English and Continental mathematicians on the invention
of the calculus. The question was, did Leibniz invent it
independently of Newton, or was he a plagiarist?
\index{Newton, controversy with Leibniz|(}%

We must begin with the early correspondence between the
parties appearing in this dispute. Newton had begun using
his notation of fluxions in 1666.\Reference{41} In~1669 Barrow sent Collins
\index{Barrow!ref.\ to}%
\index{Collins}%
Newton's tract, \textit{De Analysi per Equationes}, etc.

The first visit of Leibniz to London extended from the 11th
of January until March,~1673. He was in the habit of committing
to writing important scientific communications received
from others. In~1890 Gerhardt discovered in the royal library
\index{Gerhardt}%
at Hanover a sheet of manuscript with notes taken by Leibniz
during this journey.\Reference{40} They are headed ``Observata Philosophica
in itinere Anglicano sub initium anni~1673.'' The sheet
is divided by horizontal lines into sections. The sections
given to Chymica, Mechanica, Magnetica, Botanica, Anatomica,
Medica, Miscellanea, contain extensive memoranda, while those
devoted to mathematics have very few notes. Under Geometrica
he says only this: ``Tangentes omnium figurarum.
Figurarum geometricarum explicatio per motum puncti in
moto lati.'' We suspect from this that Leibniz had read
Barrow's lectures. Newton is referred to only under Optica.
Evidently Leibniz did not obtain a knowledge of fluxions
during this visit to London, nor is it claimed that he did by
his opponents.
%% -----File: 242.png---Folio 228-------

Various letters of Newton, Collins, and others, up to the
\index{Collins}%
beginning of~1676, state that Newton invented a method by
which tangents could be drawn without the necessity of freeing
their equations from irrational terms. Leibniz announced in
1674 to Oldenburg, then secretary of the Royal Society, that
\index{Oldenburg}%
he possessed very general analytical methods, by which he had
found theorems of great importance on the quadrature of the
circle by means of series. In answer, Oldenburg stated
Newton and James Gregory had also discovered methods of
\index{Gregory, James}%
quadratures, which extended to the circle. Leibniz desired to
have these methods communicated to him; and Newton, at
the request of Oldenburg and Collins, wrote to the former the
celebrated letters of June~13 and October~24, 1676. The first
contained the Binomial Theorem and a variety of other matters
relating to infinite series and quadratures; but nothing
directly on the method of fluxions. Leibniz in reply speaks
in the highest terms of what Newton had done, and requests
further explanation. Newton in his second letter just mentioned
explains the way in which he found the Binomial
Theorem, and also communicates his method of fluxions and
fluents in form of an anagram in which all the letters in the
sentence communicated were placed in alphabetical order.
Thus Newton says that his method of drawing tangents was
\[
6\,\textit{a\,cc\,d\,æ }\
13\,\textit{e\,ff }\
7\,\textit{i }\
3\,\textit{l }\
9\,\textit{n }\
4\,\textit{o }\
4\,\textit{q\,rr }\
4\,\textit{s }\
9\,\textit{t }\
12\,\textit{v\,x}.
\]
The sentence was, \Lat{``Data æquatione quotcunque fluentes
quantitates involvente fluxiones invenire, et vice versa.''}
(``Having any given equation involving never so many flowing
quantities, to find the fluxions, and vice versa.'') Surely this
anagram afforded no hint. Leibniz wrote a reply to Collins,
in which, without any desire of concealment, he explained the
principle, notation, and the use of the differential calculus.

The death of Oldenburg brought this correspondence to a
%% -----File: 243.png---Folio 229-------
close. Nothing material happened till~1684, when Leibniz
published his first paper on the differential calculus in the
\textit{Leipzig Acts}, so that while Newton's claim to the priority of
invention must be admitted by all, it must also be granted
that Leibniz was the first to give the full benefit of the calculus
to the world. Thus, while Newton's invention remained a
secret, communicated only to a few friends, the calculus of
Leibniz was spreading over the Continent. No rivalry or
hostility existed, as yet, between the illustrious scientists.
Newton expressed a very favourable opinion of Leibniz's
inventions, known to him through the above correspondence
with Oldenburg, in the following celebrated scholium (\textit{Principia},
\index{Newton's Principia}%
\index{Principia (Newton's)}%
first edition, 1687, Book~II., Prop.~7, scholium):---

``In letters which went between me and that most excellent
geometer, G.~G. Leibniz, ten years ago, when I signified that
I was in the knowledge of a method of determining maxima
and minima, of drawing tangents, and the like, and when I
concealed it in transposed letters involving this sentence (Data
æquatione, etc., above cited), that most distinguished man
wrote back that he had also fallen upon a method of the same
kind, and communicated his method, which hardly differed
from mine, except in his forms of words and symbols.''

As regards this passage, we shall see that Newton was afterwards
weak enough, as De Morgan says: ``First, to deny the
\index{Demorgan@{De Morgan}!ref.\ to}%
plain and obvious meaning, and secondly, to omit it entirely
from the third edition of the \textit{Principia}.'' On the Continent,
great progress was made in the calculus by Leibniz and his
coadjutors, the brothers James and John Bernoulli, and
\index{Bernoulli, James (born 1654)!ref.\ to}%
\index{Bernoulli, John (born 1667)!ref.\ to}%
Marquis de l'Hospital. In~1695 Wallis informed Newton by
\index{Lhospital@{L'Hospital}!ref.\ to}%
\index{Wallis!ref.\ to}%
letter that ``he had heard that his notions of fluxions passed
in Holland with great applause by the name of `Leibniz's
Calculus Differentialis.'\,'' Accordingly Wallis stated in the
preface to a volume of his works that the calculus differentialis
%% -----File: 244.png---Folio 230-------
was Newton's method of fluxions which had been
communicated to Leibniz in the Oldenburg letters. A review
of Wallis' works, in the \textit{Leipzig Acts} for~1696, reminded the
reader of Newton's own admission in the scholium above
cited.

For fifteen years Leibniz had enjoyed unchallenged the
honour of being the inventor of his calculus. But in 1699 Fato
de Duillier, a Swiss, who had settled in England, stated in a
\index{Duillier}%
mathematical paper, presented to the Royal Society, his conviction
that Newton was the first inventor; adding that,
whether Leibniz, the second inventor, had borrowed anything
from the other, he would leave to the judgment of those who
had seen the letters and manuscripts of Newton. This was
the first distinct insinuation of plagiarism. It would seem that
the English mathematicians had for some time been cherishing
suspicions unfavourable to Leibniz. A feeling had doubtless
long prevailed that Leibniz, during his second visit to London
in~1676, had or might have seen among the papers of Collins,
\index{Collins}%
Newton's \textit{Analysis per æquationes}, etc., which contained applications
of the fluxionary method, but no systematic development
or explanation of it. Leibniz certainly did see at least
part of this tract. During the week spent in London, he took
note of whatever interested him among the letters and papers
of Collins. His memoranda discovered by Gerhardt in 1849 in
\index{Gerhardt}%
the Hanover library fill two sheets.\Reference{40} The one bearing on our
question is headed ``Excerpta ex tractatu Newtoni Msc.~de
Analysi per æquationes numero terminorum infinitas.'' The
notes are very brief, excepting those \textit{De Resolutione æquationum
affectarum}, of which there is an almost complete copy.
This part was evidently new to him. If he examined
Newton's entire tract, the other parts did not particularly
impress him. From it he seems to have gained nothing pertaining
to the infinitesimal calculus. By the previous introduction
%% -----File: 245.png---Folio 231-------
of his own algorithm he had made greater progress
than by what came to his knowledge in London. Nothing
mathematical that he had received engaged his thoughts in
the immediate future, for on his way back to Holland he composed
a lengthy dialogue on mechanical subjects.

Duillier's insinuations lighted up a flame of discord which a
whole century was hardly sufficient to extinguish. Leibniz,
who had never contested the priority of Newton's discovery,
and who appeared to be quite satisfied with Newton's admission
in his scholium, now appears for the first time in the
controversy. He made an animated reply in the \textit{Leipzig Acts},
and complained to the Royal Society of the injustice done him.

Here the affair rested for some time. In the \textit{Quadrature of
Curves}, published~1704, for the first time, a formal exposition
of the method and notation of fluxions was made public. In~1705
appeared an unfavourable review of this in the \textit{Leipzig
Acts}, stating that Newton uses and always has used fluxions
for the differences of Leibniz. This was considered by Newton's
friends an imputation of plagiarism on the part of their
chief, but this interpretation was always strenuously resisted
by Leibniz. Keill, professor of astronomy at Oxford, undertook
\index{Keill}%
with more zeal than judgment the defence of Newton.
In a paper inserted in the \textit{Philosophical Transactions} of~1708,
he claimed that Newton was the first inventor of fluxions and
``that the same calculus was afterward published by Leibniz,
the name and the mode of notation being changed.'' Leibniz
complained to the secretary of the Royal Society of bad treatment
and requested the interference of that body to induce
Keill to disavow the intention of imputing fraud. Keill was
not made to retract his accusation; on the contrary, was
authorised by Newton and the Royal Society to explain and
defend his statement. This he did in a long letter. Leibniz
thereupon complained that the charge was now more open than
%% -----File: 246.png---Folio 232-------
before, and appealed for justice to the Royal Society and to
Newton himself. The Royal Society, thus appealed to as a
judge, appointed a committee which collected and reported
upon a large mass of documents---mostly letters from and to
Newton, Leibniz, Wallis, Collins, etc. This report, called the
\index{Collins}%
\textit{Commercium Epistolicum}, appeared in the year~1712 and again
\index{Commercium epistolicum}%
in~1725, with a Recensio prefixed, and additional notes by Keill.
\index{Keill}%
The final conclusion in the \textit{Commercium Epistolicum} was
that Newton was the first inventor. But this was not to the
point. The question was not whether Newton was the first
inventor, but whether Leibniz had stolen the method. The
committee had not formally ventured to assert their belief
that Leibniz was a plagiarist. Yet there runs throughout the
document a desire of proving Leibniz guilty of more than
they meant positively to affirm. Leibniz protested only in
private letters against the proceeding of the Royal Society,
declaring that he would not answer an argument so weak.
John Bernoulli, in a letter to Leibniz, which was published
\index{Bernoulli, John (born 1667)!ref.\ to}%
later in an anonymous tract, is as decidedly unfair towards
Newton as the friends of the latter had been towards Leibniz.
Keill replied, and then Newton and Leibniz appear as mutual
accusers in several letters addressed to third parties. In a
letter to Conti, April~9, 1716, Leibniz again reminded Newton
of the admission he had made in the scholium, which he was
now desirous of disavowing; Leibniz also states that he
always believed Newton, but that, seeing him connive at
accusations which he must have known to be false, it was
natural that he (Leibniz) should begin to doubt. Newton
did not reply to this letter, but circulated some remarks among
his friends which he published immediately after hearing
of the death of Leibniz, November~14, 1716. This paper
of Newton gives the following explanation pertaining to the
scholium in question: ``He [Leibniz] pretends that in my
%% -----File: 247.png---Folio 233-------
\index{Differential calculus!controversy between Newton and Leibniz|)}%
\index{Fluxional controversy|)}%
book of principles I allowed him the invention of the calculus
differentialis, independently of my own; and that to attribute
this invention to myself is contrary to my knowledge there
avowed. But in the paragraph there referred unto I do not
find one word to this purpose.'' In the third edition of the
\textit{Principia}, 1726, Newton omitted the scholium and substituted
\index{Newton's Principia}%
\index{Principia (Newton's)}%
in its place another, in which the name of Leibniz does not
appear.

National pride and party feeling long prevented the adoption
of impartial opinions in England, but now it is generally admitted
by nearly all familiar with the matter, that Leibniz
really was an independent inventor. Perhaps the most telling
evidence to show that Leibniz was an independent inventor
is found in the study of his mathematical papers (collected
and edited by C.~I. Gerhardt, in six volumes, Berlin, 1849--1860),
\index{Gerhardt}%
which point out a gradual and natural evolution of the
rules of the calculus in his own mind. ``There was throughout
the whole dispute,'' says De~Morgan, ``a confusion between
\index{Demorgan@{De Morgan}!ref.\ to}%
the knowledge of fluxions or differentials and that of a \emph{calculus}
of fluxions or differentials; that is, a digested method with
general rules.''

This controversy is to be regretted on account of the long
and bitter alienation which it produced between English and
Continental mathematicians. It stopped almost completely
all interchange of ideas on scientific subjects. The English
adhered closely to Newton's methods and, until about~1820,
\index{Newton, controversy with Leibniz|)}%
remained, in most cases, ignorant of the brilliant mathematical
discoveries that were being made on the Continent. The loss
in point of scientific advantage was almost entirely on the
side of Britain. The only way in which this dispute may be
said, in a small measure, to have furthered the progress of
mathematics, is through the challenge problems by which
each side attempted to annoy its adversaries.
%% -----File: 248.png---Folio 234-------
\index{Brachistochrone (line of swiftest descent)}%
\index{Curve of swiftest descent}%

The recurring practice of issuing challenge problems was
inaugurated at this time by Leibniz. They were, at first, not
intended as defiances, but merely as exercises in the new calculus.
Such was the problem of the isochronous curve (to
\index{Isochronous curve}%
find the curve along which a body falls with uniform velocity),
proposed by him to the Cartesians in~1687, and solved by
James Bernoulli, himself, and John Bernoulli. James Bernoulli
\index{Bernoulli, John (born 1667)!ref.\ to}%
proposed in the Leipzig \textit{Journal} the question to find the
curve (the catenary) formed by a chain of uniform weight
\index{Catenary}%
suspended freely from its ends. It was resolved by Huygens,
\index{Huygens!ref.\ to}%
Leibniz, and himself. In~1697 John Bernoulli challenged the
best mathematicians in Europe to solve the difficult problem,
to find the curve (the cycloid) along which a body falls from
\index{Cycloid}%
one point to another in the shortest possible time. Leibniz
solved it the day he received it. Newton, de~l'Hospital, and
\index{Lhospital@{L'Hospital}!ref.\ to}%
the two Bernoullis gave solutions. Newton's appeared anonymously
in the \textit{Philosophical Transactions}, but John Bernoulli
recognised in it his powerful mind, ``anquam,'' he says, ``ex
ungue leonem.'' The problem of orthogonal trajectories (a
\index{Trajectories}%
system of curves described by a known law being given, to
describe a curve which shall cut them all at right angles) had
been long proposed in the \textit{Acta Eruditorum}, but failed at
first to receive much attention. It was again proposed in~1716
by Leibniz, to feel the pulse of the English mathematicians.

This may be considered as the first defiance problem professedly
aimed at the English. Newton solved it the same
evening on which it was delivered to him, although he was
much fatigued by the day's work at the mint. His solution,
as published, was a general plan of an investigation rather
than an actual solution, and was, on that account, criticised by
Bernoulli as being of no value. Brook Taylor undertook the
\index{Taylor, B.!ref.\ to}%
defence of it, but ended by using very reprehensible language.
%% -----File: 249.png---Folio 235-------
Bernoulli was not to be outdone in incivility, and made a
bitter reply. Not long afterwards Taylor sent an open defiance
to Continental mathematicians of a problem on the
integration of a fluxion of complicated form which was known
to very few geometers in England and supposed to be beyond
the power of their adversaries. The selection was injudicious,
for Bernoulli had long before explained the method of this
and similar integrations. It served only to display the skill
and augment the triumph of the followers of Leibniz. The
last and most unskilful challenge was by John Keill. The
\index{Keill}%
problem was to find the path of a projectile in a medium
which resists proportionally to the square of the velocity.
Without first making sure that he himself could solve it,
Keill boldly challenged Bernoulli to produce a solution. The
latter resolved the question in very short time, not only for a
resistance proportional to the square, but to any power of the
velocity. Suspecting the weakness of the adversary, he repeatedly
offered to send his solution to a confidential person
in London, provided Keill would do the same. Keill never
made a reply, and Bernoulli abused him and cruelly exulted
over him.\Reference{26}

The explanations of the fundamental principles of the calculus,
as given by Newton and Leibniz, lacked clearness and
rigour. For that reason it met with opposition from several
quarters. In~1694 Bernard Nieuwentyt of Holland denied
\index{Nieuwentyt}%
the existence of differentials of higher orders and objected to
the practice of neglecting infinitely small quantities. These
objections Leibniz was not able to meet satisfactorily. In his
reply he said the value of $\dfrac{dy}{dx}$ in geometry could be expressed
as the ratio of finite quantities. In the interpretation of $dx$~and~$dy$
Leibniz vacillated. At one time they appear in his
\index{Leibniz|)}%
writings as finite lines; then they are called infinitely small
%% -----File: 250.png---Folio 236-------
quantities, and again, \emph{quantitates inassignabiles}, which spring
from \emph{quantitates assignabiles} by the law of continuity. In this
last presentation Leibniz approached nearest to Newton.

In England the principles of fluxions were boldly attacked
by Bishop Berkeley, the eminent metaphysician, who argued
\index{Berkeley}%
with great acuteness, contending, among other things, that
the fundamental idea of supposing a finite ratio to exist
between terms absolutely evanescent---``the ghosts of departed
quantities,'' as he called them---was absurd and unintelligible.
The reply made by Jurin failed to remove all the
\index{Jurin}%
objections. Berkeley was the first to point out what was
again shown later by Lazare Carnot, that correct answers were
\index{Carnot, Lazare!ref.\ to}%
reached by a ``compensation of errors.'' Berkeley's attack
was not devoid of good results, for it was the immediate cause
of the work on fluxions by Maclaurin. In France \textit{Michel
\index{Maclaurin!ref.\ to}%
Rolle} rejected the differential calculus and had a controversy
\index{Differential calculus|(}%
\index{Differential calculus!philosophy of}%
\index{Rolle!ref.\ to}%
with \textit{Varignon} on the subject.
\index{Varignon!ref.\ to}%

Among the most vigorous promoters of the calculus on the
Continent were the Bernoullis. They and Euler made Basel
in Switzerland famous as the cradle of great mathematicians.
The family of Bernoullis furnished in course of a century
eight members who distinguished themselves in mathematics.
We subjoin the following genealogical table:---
\begin{center}
\begin{minipage}{\linewidth}
\footnotesize
\noindent$\overbrace{
\begin{tabular}{@{}lll@{}}
\textbf{Jacob}, 1654--1705 & \textit{Nicolaus} & \textbf{Johann}, 1667--1748 \\
& \qquad$|$ & \qquad$|$ \\
& \textit{Nicolaus}, 1687--1759 & \textit{Nicolaus}, 1695--1726 \\
& & \textbf{Daniel}, 1700--1782 \\
& & \textit{Johann}, 1710--1790 \\
& \multicolumn{2}{c}{%
\qquad\makebox[0pt][c]{$\overbrace{
\begin{tabular}{@{}ccc@{}}
\textit{Daniel} & \textit{Johann}, 1744--1807 & \textit{Jacob}, 1758--1789
\end{tabular}}^{}$}}
\end{tabular}}^{\mbox{\small\textit{Nicolaus Bernoulli}, the Father}}$
\end{minipage}
\end{center}
\index{Bernoullis, genealogical table of}%
Most celebrated were the two brothers Jacob (James) and
Johann (John), and Daniel, the son of John. James and
%% -----File: 251.png---Folio 237-------
\index{Integral calculus!origin of term}%
\index{Isoperimetrical figures}%
\index{Mechanics!Bernoullis}%
\index{Probability}%
John were staunch friends of Leibniz and worked hand in
\index{Leibniz!ref.\ to}%
hand with him. \textbf{James Bernoulli} (1654--1705) was born in
\index{Bernoulli, James (born 1654)}%
\index{Bernoulli, John (born 1667)!ref.\ to}%
Basel. Becoming interested in the calculus, he mastered it
without aid from a teacher. From 1687 until his death he
occupied the mathematical chair at the University of Basel.
He was the first to give a solution to Leibniz's problem of the
isochronous curve. In his solution, published in the \textit{Acta
Eruditorum}, 1690, we meet for the first time with the word
\emph{integral}. Leibniz had called the integral calculus \textit{calculus
summatorius}, but in 1696 the term \textit{calculus integralis} was
agreed upon between Leibniz and John Bernoulli. James
proposed the problem of the catenary, then proved the correctness
\index{Catenary}%
of Leibniz's construction of this curve, and solved the
more complicated problems, supposing the string to be (1)~of
variable density, (2)~extensible, (3)~acted upon at each point
by a force directed to a fixed centre. Of these problems he
published answers without explanations, while his brother
John gave in addition their theory. He determined the shape
of the ``elastic curve'' formed by an elastic plate or rod fixed
\index{Elastic curve}%
at one end and bent by a weight applied to the other end; of
the ``lintearia,'' a flexible rectangular plate with two sides
\index{Lintearia}%
fixed horizontally at the same height, filled with a liquid; of
the ``volaria,'' a rectangular sail filled with wind. He studied
\index{Volaria}%
the loxodromic and logarithmic spirals, in the last of which
\index{Spirals}%
he took particular delight from its remarkable property of
reproducing itself under a variety of conditions. Following
the example of Archimedes, he willed that the curve be engraved
upon his \Change{tombstone}{tomb-stone} with the inscription ``\textit{eadem mutata
resurgo}.'' In~1696 he proposed the famous problem of isoperimetrical
figures, and in 1701 published his own solution. He
wrote a work on \textit{Ars Conjectandi}, which is a development of
the calculus of probabilities and contains the investigation
now called ``Bernoulli's theorem'' and the so-called ``numbers
\index{Bernoulli's theorem}%
%% -----File: 252.png---Folio 238-------
\index{Brachistochrone (line of swiftest descent)}%
\index{Caustics}%
\index{Mechanics!Bernoullis}%
\index{Numbers of Bernoulli}%
of Bernoulli,'' which are in fact (though not so considered by
\index{Bernoulli, Daniel}%
\index{Bernoulli, Nicolaus (born 1695)}%
\index{Bernoulli, James (born 1654)}%
\index{Bernoulli, John (born 1667)}%
him) the coefficients of $\dfrac{x^n}{n!}$ in the expansion of $(e^x - 1)^{-1}$. Of
his collected works, in three volumes, one was printed in~1713,
the other two in~1744.

\textbf{John Bernoulli} (1667--1748) was initiated into mathematics
by his brother. He afterwards visited France, where he met
Malebranche, Cassini, De Lahire, Varignon, and de l'Hospital.
For ten years he occupied the mathematical chair at Gröningen
and then succeeded his brother at Basel. He was one of the
most enthusiastic teachers and most successful original investigators
of his time. He was a member of almost every learned
society in Europe. His controversies were almost as numerous
as his discoveries. He was ardent in his friendships, but
unfair, mean, and violent toward all who incurred his dislike---even
his own brother and son. He had a bitter dispute
with James on the isoperimetrical problem. James convicted
him of several paralogisms. After his brother's death he
attempted to substitute a disguised solution of the former for
an incorrect one of his own. John admired the merits of
Leibniz and Euler, but was blind to those of Newton. He
\index{Newton!ref.\ to}%
immensely enriched the integral calculus by his labours.
Among his discoveries are the exponential calculus, the line
of swiftest descent, and its beautiful relation to the path
\index{Curve of swiftest descent}%
described by a ray passing through strata of variable density.
He treated trigonometry by the analytical method, studied
\index{Trigonometry}%
caustic curves and trajectories. Several times he was given
\index{Trajectories}%
prizes by the Academy of Science in Paris.

Of his sons, \textbf{Nicholas} and \textbf{Daniel} were appointed professors
of mathematics at the same time in the Academy of St.~Petersburg.
The former soon died in the prime of life; the
latter returned to Basel in~1733, where he assumed the chair
of experimental philosophy. His first mathematical publication
%% -----File: 253.png---Folio 239-------
\index{Differential equations}%
was the solution of a differential equation proposed by
Riccati. He wrote a work on hydrodynamics. His investigations
\index{Hydrodynamics}%
\index{Riccati!ref.\ to}%
on probability are remarkable for their boldness and
\index{Probability}%
originality. He proposed the theory of \emph{moral expectation},
\index{Moral expectation}%
which he thought would give results more in accordance with
our ordinary notions than the theory of \emph{mathematical probability}.
His ``moral expectation'' has become classic, but no
one ever makes use of it. He applies the theory of probability
to insurance; to determine the mortality caused by small-pox
\index{Insurance}%
at various stages of life; to determine the number of survivors
at a given age from a given number of births; to determine
how much inoculation lengthens the average duration of life.
He showed how the differential calculus could be used in the
theory of probability. He and Euler enjoyed the honour of
\index{Euler!ref.\ to}%
having gained or shared no less than ten prizes from the
Academy of Sciences in Paris.

\textbf{Johann Bernoulli} (born 1710) succeeded his father in the
\index{Bernoulli, Nicolaus (born 1687)}%
\index{Bernoulli, James (born 1758)}%
\index{Bernoulli, John (born 1710)}%
\index{Bernoulli, John (born 1744)}%
professorship of mathematics at Basel. He captured three
prizes (on the capstan, the propagation of light, and the
magnet) from the Academy of Sciences at Paris. \textbf{Nicolaus
Bernoulli} (born 1687) held for a time the mathematical chair
at Padua which Galileo had once filled. \textbf{Johann Bernoulli}
(born 1744) at the age of nineteen was appointed astronomer
royal at Berlin, and afterwards director of the mathematical
department of the Academy. His brother \textit{Jacob} took upon
himself the duties of the chair of experimental physics at
Basel, previously performed by his uncle Jacob, and later
was appointed mathematical professor in the Academy at St.\
Petersburg.

Brief mention will now be made of some other mathematicians
belonging to the period of Newton, Leibniz, and the
elder Bernoullis.

\textbf{Guillaume François Antoine l'Hospital} (1661--1704), a pupil
\index{Hospital,~l'}%
\index{Lhospital@{L'Hospital}}%
%% -----File: 254.png---Folio 240-------
\index{Analytic geometry}%
\index{Cycloid}%
\index{Equations!theory of}%
\index{Hospital,~l'}%
\index{Laplace!ref.\ to}%
\index{Lhospital@{L'Hospital}}%
of John Bernoulli, has already been mentioned as taking
part in the challenges issued by Leibniz and the Bernoullis.
He helped powerfully in making the calculus of Leibniz better
known to the mass of mathematicians by the publication of a
treatise thereon in 1696. This contains for the first time the
method of finding the limiting value of a fraction whose two
terms tend toward zero at the same time.

Another zealous French advocate of the calculus was \textbf{Pierre
Varignon} (1654--1722). \textbf{Joseph Saurin} (1659--1737) solved the
\index{Saurin}%
\index{Varignon}%
delicate problem of how to determine the tangents at the
multiple points of algebraic curves. \textbf{François Nicole} (1683--1758)
\index{Curves!theory of}%
\index{Nicole}%
in 1717 issued the first systematic treatise on finite
differences, in which he finds the sums of a considerable
\Addendlabel{240}%
number of interesting series. He wrote also on roulettes,
particularly spherical epicycloids, and their rectification. Also
interested in finite differences was \textbf{Pierre Raymond de~Montmort}
\index{Finite differences}%
\index{Montmort, de}%
(1678--1719). His chief writings, on the theory of probability,
\index{Probability}%
served to stimulate his more distinguished successor, De~Moivre.
\index{Demoivre@{De Moivre}}%
\index{Moivre, de}%
\textbf{Jean Paul de~Gua} (1713--1785) gave the demonstration
\index{Gua, de}%
of Descartes' rule of signs, now given in books. This skilful
\index{Descartes!ref.\ to}%
geometer wrote in 1740 a work on analytical geometry, the
\index{Geometry!modern synthetic}%
object of which was to show that most investigations on curves
could be carried on with the analysis of Descartes quite as
easily as with the calculus. He shows how to find the tangents,
asymptotes, and various singular points of curves of all
degrees, and proved by perspective that several of these points
can be at infinity. A mathematician who clung to the methods
of the ancients was \textbf{Philippe de~Lahire} (1640--1718), a pupil of
\index{Lahire, de}%
Desargues. His work on conic sections is purely synthetic,
\index{Desargues!ref.\ to}%
but differs from ancient treatises in deducing the properties of
conics from those of the circle in the same manner as did
Desargues and Pascal. His innovations stand in close relation
\index{Pascal!ref.\ to}%
with modern synthetic geometry. He wrote on roulettes, on
%% -----File: 255.png---Folio 241-------
\index{Caustics}%
\index{Magic squares}%
\index{p@{$\pi$}: values for!Fagnano's}%
\index{Tchirnhausen}%
graphical methods, epicycloids, conchoids, and on magic
squares. \textbf{Michel Rolle} (1652--1719) is the author of a theorem
\index{Rolle}%
named after him.

Of Italian mathematicians, Riccati and Fagnano must not
\index{Fagnano}%
\index{Riccati}%
remain unmentioned. \textbf{Jacopo Francesco, Count Riccati} (1676--1754)
is best known in connection with his problem, called
Riccati's equation, published in the \textit{Acta Eruditorum} in~1724.
He succeeded in integrating this differential equation for some
special cases. A geometrician of remarkable power was \textbf{Giulio
Carlo, Count de~Fagnano} (1682--1766). He discovered the following
formula, $\pi = 2i \log \dfrac{1 - i}{1 + i}$, in which he anticipated Euler
\index{Euler!ref.\ to}%
in the use of imaginary exponents and logarithms. His studies
\index{Exponents}%
\index{Imaginary quantities}%
on the rectification of the ellipse and hyperbola are the starting-points
of the theory of elliptic functions. He showed, for
\index{Elliptic functions}%
instance, that two arcs of an ellipse can be found in an indefinite
number of ways, whose difference is expressible by a
right line.

In Germany the only noted contemporary of Leibniz is
\index{Leibniz!ref.\ to}%
\textbf{Ehrenfried Walter Tschirnhausen} (1651--1708), who discovered
the caustic of reflection, experimented on metallic reflectors
and large burning-glasses, and gave us a method of transforming
equations named after him. Believing that the most
\index{Equations!theory of}%
simple methods (like those of the ancients) are the most
correct, he concluded that in the researches relating to the
properties of curves the calculus might as well be dispensed
with.

After the death of Leibniz there was in Germany not a
single mathematician of note. \textbf{Christian Wolf} (1679--1754),
\index{Wolf, C.}%
professor at Halle, was ambitious to figure as successor of
Leibniz, but he ``forced the ingenious ideas of Leibniz into a
pedantic scholasticism, and had the unenviable reputation of
having presented the elements of the arithmetic, algebra, and
%% -----File: 256.png---Folio 242-------
\index{Curves!theory of}%
\index{Differential calculus|)}%
\index{Strings, vibrating}%
\index{Vibrating strings}%
analysis developed since the time of the Renaissance in the
form of Euclid,---of course only in outward form, for into the
spirit of them he was quite unable to penetrate.''\Reference{16}

The contemporaries and immediate successors of Newton in
Great Britain were men of no mean merit. We have reference
to Cotes, Taylor, Maclaurin, and De~Moivre. We are
\index{Cotes}%
\index{Demoivre@{De Moivre}}%
\index{Moivre, de}%
\index{Taylor, B.}%
told that at the death of \textbf{Roger Cotes} (1682--1716), Newton
exclaimed, ``If Cotes had lived, we might have known something.''
It was at the request of Dr.~Bentley that Cotes
undertook the publication of the second edition of Newton's
\textit{Principia}. His mathematical papers were published after his
\index{Newton's Principia}%
\index{Principia (Newton's)}%
death by Robert Smith, his successor in the Plumbian professorship
\index{Smith, R.}%
at Trinity College. The title of the work, \textit{Harmonia
Mensurarum}, was suggested by the following theorem
contained in it: If on each radius vector, through a fixed point~$O$,
there be taken a point~$R$, such that the reciprocal of $OR$ be
the arithmetic mean of the reciprocals of $OR_1, OR_2, \ldots OR_n$,
then the locus of~$R$ will be a straight line. In this work
progress was made in the application of logarithms and the
\index{Logarithms}%
properties of the circle to the calculus of fluents. To Cotes
we owe a theorem in trigonometry which depends on the
\index{Trigonometry}%
forming of factors of~$x^n - 1$. Chief among the admirers of
Newton were Taylor and Maclaurin. The quarrel between
English and Continental mathematicians caused them to work
quite independently of their great contemporaries across the
Channel.

\textbf{Brook Taylor} (1685--1731) was interested in many branches
of learning, and in the latter part of his life engaged mainly in
religious and philosophic speculations. His principal work,
\textit{Methodus incrementorum directa et inversa}, London, 1715--1717,
added a new branch to mathematics, now called ``finite differences.''
\index{Finite differences}%
He made many important applications of it, particularly
to the study of the form of movement of vibrating
%% -----File: 257.png---Folio 243-------
\index{Centre!of oscillation}%
\index{Mechanics!Taylor}%
strings, first reduced to mechanical principles by him. This
work contains also ``Taylor's theorem,'' the importance of
\index{Taylor's theorem}%
which was not recognised by analysts for over fifty years,
until Lagrange pointed out its power. His proof of it does not
consider the question of convergency, and is quite worthless.
The first rigorous proof was given a century later by Cauchy.
\index{Cauchy!ref.\ to}%
Taylor's work contains the first correct explanation of astronomical
refraction. He wrote also a work on linear perspective,
a treatise which, like his other writings, suffers for
want of fulness and clearness of expression. At the age of
twenty-three he gave a remarkable solution of the problem of
the centre of oscillation, published in~1714. His claim to
\index{Oscillation, centre of}%
priority was unjustly disputed by John Bernoulli.
\index{Bernoulli, John (born 1667)!ref.\ to}%

\textbf{Colin Maclaurin} (1698--1746) was elected professor of mathematics
\index{Maclaurin}%
at Aberdeen at the age of nineteen by competitive
examination, and in 1725 succeeded James Gregory at the University
\index{Gregory, James}%
of Edinburgh. He enjoyed the friendship of Newton,
\index{Newton!ref.\ to}%
and, inspired by Newton's discoveries, he published in 1719 his
\textit{Geometria Organica}, containing a new and remarkable mode
of generating conics, known by his name. A second tract,
\textit{De Linearum geometricarum Proprietatibus}, 1720, is remarkable
for the elegance of its demonstrations. It is based upon two
theorems: the first is the theorem of Cotes; the second is
\index{Cotes!ref.\ to}%
Maclaurin's: If through any point~$O$ a line be drawn meeting
the curve in $n$~points, and at these points tangents be drawn,
and if any other line through~$O$ cut the curve in $R_1$, $R_2$, etc.,
and the system of $n$~tangents in $r_1$, $r_2$, etc., then $\sum \dfrac{1}{OR} = \sum \dfrac{1}{Or}$.
This and Cotes' theorem are generalisations of theorems of
Newton. Maclaurin uses these in his treatment of curves of
\index{Curves!theory of}%
the second and third degree, culminating in the remarkable
theorem that if a quadrangle has its vertices and the two
points of intersection of its opposite sides upon a curve of the
%% -----File: 258.png---Folio 244-------
third degree, then the tangents drawn at two opposite vertices
cut each other on the curve. He deduced independently
Pascal's theorem on the hexagram. The following is his extension
of this theorem (\textit{Phil.\ Trans.}, 1735): If a polygon
move so that each of its sides passes through a fixed point,
and if all its summits except one describe curves of the degrees
$m$,~$n$, $p$,~etc., respectively, then the free summit moves on a
curve of the degree $2 mnp\cdots$,\DPnote{** PP: Why dots?} which reduces to $mnp \cdots$ when
the fixed points all lie on a straight line. Maclaurin wrote on
\index{Maclaurin!ref.\ to}%
pedal curves. He is the author of an \textit{Algebra}. The object of
his treatise on \textit{Fluxions} was to found the doctrine of fluxions
on geometric demonstrations after the manner of the ancients,
and thus, by rigorous exposition, answer such attacks as Berkeley's
that the doctrine rested on false reasoning. The \textit{Fluxions}
contained for the first time the correct way of distinguishing
between maxima and minima, and explained their use in the
\index{Maxima and minima}%
theory of multiple points. ``Maclaurin's theorem'' was previously
given by James Stirling, and is but a particular case
\index{Stirling}%
of ``Taylor's theorem.'' Appended to the treatise on \textit{Fluxions}
is the solution of a number of beautiful geometric, mechanical,
and astronomical problems, in which he employs ancient
methods with such consummate skill as to induce Clairaut to
\index{Clairaut!ref.\ to}%
abandon analytic methods and to attack the problem of the
figure of the earth by pure geometry. His solutions commanded
the liveliest admiration of Lagrange. Maclaurin investigated
\index{Lagrange!ref.\ to}%
the attraction of the ellipsoid of revolution, and
showed that a homogeneous liquid mass revolving uniformly
around an axis under the action of gravity must assume the
form of an ellipsoid of revolution. Newton had given this
\index{Newton!ref.\ to}%
theorem without proof. Notwithstanding the genius of Maclaurin,
his influence on the progress of mathematics in Great
Britain was unfortunate; for, by his example, he induced his
countrymen to neglect analysis and to be indifferent to the
%% -----File: 259.png---Folio 245-------
wonderful progress in the higher analysis made on the Continent.

It remains for us to speak of \textbf{Abraham de~Moivre} (1667--1754),
\index{Demoivre@{De Moivre}}%
\index{Moivre, de}%
who was of French descent, but was compelled to leave France
at the age of eighteen, on the Revocation of the Edict of Nantes.
He settled in London, where he gave lessons in mathematics.
He lived to the advanced age of eighty-seven and sank into a
state of almost total lethargy. His subsistence was latterly
dependent on the solution of questions on games of chance and
problems on probabilities, which he was in the habit of giving
at a tavern in St.~Martin's Lane. Shortly before his death he
declared that it was necessary for him to sleep ten or twenty
minutes longer every day. The day after he had reached the
total of over twenty-three hours, he slept exactly twenty-four
hours and then passed away in his sleep. De~Moivre enjoyed
the friendship of Newton and Halley. His power as a mathematician
lay in analytic rather than geometric investigation.
He revolutionised higher trigonometry by the discovery of the
\index{Trigonometry}%
theorem known by his name and by extending the theorems on
the multiplication and division of sectors from the circle to the
hyperbola. His work on the theory of probability surpasses
\index{Probability}%
anything done by any other mathematician except Laplace.
\index{Laplace!ref.\ to}%
His principal contributions are his investigations respecting
the Duration of Play, his Theory of Recurring Series, and
\index{Series}%
his extension of the value of Bernoulli's theorem by the aid
of Stirling's theorem.\Reference{42} His chief works are the \textit{Doctrine of
Chances}, 1716, the \textit{Miscellanea Analytica}, 1730, and his papers
in the \textit{Philosophical Transactions}.
%% -----File: 260.png---Folio 246-------


\Section{Euler, Lagrange, and Laplace}
\index{Euler!ref.\ to}%
\index{Lagrange!ref.\ to}%

During the epoch of ninety years from 1730 to 1820 the French
and Swiss cultivated mathematics with most brilliant success.
No previous period had shown such an array of illustrious
names. At this time Switzerland had her Euler; France, her
Lagrange, Laplace, Legendre, and Monge. The mediocrity of
French mathematics which marked the time of Louis~XIV.
was now followed by one of the very brightest periods of all
history. England and Germany, on the other hand, which
during the unproductive period in France had their Newton
and Leibniz, could now boast of no great mathematician.
France now waved the mathematical sceptre. Mathematical
studies among the English and German people had sunk to
the lowest ebb. Among them the direction of original research
was ill-chosen. The former adhered with excessive partiality
to ancient geometrical methods; the latter produced the combinatorial
school, which brought forth nothing of value.

The labours of Euler, Lagrange, and Laplace lay in higher
analysis, and this they developed to a wonderful degree. By
them analysis came to be completely severed from geometry.
During the preceding period the effort of mathematicians
not only in England, but, to some extent, even on the continent,
had been directed toward the solution of problems
clothed in geometric garb, and the results of calculation
were usually reduced to geometric form. A change now
took place. Euler brought about an emancipation of the
analytical calculus from geometry and established it as an
independent science. Lagrange and Laplace scrupulously
adhered to this separation. Building on the broad foundation
laid for higher analysis and mechanics by Newton
and Leibniz, Euler, with matchless fertility of mind, erected
%% -----File: 261.png---Folio 247-------
an elaborate structure. There are few great ideas pursued
by succeeding analysts which were not suggested by Euler,
or of which he did not share the honour of invention.
With, perhaps, less exuberance of invention, but with more
comprehensive genius and profounder reasoning, Lagrange
\index{Lagrange!ref.\ to}%
developed the infinitesimal calculus and put analytical
\index{Calculus!of variations}%
mechanics into the form in which we now know it. Laplace
applied the calculus and mechanics to the elaboration
of the theory of universal gravitation, and thus, largely extending
and supplementing the labours of Newton, gave a full
analytical discussion of the solar system. He also wrote an
epoch-marking work on Probability. Among the analytical
branches created during this period are the calculus of Variations
by Euler and Lagrange, Spherical Harmonics by Laplace
\index{Spherical Harmonics}%
and Legendre, and Elliptic Integrals by Legendre.
\index{Elliptic integrals}%
\index{Legendre!ref.\ to}%

Comparing the growth of analysis at this time with the
growth during the time of Gauss, Cauchy, and recent mathematicians,
\index{Cauchy!ref.\ to}%
\index{Gauss!ref.\ to}%
we observe an important difference. During the
former period we witness mainly a development with reference
to \emph{form}. Placing almost implicit confidence in results of
calculation, mathematicians did not always pause to discover
rigorous proofs, and were thus led to general propositions,
some of which have since been found to be true in only special
cases. The Combinatorial School in Germany carried this
\index{Combinatorial School}%
tendency to the greatest extreme; they worshipped formalism
and paid no attention to the actual contents of formulæ. But
in recent times there has been added to the dexterity in the
formal treatment of problems, a much-needed rigour of demonstration.
A good example of this increased rigour is seen in
the present use of infinite series as compared to that of Euler,
\index{Infinite series}%
and of Lagrange in his earlier works.

The ostracism of geometry, brought about by the master-minds %[** PP: Only instance]
of this period, could not last permanently. Indeed, a
%% -----File: 262.png---Folio 248-------
new geometric school sprang into existence in France before
the close of this period. Lagrange would not permit a single
\index{Lagrange!ref.\ to}%
diagram to appear in his \textit{Mécanique analytique}, but thirteen
years before his death, Monge published his epoch-making
\index{Monge!ref.\ to}%
\textit{Géometrie descriptive}.

\textbf{Leonhard Euler} (1707--1783) was born in Basel. His father,
\index{Euler|(}%
a minister, gave him his first instruction in mathematics and
then sent him to the University of Basel, where he became a
favourite pupil of John Bernoulli. In his nineteenth year he
composed a dissertation on the masting of ships, which received
the second prize from the French Academy of Sciences.
When John Bernoulli's two sons, Daniel and Nicolaus, went to
Russia, they induced Catharine~I., in 1727, to invite their friend
Euler to St.~Petersburg, where Daniel, in 1733, was assigned to
the chair of mathematics. In 1735 the solving of an astronomical
problem, proposed by the Academy, for which several
eminent mathematicians had demanded some months' time,
was achieved in three days by Euler with aid of improved
methods of his own. But the effort threw him into a fever
and deprived him of the use of his right eye. With still
superior methods this same problem was solved later by the
illustrious Gauss in one hour!\Reference{47} The despotism of Anne~I.
\index{Gauss!ref.\ to}%
caused the gentle Euler to shrink from public affairs and to
devote all his time to science. After his call to Berlin by
Frederick the Great in 1747, the queen of Prussia, who
received him kindly, wondered how so distinguished a scholar
should be so timid and reticent. Euler naïvely replied,
``Madam, it is because I come from a country where, when one
speaks, one is hanged.'' In 1766 he with difficulty obtained
permission to depart from Berlin to accept a call by Catharine~II.
to St.~Petersburg. Soon after his return to Russia he
became blind, but this did not stop his wonderful literary
productiveness, which continued for seventeen years, until the
%% -----File: 263.png---Folio 249-------
\index{Beta function}%
\index{Gamma function}%
\index{Isoperimetrical figures}%
\index{Notation!trigonometry}%
day of his death.\Reference{45} He dictated to his servant his \textit{Anleitung
zur Algebra}, 1770, which, though purely elementary, is meritorious
as one of the earliest attempts to put the fundamental
processes on a sound basis.

Euler wrote an immense number of works, chief of which
are the following: \textit{Introductio in analysin infinitorum}, 1748,
a work that caused a revolution in analytical mathematics, a
subject which had hitherto never been presented in so general
and systematic manner; \textit{Institutiones calculi differentialis}, 1755,
and \textit{Institutiones calculi integralis}, 1768--1770, which were the
most complete and accurate works on the calculus of that time,
\index{Calculus!of variations}%
and contained not only a full summary of everything then
known on this subject, but also the Beta and Gamma Functions
and other original investigations; \textit{Methodus inveniendi
lineas curvas maximi minimive proprietate gaudentes}, 1744,
which, displaying an amount of mathematical genius seldom
rivalled, contained his researches on the calculus of variations
(a subject afterwards improved by Lagrange), to the invention
of which Euler was led by the study of isoperimetrical curves,
the brachistochrone in a resisting medium, and the theory of
geodesics (subjects which had previously engaged the attention
\index{Geodesics}%
of the elder Bernoullis and others); the \textit{Theoria motuum planetarum
et cometarum}, 1744, \textit{Theoria motus lunæ}, 1753, \textit{Theoria
motuum lunæ}, 1772, are his chief works on astronomy; \textit{Ses
lettres à une princesse d'Allemagne sur quelques sujets de
Physique et de Philosophie}, 1770, was a work which enjoyed
great popularity.

We proceed to mention the principal innovations and inventions
of Euler. He treated trigonometry as a branch of
\index{Trigonometry}%
analysis, introduced (simultaneously with Thomas Simpson in
\index{Simpson}%
England) the now current abbreviations for trigonometric
functions, and simplified formulæ by the simple expedient
of designating the angles of a triangle by $A$,~$B$,~$C$, and the
%% -----File: 264.png---Folio 250-------
\index{Bezout@{Bézout}!ref.\ to}%
opposite sides by $a$,~$b$,~$c$, respectively. He pointed out the
relation between trigonometric and exponential functions. In
a paper of 1737 we first meet the symbol~$\pi$ to denote $3.14159\ldots$.\Reference{21}
\index{p@{$\pi$}: values for!selection of letter~$\pi$}%
\Addendlabel{250}%
Euler laid down the rules for the transformation of co-ordinates
\index{Euler!ref.\ to}%
in space, gave a methodic analytic treatment of plane curves
and of surfaces of the second order. He was the first to
\index{Surfaces, theory of}%
discuss the equation of the second degree in three variables,
and to classify the surfaces represented by it. By criteria
analogous to those used in the classification of conics he
obtained five species. He devised a method of solving \Change{bi-quadratic}{biquadratic}
equations by assuming $x = \sqrt{\smash[b]{p}} + \sqrt{\smash[b]{q}} + \sqrt{r}$, with the
\index{Equations!solution of}%
\index{Equations!theory of}%
hope that it would lead him to a general solution of algebraic
equations. The method of elimination by solving a series of
\index{Elimination}%
linear equations (invented independently by \DPtypo{Bezout}{Bézout}) and the
method of elimination by symmetric functions, are due to him.\Reference{20}
\index{Symmetric functions}%
Far reaching are Euler's researches on logarithms. Leibniz
\index{Leibniz!ref.\ to}%
\index{Logarithms}%
and John Bernoulli once argued the question whether a
\index{Bernoulli, John (born 1667)!ref.\ to}%
negative number has a logarithm. Bernoulli claimed that
since $(-a)^2 = (+a)^2$, we have $\log(-a)^2= \log(+a)^2$ and
$2 \log(-a) = 2 \log(+a)$, and finally $\log(-a) = \log(+a)$.
Euler proved that $a$ has really an infinite number of logarithms,
all of which are imaginary when $a$ is negative, and all
except one when $a$ is positive. He then explained how
$\log(-a)^2$ might equal $\log(+a)^2$, and yet $\log(-a)$ not
equal $\log(+a)$.

The subject of infinite series received new life from him.
\index{Infinite series}%
To his researches on series we owe the creation of the theory of
definite integrals by the development of the so-called \emph{Eulerian
integrals}. He warns his readers occasionally against the use
of divergent series, but is nevertheless very careless himself.
The rigid treatment to which infinite series are subjected now
was then undreamed of. No clear notions existed as to what
constitutes a convergent series. Neither Leibniz nor Jacob
%% -----File: 265.png---Folio 251-------
and John Bernoulli had entertained any serious doubt of the
\index{Bernoulli, Nicolaus (born 1687)}%
\index{Bernoulli, James (born 1654)!ref.\ to}%
\index{Bernoulli, John (born 1667)!ref.\ to}%
correctness of the expression $\frac{1}{2} = 1 - 1 + 1 - 1 +\dotsb$. Guido
Grandi went so far as to conclude from this that $\frac{1}{2} = 0 + 0 +
\index{Grandi}%
0 +\dotsb$. In the treatment of series Leibniz advanced a metaphysical
\index{Leibniz!ref.\ to}%
method of proof which held sway over the minds of
the elder Bernoullis, and even of Euler.\Reference{46} The tendency of
that reasoning was to justify results which seem to us now
highly absurd. The looseness of treatment can best be seen
from examples. The very paper in which Euler cautions
against divergent series contains the proof that
\begin{gather*}
\dotsb \frac{1}{n^2} + \frac{1}{n} + 1 + n + n^2 +\dotsb = 0 \text{ as follows:} \\
n+n^2+\dotsb = \frac{n}{1-n},\quad 1+\frac{1}{n}+\frac{1}{n^2}+\dotsb=\frac{n}{n-1};
\end{gather*}
these added give zero. Euler has no hesitation to write
$1-3+5-7+\dotsb=0$, and no one objected to such results
excepting Nicolaus Bernoulli, the nephew of John and Jacob.
Strange to say, Euler finally succeeded in converting Nicolaus
Bernoulli to his own erroneous views. At the present time
it is difficult to believe that Euler should have confidently
written $\sin\phi - 2 \sin 2\phi + 3 \sin 3\phi - 4 \sin 4\phi+\dotsb = 0$, but
such examples afford striking illustrations of the want of
scientific basis of certain parts of analysis at that time.
Euler's proof of the binomial formula for negative and
\index{Binomial formula}%
fractional exponents, which has been reproduced in elementary
text-books of even recent years, is faulty. A remarkable
development, due to Euler, is what he named the hypergeometric
series, the summation of which he observed to be
dependent upon the integration of a linear differential equation
of the second order, but it remained for Gauss to point
\index{Gauss!ref.\ to}%
out that for special values of its letters, this series represented
nearly all functions then known.

Euler developed the calculus of finite differences in the first
\index{Finite differences}%
%% -----File: 266.png---Folio 252-------
\index{Numbers!theory of numbers}%
\index{Theory of numbers}%
chapters of his \textit{Institutiones calculi differentialis}, and then
deduced the differential calculus from it. He established a
theorem on homogeneous functions, known by his name, and
\index{Addition theorem of elliptic integrals}%
contributed largely to the theory of differential equations, a
\index{Differential equations}%
subject which had received the attention of Newton, Leibniz,
\index{Leibniz!ref.\ to}%
\index{Newton!ref.\ to}%
and the Bernoullis, but was still undeveloped. Clairaut,
\index{Clairaut!ref.\ to}%
Fontaine, and Euler about the same time observed criteria of
\index{Fontaine}%
integrability, but Euler in addition showed how to employ
them to determine integrating factors. The principles on
which the criteria rested involved some degree of obscurity.
The celebrated addition-theorem for elliptic integrals was first
\index{Elliptic integrals}%
established by Euler. He invented a new algorithm for
continued fractions, which he employed in the solution of
\index{Continued fractions}%
\index{Fractions!continued}%
the indeterminate equation $ax + by = c$. We now know that
substantially the same solution of this equation was given
$1000$ years earlier, by the Hindoos. By giving the factors of
the number $2^{2^n} + 1$ when $n = 5$, he pointed out that this expression
did not always represent primes, as was supposed by
Fermat. He first supplied the proof to ``Fermat's theorem,''
\index{Fermat!ref.\ to}%
\index{Fermat's theorem}%
and to a second theorem of Fermat, which states that every
prime of the form $4n + 1$ is expressible as the sum of two
squares in one and only one way. A third theorem of Fermat,
that $x^n + y^n = z^n$, has no integral solution for values of $n$
greater than~$2$, was proved by Euler to be correct when $n = 3$.
Euler discovered four theorems which taken together make
out the great law of quadratic reciprocity, a law independently
\index{Quadratic reciprocity}%
discovered by Legendre.\Reference{48} Euler enunciated and proved a
\index{Legendre!ref.\ to}%
well-known theorem, giving the relation between the number
of vertices, faces, and edges of certain polyhedra, which,
however, appears to have been known to Descartes. The
powers of Euler were directed also towards the fascinating
subject of the theory of probability, in which he solved some
\index{Probability}%
difficult problems.
%% -----File: 267.png---Folio 253-------
\index{Action, least}%

Of no little importance are Euler's labours in analytical
mechanics. Says Whewell: ``The person who did most to
\index{Mechanics!Euler}%
\index{Whewell}%
give to analysis the generality and symmetry which are now
its pride, was also the person who made mechanics analytical;
I mean Euler.''\Reference{11} He worked out the theory of the rotation of
a body around a fixed point, established the general equations
of motion of a free body, and the general equation of hydrodynamics.
He solved an immense number and variety of mechanical
problems, which arose in his mind on all occasions. Thus,
on reading Virgil's lines, ``The anchor drops, the rushing keel
is staid,'' he could not help inquiring what would be the
ship's motion in such a case. About the same time as Daniel
Bernoulli he published the \textit{Principle of the Conservation of
\index{Conservation!of areas}%
Areas} and defended the principle of ``least action,'' advanced
\index{Areas, conservation of}%
\index{Least action}%
by Maupertius. He wrote also on tides and on sound.
\index{Maupertius}%

Astronomy owes to Euler the method of the variation of
\index{Astronomy!more recent researches}%
arbitrary constants. By it he attacked the problem of perturbations,
explaining, in case of two planets, the secular variations
of eccentricities, nodes, etc. He was one of the first
to take up with success the theory of the moon's motion by
giving approximate solutions to the ``problem of three bodies.''
\index{Problem of three bodies}%
\index{Three bodies, problem of}%
He laid a sound basis for the calculation of tables of the moon.
These researches on the moon's motion, which captured two
prizes, were carried on while he was blind, with the assistance
of his sons and two of his pupils.

Most of his memoirs are contained in the transactions of
the Academy of Sciences at St.~Petersburg, and in those of
the Academy at Berlin. From 1728 to 1783 a large portion
of the Petropolitan transactions were filled by his writings.
He had engaged to furnish the Petersburg Academy with
memoirs in sufficient number to enrich its acts for twenty
years---a promise more than fulfilled, for down to 1818 the
volumes usually contained one or more papers of his. It has
%% -----File: 268.png---Folio 254-------
been said that an edition of Euler's complete works would fill
$16,000$ quarto pages. His mode of working was, first to concentrate
his powers upon a special problem, then to solve
separately all problems growing out of the first. No one
excelled him in dexterity of accommodating methods to special
problems. It is easy to see that mathematicians could not
long continue in Euler's habit of writing and publishing. The
material would soon grow to such enormous proportions as to
be unmanageable. We are not surprised to see almost the
opposite in Lagrange, his great successor. The great Frenchman
\index{Lagrange!ref.\ to}%
delighted in the general and abstract, rather than, like
Euler, in the special and concrete. His writings are condensed
and give in a nutshell what Euler narrates at great
\index{Euler|)}%
length.

\textbf{Jean-le-Rond D'Alembert} (1717--1783) was exposed, when
\index{D'Alembert|(}%
\index{D'Alembert!ref.\ to}%
\index{D'Alembert's principle}%
an infant, by his mother in a market by the church of St.\
Jean-le-Rond, near the Nôtre-Dame in Paris, from which he
derived his Christian name. He was brought up by the wife
of a poor glazier. It is said that when he began to show signs
of great talent, his mother sent for him, but received the
reply, ``You are only my step-mother; the glazier's wife is
my mother.'' His father provided him with a yearly income.
D'Alembert entered upon the study of law, but such was his
love for mathematics, that law was soon abandoned. At the
age of twenty-four his reputation as a mathematician secured
for him admission to the Academy of Sciences. In 1743
appeared his \textit{Traité de dynamique}, founded upon the important
general principle bearing his name: The impressed forces are
equivalent to the effective forces. D'Alembert's principle
seems to have been recognised before him by Fontaine, and
\index{Fontaine}%
in some measure by John Bernoulli and Newton. D'Alembert
\index{Newton!ref.\ to}%
gave it a clear mathematical form and made numerous applications
of it. It enabled the laws of motion and the reasonings
%% -----File: 269.png---Folio 255-------
\index{Hydrodynamics}%
\index{Hydrostatics}%
\index{Infinite series}%
\index{Strings, vibrating}%
\index{Vibrating strings}%
depending on them to be represented in the most general
form, in analytical language. D'Alembert applied it in 1744
in a treatise on the equilibrium and motion of fluids, in 1746
to a treatise on the general causes of winds, which obtained
a prize from the Berlin Academy. In both these treatises, as
also in one of 1747, discussing the famous problem of vibrating
chords, he was led to partial differential equations. He was
\index{Partial differential equations}%
a leader among the pioneers in the study of such equations.
To the equation $\dfrac{\partial^2 y}{\partial t^2} = a^2 \dfrac{\partial^2 y}{\partial x^2}$, arising in the problem of vibrating
chords, he gave as the general solution,
\[
y = f(x + at) + \phi(x - at),
\]
and showed that there is only one arbitrary function, if $y$ be
supposed to vanish for $x = 0$ and~$x=l$. Daniel Bernoulli,
\index{Bernoulli, Daniel!ref.\ to}%
starting with a particular integral given by Brook Taylor,
\index{Taylor, B.!ref.\ to}%
showed that this differential equation is satisfied by the
trigonometric series
\[
y = \alpha \sin \frac{\pi x}{l} · \cos \frac{\pi t}{l}
   + \beta \sin \frac{2 \pi x}{l}· \cos \frac{2 \pi t}{l} + \dotsb,
\]
and claimed this expression to be the most general solution.
Euler denied its generality, on the ground that, if true, the
doubtful conclusion would follow that the above series represents
any arbitrary function of a variable. These doubts were
dispelled by Fourier. Lagrange proceeded to find the sum
\index{Fourier!ref.\ to}%
\index{Lagrange!ref.\ to}%
of the above series, but D'Alembert rightly objected to his
process, on the ground that it involved divergent series.\Reference{46}
\index{Divergent series}%

A most beautiful result reached by D'Alembert, with aid
of his principle, was the complete solution of the problem of
the precession of the equinoxes, which had baffled the talents
of the best minds. He sent to the French Academy in 1747,
on the same day with Clairaut, a solution of the problem of
\index{Clairaut!ref.\ to}%
three bodies. This had become a question of universal interest
%% -----File: 270.png---Folio 256-------
\index{Problem of three bodies}%
to mathematicians, in which each vied to outdo all others.
The problem of two bodies, requiring the determination of
their motion when they attract each other with forces inversely
proportional to the square of the distance between
them, had been completely solved by Newton. The ``problem
of three bodies'' asks for the motion of three bodies attracting
\index{Three bodies, problem of}%
each other according to the law of gravitation. Thus far,
the complete solution of this has transcended the power of
analysis. The general differential equations of motion were
\index{Differential calculus!philosophy of}%
stated by Laplace, but the difficulty arises in their integration.
\index{Laplace!ref.\ to}%
The ``solutions'' hitherto given are merely convenient methods
of approximation in special cases when one body is the sun,
disturbing the motion of the moon around the earth, or where
a planet moves under the influence of the sun and another
planet.

In the discussion of the meaning of negative quantities, of
\index{Negative quantities}%
the fundamental processes of the calculus, and of the theory of
probability, D'Alembert paid some attention to the philosophy
of mathematics. His criticisms were not always happy. In
1754 he was made permanent secretary of the French Academy.
During the last years of his life he was mainly occupied with
the great French encyclopædia, which was begun by Diderot
and himself. D'Alembert declined, in 1762, an invitation of
\index{D'Alembert|)}%
Catharine~II. to undertake the education of her son. Frederick
the~Great pressed him to go to Berlin. He made a visit, but
declined a permanent residence there.

\textbf{Alexis Claude Clairaut} (1713--1765) was a youthful prodigy.
\index{Clairaut|(}%
He read l'Hospital's works on the infinitesimal calculus and on
conic sections at the age of ten. In 1731 was published his
\textit{Recherches sur les courbes à double courbure}, which he had ready
for the press when he was sixteen. It was a work of remarkable
elegance and secured his admission to the Academy of
Sciences when still under legal age. In 1731 he gave a proof of
%% -----File: 271.png---Folio 257-------
\index{Astronomy!more recent researches}%
\index{Cubic curves}%
the theorem enunciated by Newton, that every cubic is a projection
\index{Newton!ref.\ to}%
of one of five divergent parabolas. Clairaut formed the
\index{Divergent parabolas}%
acquaintance of Maupertius, whom he accompanied on an expedition
\index{Maupertius}%
to Lapland to measure the length of a degree of the
meridian. At that time the shape of the earth was a subject
\index{Earth!figure of}%
of serious disagreement. Newton and Huygens had concluded
\index{Huygens!ref.\ to}%
from theory that the earth was flattened at the poles. About
1713 Dominico Cassini measured an arc extending from Dunkirk
\index{Cassini, D}%
to Perpignan and arrived at the startling result that the earth
is elongated at the poles. To decide between the conflicting
opinions, measurements were renewed. Maupertius earned by
his work in Lapland the title of ``earth flattener'' by disproving
the Cassinian tenet that the earth was elongated at the
poles, and showing that Newton was right. On his return, in
1743, Clairaut published a work, \textit{Théorie de la figure de la Terre},
\index{Figure of the earth}%
which was based on the results of Maclaurin on homogeneous
ellipsoids. It contains a remarkable theorem, named after
Clairaut, that the sum of the fractions expressing the ellipticity
and the increase of gravity at the pole is equal to $2\frac{1}{2}$~times the
fraction expressing the centrifugal force at the equator, the
unit of force being represented by the force of gravity at the
equator. This theorem is independent of any hypothesis with
respect to the law of densities of the successive strata of the
earth. It embodies most of Clairaut's researches. Todhunter
says that ``in the figure of the earth no other person has
accomplished so much as Clairaut, and the subject remains at
present substantially as he left it, though the form is different.
The splendid analysis which Laplace supplied, adorned but did
not really alter the theory which started from the creative
hands of Clairaut.''

In 1752 he gained a prize of the St.~Petersburg Academy
for his paper on \textit{Théorie de la Lune}, in which for the first time
modern analysis is applied to lunar motion. This contained
%% -----File: 272.png---Folio 258-------
the explanation of the motion of the lunar apsides. This
motion, left unexplained by Newton, seemed to him at first
\index{Newton!ref.\ to}%
inexplicable by Newton's law, and he was on the point of
advancing a new hypothesis regarding gravitation, when, taking
\index{Gravitation, theory of}%
the precaution to carry his calculation to a higher degree
of approximation, he reached results agreeing with observation.
The motion of the moon was studied about the same
time by Euler and D'Alembert. Clairaut predicted that
\index{D'Alembert!ref.\ to}%
\index{Euler!ref.\ to}%
``Halley's Comet,'' then expected to return, would arrive at
\index{Halley's Comet}%
its nearest point to the sun on April~13, 1759, a date which
turned out to be one month too late. He was the first to
detect singular solutions in differential equations of the first
order but of higher degree than the first.

In their scientific labours there was between Clairaut and
D'Alembert great rivalry, often far from friendly. The growing
ambition of Clairaut to shine in society, where he was a
\index{Clairaut|)}%
great favourite, hindered his scientific work in the latter part
of his life.

\textbf{Johann Heinrich Lambert} (1728--1777), born at Mühl\-hausen
\index{Lambert|(}%
in Alsace, was the son of a poor tailor. While working at his
father's trade, he acquired through his own unaided efforts a
knowledge of elementary mathematics. At the age of thirty he
became tutor in a Swiss family and secured leisure to continue
his studies. In his travels with his pupils through Europe he
became acquainted with the leading mathematicians. In 1764
he settled in Berlin, where he became member of the Academy,
and enjoyed the society of Euler and Lagrange. He received
a small pension, and later became editor of the Berlin \textit{Ephemeris}.
His many-sided scholarship reminds one of Leibniz.
In his \textit{Cosmological Letters} he made some remarkable prophecies
regarding the stellar system. In mathematics he made
several discoveries which were extended and overshadowed by
his great contemporaries. His first research on pure mathematics
%% -----File: 273.png---Folio 259-------
\index{Differential calculus!philosophy of}%
developed in an infinite series the root~$x$ of the equation
\index{Infinite series}%
$x^m + px = q$. Since each equation of the form $ax^r + bx^s = d$
can be reduced to $x^m + px = q$ in two ways, one or the other of
the two resulting series was always found to be convergent,
and to give a value of~$x$. Lambert's results stimulated Euler,
\index{Euler!ref.\ to}%
who extended the method to an equation of four terms, and
particularly Lagrange, who found that a function of a root of
\index{Lagrange!ref.\ to}%
$a - x + \phi(x) = 0$ can be expressed by the series bearing his
name. In 1761 Lambert communicated to the Berlin Academy
a memoir, in which he proves that $\pi$ is irrational. This proof
\index{p@{$\pi$}: values for!proved to be irrational}%
is given in Note~IV. of Legendre's \textit{Géometrie}, where it is
\index{Legendre!ref.\ to}%
extended to~$\pi^2$. To the genius of Lambert we owe the introduction
\index{Lambert|)}%
into trigonometry of hyperbolic functions, which he
\index{Trigonometry}%
designated by $\textit{sinh}\,x$, $\textit{cosh}\,x$,~etc. His \textit{Freye Perspective}, 1759
and 1773, contains researches on descriptive geometry, and
entitle him to the honour of being the forerunner of Monge.
\index{Monge!ref.\ to}%
In his effort to simplify the calculation of cometary orbits, he
was led geometrically to some remarkable theorems on conics,
for instance this: ``If in two ellipses having a common major
axis we take two such arcs that their chords are equal, and
that also the sums of the radii vectores, drawn respectively
from the foci to the extremities of these arcs, are equal to
each other, then the sectors formed in each ellipse by the arc
and the two radii vectores are to each other as the square
roots of the parameters of the ellipses.''\Reference{13}

\textbf{John Landen} (1719--1790) was an English mathematician
\index{Landen}%
whose writings served as the starting-point of investigations
by Euler, Lagrange, and Legendre. Landen's capital discovery,
contained in a memoir of 1755, was that every arc of the
hyperbola is immediately rectified by means of two arcs of an
ellipse. In his ``residual analysis'' he attempted to obviate
the metaphysical difficulties of fluxions by adopting a purely
algebraic method. Lagrange's \textit{Calcul des Fonctions} is based
%% -----File: 274.png---Folio 260-------
\index{Witch of Agnesi}%
upon this idea. Landen showed how the algebraic expression
for the roots of a cubic equation could be derived by application
of the differential and integral calculus. Most of the
time of this suggestive writer was spent in the pursuits of
active life.

\textbf{Étienne Bézout} (1730--1783) was a French writer of popular
\index{Bezout@{Bézout}}%
mathematical school-books. In his \textit{Théorie générale des Équations
Algébriques}, 1779, he gave the method of elimination by
\index{Bezout's@{Bézout's method of elimination}}%
linear equations (invented also by Euler). This method was
\index{Equations!solution of}%
first published by him in a memoir of 1764, in which he uses
determinants, without, however, entering upon their theory.
A beautiful theorem as to the degree of the resultant goes by
his name.

\textbf{Louis Arbogaste} (1759--1803) of Alsace was professor of
\index{Arbogaste}%
mathematics at Strasburg. His chief work, the \textit{Calcul des
Dérivations}, 1800, gives the method known by his name, by
which the successive coefficients of a development are derived
from one another when the expression is complicated. De~Morgan
\index{Demorgan@{De Morgan}!ref.\ to}%
has pointed out that the true nature of derivation
is differentiation accompanied by integration. In this book
for the first time are the symbols of operation separated from
those of quantity. The notation $D_x y$ for $dy/dx$ is due to him.
\index{Notation!differential calculus}%

\textbf{Maria Gaetana Agnesi} (1718--1799) of Milan, distinguished as
\index{Agnesi}%
a linguist, mathematician, and philosopher, filled the mathematical
chair at the University of Bologna during her father's
sickness. In 1748 she published her \textit{Instituzioni Analitiche},
which was translated into English in 1801. The ``witch of
Agnesi'' or ``versiera'' is a plane curve containing a straight
line, $x = 0$, and a cubic $\left(\dfrac{y}{c}\right)^2 + 1 = \dfrac{c}{x}$.

\textbf{Joseph Louis Lagrange} (1736--1813), one of the greatest
\index{Lagrange|(}%
mathematicians of all times, was born at Turin and died at
Paris. He was of French extraction. His father, who had
%% -----File: 275.png---Folio 261-------
\index{Isoperimetrical figures}%
charge of the Sardinian military chest, was once wealthy, but
lost all he had in speculation. Lagrange considered this loss
his good fortune, for otherwise he might not have made mathematics
the pursuit of his life. While at the college in Turin
his genius did not at once take its true bent. Cicero and Virgil
at first attracted him more than Archimedes and Newton.
He soon came to admire the geometry of the ancients, but the
perusal of a tract of Halley roused his enthusiasm for the
\index{Halley}%
analytical method, in the development of which he was destined
to reap undying glory. He now applied himself to
mathematics, and in his seventeenth year he became professor
of mathematics in the royal military academy at Turin.
Without assistance or guidance he entered upon a course of
study which in two years placed him on a level with the
greatest of his contemporaries. With aid of his pupils he
established a society which subsequently developed into the
Turin Academy. In the first five volumes of its transactions
appear most of his earlier papers. At the age of nineteen he
communicated to Euler a general method of dealing with
\index{Euler!ref.\ to}%
``isoperimetrical problems,'' known now as the Calculus of
\index{Calculus!of variations}%
Variations. This commanded Euler's lively admiration, and
he courteously withheld for a time from publication some
researches of his own on this subject, so that the youthful
Lagrange might complete \emph{his} investigations and claim the
invention. Lagrange did quite as much as Euler towards the
creation of the Calculus of Variations. As it came from Euler
it lacked an analytic foundation, and this Lagrange supplied.
He separated the principles of this calculus from geometric
considerations by which his predecessor had derived them.
Euler had assumed as fixed the limits of the integral, \ie~the
extremities of the curve to be determined, but Lagrange
removed this restriction and allowed all co-ordinates of the
curve to vary at the same time. Euler introduced in 1766 the
%% -----File: 276.png---Folio 262-------
\index{Acoustics}%
\index{Astronomy!more recent researches}%
name ``calculus of variations,'' and did much to improve this
science along the lines marked out by Lagrange.

Another subject engaging the attention of Lagrange at
Turin was the propagation of sound. In his papers on this
subject in the \textit{Miscellanea Taurinensia}, the young mathematician
appears as the critic of Newton, and the arbiter between
\index{Newton!ref.\ to}%
Euler and D'Alembert. By considering only the particles
\index{D'Alembert!ref.\ to}%
\index{Euler!ref.\ to}%
which are in a straight line, he reduced the problem to the
same partial differential equation that represents the motions
of vibrating strings. The general integral of this was found by
\index{Strings, vibrating}%
\index{Vibrating strings}%
D'Alembert to contain two arbitrary functions, and the question
\index{Functions!arbitrary}%
now came to be discussed whether an arbitrary function
may be discontinuous. D'Alembert maintained the negative
against Euler, Daniel Bernoulli, and finally Lagrange,---arguing
\index{Bernoulli, Daniel!ref.\ to}%
that in order to determine the position of a point of the
chord at a time~$t$, the initial position of the chord must be
continuous. Lagrange settled the question in the affirmative.

By constant application during nine years, Lagrange, at the
age of twenty-six, stood at the summit of European fame.
But his intense studies had seriously weakened a constitution
never robust, and though his physicians induced him to take
rest and exercise, his nervous system never fully recovered its
tone, and he was thenceforth subject to fits of melancholy.

In 1764 the French Academy proposed as the subject of
a prize the theory of the libration of the moon. It demanded
an explanation, on the principle of universal gravitation, why
the moon always turns, with but slight variations, the same
\DPtypo{phase}{face} to the earth. Lagrange secured the prize. This success
encouraged the Academy to propose as a prize the theory
of the four satellites of Jupiter,---a problem of six bodies,
more difficult than the one of three bodies previously solved
by Clairaut, D'Alembert, and Euler. Lagrange overcame the
\index{Clairaut!ref.\ to}%
difficulties, but the shortness of time did not permit him to
%% -----File: 277.png---Folio 263-------
exhaust the subject. Twenty-four years afterwards it was
completed by Laplace. Later astronomical investigations of
\index{Laplace!ref.\ to}%
Lagrange are on cometary perturbations (1778 and 1783), on
Kepler's problem, and on a new method of solving the problem
\index{Kepler!ref.\ to}%
of three bodies.

Being anxious to make the personal acquaintance of leading
mathematicians, Lagrange visited Paris, where he enjoyed the
stimulating delight of conversing with Clairaut, D'Alembert,
Condorcet, the Abbé Marie, and others. He had planned a
visit to London, but he fell dangerously ill after a dinner in
Paris, and was compelled to return to Turin. In 1766 Euler
left Berlin for St.~Petersburg, and he pointed out Lagrange as
the only man capable of filling the place. D'Alembert recommended
him at the same time. Frederick the Great thereupon
sent a message to Turin, expressing the wish of ``the
greatest king of Europe'' to have ``the greatest mathematician''
at his court. Lagrange went to Berlin, and staid\DPnote{** PP: [sic], archaic form} there
twenty years. Finding all his colleagues married, and being
assured by their wives that the marital state alone is happy,
he married. The union was not a happy one. His wife
soon died. Frederick the Great held him in high esteem,
and frequently conversed with him on the advantages of perfect
regularity of life. This led Lagrange to cultivate regular
habits. He worked no longer each day than experience taught
him he could without breaking down. His papers were carefully
thought out before he began writing, and when he wrote
he did so without a single correction.

During the twenty years in Berlin he crowded the transactions
of the Berlin Academy with memoirs, and wrote also
the epoch-making work called the \textit{Mécanique Analytique}. He
enriched algebra by researches on the solution of equations.
\index{Equations!solution of}%
There are two methods of solving directly algebraic equations,---that
of substitution and that of combination. The
%% -----File: 278.png---Folio 264-------
\index{Warring}%
former method was developed by Ferrari, Vieta, Tchirnhausen,
\index{Ferrari!ref.\ to}%
\index{Tchirnhausen!ref.\ to}%
\index{Vieta!ref.\ to}%
Euler, Bézout, and Lagrange; the latter by Vandermonde and
\index{Bezout@{Bézout}!ref.\ to}%
\index{Euler!ref.\ to}%
\index{Vandermonde!ref.\ to}%
Lagrange.\Reference{20} In the method of substitution the original forms
are so transformed that the determination of the roots is made
to depend upon simpler functions (resolvents). In the method
of combination auxiliary quantities are substituted for certain
simple combinations (``types'') of the unknown roots of the
equation, and auxiliary equations (resolvents) are obtained for
\index{Equations!numerical}%
these quantities with aid of the coefficients of the given equation.
Lagrange traced all known algebraic solutions of equations
to the uniform principle consisting in the formation and
solution of equations of lower degree whose roots are linear
functions of the required roots, and of the roots of unity. He
showed that the quintic cannot be reduced in this way, its
resolvent being of the sixth degree. His researches on the
theory of equations were continued after he left Berlin. In
the \textit{Résolution des équations numériques} (1798) he gave a
method of approximating to the real roots of numerical equations
by continued fractions. Among other things, it contains
also a proof that every equation must have a root,---a theorem
which appears before this to have been considered self-evident.
Other proofs of this were given by Argand, Gauss, and Cauchy.
\index{Argand!ref.\ to}%
\index{Cauchy!ref.\ to}%
\index{Gauss!ref.\ to}%
In a note to the above work Lagrange uses Fermat's theorem
\index{Fermat!ref.\ to}%
and certain suggestions of Gauss in effecting a complete algebraic
solution of any binomial equation.

While in Berlin Lagrange published several papers on the
theory of numbers. In~1769 he gave a solution in integers of
\index{Numbers!theory of numbers}%
\index{Theory of numbers}%
indeterminate equations of the second degree, which resembles
the Hindoo cyclic method; he was the first to prove, in~1771,
``Wilson's theorem,'' enunciated by an Englishman, John
\index{Wilson}%
\index{Wilson's theorem}%
Wilson, and first published by Waring in his \textit{Meditationes
Algebraicæ}; he investigated in 1775 under what conditions
$±2$~and~$±5$ ($-1$~and~$±3$ having been discussed by Euler)
%% -----File: 279.png---Folio 265-------
are quadratic residues, or non-residues of odd prime numbers,~$q$;
he proved in 1770 Méziriac's theorem that every integer is
\index{Meziriac@{Méziriac}!ref.\ to}%
equal to the sum of four, or a less number, of squares. He
proved Fermat's theorem on $x^n + y^n = z^n$, for the case $n = 4$,
\index{Fermat!ref.\ to}%
also Fermat's theorem that, if $a^2 + b^2 = c^2$, then $ab$ is not a
square.

In his memoir on Pyramids, 1773, Lagrange made considerable
use of determinants of the third order, and demonstrated
\index{Determinants}%
that the square of a determinant is itself a determinant. He
never, however, dealt explicitly and directly with determinants;
he simply obtained accidentally identities which are
now recognised as relations between determinants.

Lagrange wrote much on differential equations. Though
\index{Differential equations}%
the subject of contemplation by the greatest mathematicians
(Euler, D'Alembert, Clairaut, Lagrange, Laplace), yet more
\index{D'Alembert!ref.\ to}%
\index{Euler!ref.\ to}%
than other branches of mathematics did they resist the systematic
application of fixed methods and principles. Lagrange
established criteria for singular solutions (\textit{Calcul des Fonctions},
\index{Singular solutions}%
Lessons~14--17), which are, however, erroneous. He was the
first to point out the geometrical significance of such solutions.
He generalised Euler's researches on total differential equations
of two variables, and of the ninth order; he gave a solution
of partial differential equations of the first order (\textit{Berlin
Memoirs}, 1772 and 1774), and spoke of their singular solutions,
extending their solution in \textit{Memoirs} of 1779 and 1785 to equations
of any number of variables. The discussion on partial
differential equations of the second order, carried on by
D'Alembert, Euler, and Lagrange, has already been referred
to in our account of D'Alembert.

While in Berlin, Lagrange wrote the ``\textit{Mécanique Analytique},''
the greatest of his works (Paris, 1788). From the principle
of virtual velocities he deduced, with aid of the calculus of
\index{Calculus!of variations}%
\index{Virtual velocities}%
variations, the whole system of mechanics so elegantly and
%% -----File: 280.png---Folio 266-------
harmoniously that it may fitly be called, in Sir William
Rowan Hamilton's words, ``a kind of scientific poem.'' It is a
\index{Hamilton, W.~R.!ref.\ to}%
most consummate example of analytic generality. Geometrical
figures are nowhere allowed. ``On ne trouvera point de figures
dans cet ouvrage'' (Preface). The two divisions of mechanics---statics
\index{Mechanics!Lagrange}%
and dynamics---are in the first four sections of each
carried out analogously, and each is prefaced by a historic
sketch of principles. Lagrange formulated the principle of
least action. In their original form, the equations of motion
\index{Least action}%
involve the co-ordinates $x$,~$y$,~$z$, of the different particles $m$ or
$dm$ of the system. But $x$,~$y$,~$z$, are in general not independent,
and Lagrange introduced in place of them any variables $\xi$,~$\psi$,~$\phi$,
whatever, determining the position of the point at the time.
These may be taken to be independent. The equations of
motion may now assume the form
\[
\frac{d}{dt}\, \frac{dT}{d\xi'} - \frac{dT}{d\xi} + \Xi = 0;
\]
or when $\Xi$,~$\Change{\psi}{\Psi}$, $\Change{\phi}{\Phi},\ldots$ are the partial differential coefficients
with respect to $\xi$,~$\psi$, $\phi,\ldots$ of one and the same function~$V$,
then the form
\[
\frac{d}{dt}\, \frac{dT}{d\xi'} - \frac{dT}{d\xi} + \frac{dV}{d\xi} = 0.
\]
The latter is \textit{par excellence} the Lagrangian form of the equations
of motion. With Lagrange originated the remark that
mechanics may be regarded as a geometry of four dimensions.
To him falls the honour of the introduction of the potential
into dynamics.\Reference{49} Lagrange was anxious to have his \textit{Mécanique
Analytique} published in Paris. The work was ready for print
in 1786, but not till 1788 could he find a publisher, and then
only with the condition that after a few years he would purchase
all the unsold copies. The work was edited by
Legendre.
\index{Legendre!ref.\ to}%
%% -----File: 281.png---Folio 267-------

After the death of Frederick the Great, men of science
were no longer respected in Germany, and Lagrange accepted
an invitation of Louis~XVI. to migrate to Paris. The French
queen treated him with regard, and lodging was procured for
him in the Louvre. But he was seized with a long attack of
melancholy which destroyed his taste for mathematics. For
two years his printed copy of the \textit{Mécanique}, fresh from the
press,---the work of a quarter of a century,---lay unopened on
his desk. Through Lavoisier he became interested in chemistry,
which he found ``as easy as algebra.'' The disastrous
\index{Algebra!Lagrange}%
crisis of the French Revolution aroused him again to activity.
About this time the young and accomplished daughter of the
astronomer Lemonnier took compassion on the sad, lonely
\index{Lemonnier}%
Lagrange, and insisted upon marrying him. Her devotion to
him constituted the one tie to life which at the approach of
death he found it hard to break.

He was made one of the commissioners to establish weights
and measures having units founded on nature. Lagrange
strongly favoured the decimal subdivision, the general idea of
which was obtained from a work of Thomas Williams, London,
\index{Williams}%
1788. Such was the moderation of Lagrange's character, and
such the universal respect for him, that he was retained as president
of the commission on weights and measures even after it
had been \emph{purified} by the Jacobins by striking out the names
of Lavoisier, Laplace, and others. Lagrange took alarm at the
fate of Lavoisier, and planned to return to Berlin, but at the
establishment of the \textit{École Normale} in 1795 in Paris, he was
induced to accept a professorship. Scarcely had he time to
elucidate the foundations of arithmetic and algebra to young
pupils, when the school was closed. His additions to the
algebra of Euler were prepared at this time. In 1797 the
\index{Euler!ref.\ to}%
\textit{École Polytechnique} was founded, with Lagrange as one of
the professors. The earliest triumph of this institution was
%% -----File: 282.png---Folio 268-------
\index{Functions!theory of}%
\index{Theory of functions}%
the restoration of Lagrange to analysis. His mathematical
activity burst out anew. He brought forth the \textit{Théorie des
fonctions analytiques} (1797), \textit{Leçons sur le calcul des fonctions},
a treatise on the same lines as the preceding (1801), and the
\textit{Résolution des équations \DPtypo{numeriques}{numériques}} (1798). In 1810 he
began a thorough revision of his \textit{Mécanique analytique}, but
he died before its completion.

The \textit{Théorie des fonctions}, the germ of which is found in a
memoir of his of 1772, aimed to place the principles of the
calculus upon a sound foundation by relieving the mind of the
difficult conception of a limit or infinitesimal. John Landen's
\index{Landen!ref.\ to}%
residual calculus, professing a similar object, was unknown to
him. Lagrange attempted to prove Taylor's theorem (the
\index{Taylor's theorem}%
power of which he was the first to point out) by simple algebra,
and then to develop the entire calculus from that theorem.
The principles of the calculus were in his day involved in
philosophic difficulties of a serious nature. The infinitesimals
of Leibniz had no satisfactory metaphysical basis. In the
\index{Leibniz!ref.\ to}%
differential calculus of Euler they were treated as absolute
\index{Differential calculus!philosophy of}%
\index{Euler!ref.\ to}%
zeros. In Newton's limiting ratio, the magnitudes of which it
\index{Newton!ref.\ to}%
is the ratio cannot be found, for at the moment when they
should be caught and equated, there is neither arc nor chord.
The chord and arc were not taken by Newton as equal before
vanishing, nor after vanishing, but \emph{when} they vanish. ``That
method,'' said Lagrange, ``has the great inconvenience of considering
quantities in the state in which they cease, so to
speak, to be quantities; for though we can always well conceive
the ratios of two quantities, as long as they remain
finite, that ratio offers to the mind no clear and precise idea,
as soon as its terms become both nothing at the same time.''
D'Alembert's method of limits was much the same as the
\index{D'Alembert!ref.\ to|(}%
\index{Limits, method of}%
method of prime and ultimate ratios. D'Alembert taught
\index{Prime and ultimate ratios}%
that a variable actually reached its limit. When Lagrange
%% -----File: 283.png---Folio 269-------
endeavoured to free the calculus of its metaphysical difficulties,
by resorting to common algebra, he avoided the whirlpool of
Charybdis only to suffer wreck against the rocks of Scylla.
The algebra of his day, as handed down to him by Euler, was
founded on a false view of infinity. No correct theory of
\index{Infinity}%
infinite series had then been established. Lagrange proposed
\index{Infinite series}%
to define the differential coefficient of $f(x)$ with respect to~$x$
as the coefficient of~$h$ in the expansion of $f(x + h)$ by Taylor's
theorem, and thus to avoid all reference to limits. But he
used infinite series without ascertaining that they were convergent,
and his proof that $f(x + h)$ can always be expanded
in a series of ascending powers of~$h$, labours under serious
defects. Though Lagrange's method of developing the calculus
was at first greatly applauded, its defects were fatal, and to-day
his ``method of derivatives,'' as it was called, has been generally
\index{Derivatives, method of}%
abandoned. He introduced a notation of his own, but
\index{Notation!differential calculus}%
it was inconvenient, and was abandoned by him in the second
edition of his \textit{Mécanique}, in which he used infinitesimals. The
primary object of the \textit{Théorie des fonctions} was not attained,
but its secondary results were far-reaching. It was a purely
abstract mode of regarding functions, apart from geometrical
\index{Functions!theory of}%
or mechanical considerations. In the further development
of higher analysis a function became the leading idea, and
Lagrange's work may be regarded as the starting-point of the
theory of functions as developed by Cauchy, Riemann, Weierstrass,
\index{Theory of functions}%
and others.

In the treatment of infinite series Lagrange displayed in
his earlier writings that laxity common to all mathematicians
of his time, excepting Nicolaus Bernoulli~II. and D'Alembert.
\index{Bernoulli, Nicolaus (born 1687)}%
But his later articles mark the beginning of a period of greater
rigour. Thus, in the \textit{Calcul de fonctions} he gives his theorem
on the limits of Taylor's theorem. Lagrange's mathematical
\index{Taylor's theorem}%
researches extended to subjects which have not been mentioned
%% -----File: 284.png---Folio 270-------
\index{Acoustics}%
\index{Probability}%
here---such as probabilities, finite differences, ascending
\index{Finite differences}%
continued fractions, elliptic integrals. Everywhere his
\index{Continued fractions}%
\index{Fractions!continued}%
wonderful powers of generalisation and abstraction are made
manifest. In that respect he stood without a peer, but
his great contemporary, Laplace, surpassed him in practical
\index{Laplace|(}%
sagacity. Lagrange was content to leave the application of
his general results to others, and some of the most important
researches of Laplace (particularly those on the velocity of
sound and on the secular acceleration of the moon) are implicitly
\index{Sound, velocity of}%
contained in Lagrange's works.

Lagrange was an extremely modest man, eager to avoid
\index{Lagrange|)}%
controversy, and even timid in conversation. He spoke in
tones of doubt, and his first words generally were, ``Je ne
sais pas.'' He would never allow his portrait to be taken,
and the only ones that were secured were sketched without
his knowledge by persons attending the meetings of the
Institute.

\textbf{Pierre Simon Laplace} (1749--1827) was born at Beau\-mont-en-Auge
in Normandy. Very little is known of his early
life. When at the height of his fame he was loath to speak
of his boyhood, spent in poverty. His father was a small
farmer. Some rich neighbours who recognised the boy's
talent assisted him in securing an education. As an extern
he attended the military school in Beaumont, where at an
early age he became teacher of mathematics. At eighteen
he went to Paris, armed with letters of recommendation to
D'Alembert, who was then at the height of his fame. The
letters remained unnoticed, but young Laplace, undaunted,
wrote the great geometer a letter on the principles of mechanics,
which brought the following enthusiastic response:
``You needed no introduction; you have recommended yourself;
my support is your due.'' D'Alembert secured him a
\index{D'Alembert!ref.\ to|)}%
position at the \textit{École Militaire} of Paris as professor of mathematics.
%% -----File: 285.png---Folio 271-------
\index{Circle!degrees of}%
His future was now assured, and he entered upon
those profound researches which brought him the title of ``the
Newton of France.'' With wonderful mastery of analysis,
Laplace attacked the pending problems in the application
of the law of gravitation to celestial motions. During the
\index{Gravitation, theory of}%
succeeding fifteen years appeared most of his original contributions
to astronomy. His career was one of almost uninterrupted
\index{Astronomy!more recent researches|(}%
prosperity. In 1784 he succeeded Bézout as examiner
to the royal artillery, and the following year he became member
of the Academy of Sciences. He was made president of
the Bureau of Longitude; he aided in the introduction of the
decimal system, and taught, with Lagrange, mathematics in
the \textit{École Normale}. When, during the Revolution, there arose
a cry for the reform of everything, even of the calendar,
\index{Calendar}%
Laplace suggested the adoption of an era beginning with the
year~1250, when, according to his calculation, the major axis
of the earth's orbit had been perpendicular to the equinoctial
line. The year was to begin with the vernal equinox, and the
zero meridian was to be located east of Paris by $185.30$~degrees
of the centesimal division of the quadrant, for by this meridian
\index{Division of the circle}%
the beginning of his proposed era fell at midnight. But the
revolutionists rejected this scheme, and made the start of the
new era coincide with the beginning of the glorious French
Republic.\Reference{50}

Laplace was justly admired throughout Europe as a most
sagacious and profound scientist, but, unhappily for his reputation,
he strove not only after greatness in science, but also
after political honours. The political career of this eminent
scientist was stained by servility and suppleness. After the
18th of Brumaire, the day when Napoleon was made emperor,
Laplace's ardour for republican principles suddenly gave way
to a great devotion to the emperor. Napoleon rewarded this
devotion by giving him the post of minister of the interior,
%% -----File: 286.png---Folio 272-------
but dismissed him after six months for incapacity. Said
Napoleon, "Laplace ne saisissait aucune question sous son
véritable point de vue; il cherchait des subtilités partout,
n'avait que des idées problematiques, et portait enfin l'esprit
des infiniment petits jusque dans l'administration." Desirous
to retain his allegiance, Napoleon elevated him to the Senate
and bestowed various other honours upon him. Nevertheless,
he cheerfully gave his voice in 1814 to the dethronement of
his patron and hastened to tender his services to the Bourbons,
thereby earning the title of marquis. This pettiness of his
character is seen in his writings. The first edition of the
\textit{Système du monde} was dedicated to the Council of Five Hundred.
To the third volume of the \textit{Mécanique Céleste} is prefixed
a note that of all the truths contained in the book, that most
precious to the author was the declaration he thus made of
gratitude and devotion to the peace-maker of Europe. After
this outburst of affection, we are surprised to find in the editions
of the \textit{Théorie analytique des probabilités}, which appeared after
the Restoration, that the original dedication to the emperor is
suppressed.

Though supple and servile in politics, it must be said that
in religion and science Laplace never misrepresented or concealed
his own convictions however distasteful they might be
to others. In mathematics and astronomy his genius shines
with a lustre excelled by few. Three great works did he give
to the scientific world,---the \textit{Mécanique Céleste}, the \textit{Exposition
du système du monde}, and the \textit{Théorie analytique des \DPtypo{probabilities}{probabilités}}.
Besides these he contributed important memoirs to the
French Academy.

We first pass in brief review his astronomical researches.
In~1773 he brought out a paper in which he proved that the
mean motions or mean distances of planets are invariable or
merely subject to small periodic changes. This was the first
%% -----File: 287.png---Folio 273-------
and most important step in establishing the stability of the
solar system.\Reference{51} To Newton and also to Euler it had seemed
\index{Euler!ref.\ to}%
doubtful whether forces so numerous, so variable in position,
so different in intensity, as those in the solar system, could be
capable of maintaining permanently a condition of equilibrium.
Newton was of the opinion that a powerful hand must intervene
from time to time to repair the derangements occasioned
by the mutual action of the different bodies. This
paper was the beginning of a series of profound researches by
Lagrange and Laplace on the limits of variation of the various
\index{Lagrange!ref.\ to}%
elements of planetary orbits, in which the two great mathematicians
alternately surpassed and supplemented each other.
Laplace's first paper really grew out of researches on the
theory of Jupiter and Saturn. The behaviour of these planets
had been studied by Euler and Lagrange without receiving
satisfactory explanation. Observation revealed the existence
of a steady acceleration of the mean motions of our moon and
of Jupiter and an equally strange diminution of the mean motion
of Saturn. It looked as though Saturn might eventually
leave the planetary system, while Jupiter would fall into the
sun, and the moon upon the earth. Laplace finally succeeded
in showing, in a paper of 1784--1786, that these variations
(called the ``great inequality'') belonged to the class of ordinary
periodic perturbations, depending upon the law of attraction.
\index{Perturbations}%
The cause of so influential a perturbation was found in
the commensurability of the mean motion of the two planets.

In the study of the Jovian system, Laplace was enabled to
determine the masses of the moons. He also discovered certain
very remarkable, simple relations between the movements
of those bodies, known as ``Laws of Laplace.'' His theory of
\index{Laws of Laplace}%
these bodies was completed in papers of 1788 and~1789.
These, as well as the other papers here mentioned, were published
in the \textit{\DPtypo{Mémoirs}{Mémoires} présentés par divers savans}.
The year
%% -----File: 288.png---Folio 274-------
\index{Nebular hypothesis}%
1787 was made memorable by Laplace's announcement that
the lunar acceleration depended upon the secular changes in
the eccentricity of the earth's orbit. This removed all doubt
then existing as to the stability of the solar system. The universal
validity of the law of gravitation to explain all motion
in the solar system was established. That system, as then
known, was at last found to be a complete machine.

In 1796 Laplace published his \textit{Exposition du système du
monde}, a non-mathematical popular treatise on astronomy,
\index{Astronomy!more recent researches|)}%
ending with a sketch of the history of the science. In this
work he enunciates for the first time his celebrated nebular
hypothesis. A similar theory had been previously proposed
by Kant in~1755, and by Swedenborg; but Laplace does not
\index{Kant}%
\index{Swedenborg}%
appear to have been aware of this.

Laplace conceived the idea of writing a work which should
contain a complete analytical solution of the mechanical problem
presented by the solar system, without deriving from
observation any but indispensable data. The result was the
\textit{Mécanique Céleste}, which is a systematic presentation embracing
all the discoveries of Newton, Clairaut, D'Alembert, Euler,
Lagrange, and of Laplace himself, on celestial mechanics.
\index{Mechanics!Laplace}%
The first and second volumes of this work were published in~1799;
the third appeared in~1802, the fourth in~1805. Of the
fifth volume, Books XI.~and~XII. were published in~1823;
Books XIII.,~XIV.,~XV. in~1824, and Book~XVI. in~1825. The
first two volumes contain the general theory of the motions
and figure of celestial bodies. The third and fourth volumes
give special theories of celestial motions,---treating particularly
of motions of comets, of our moon, and of other satellites.
The fifth volume opens with a brief history of celestial
mechanics, and then gives in appendices the results of the
author's later researches. The \textit{Mécanique Céleste} was such a
master-piece, and so complete, that Laplace's successors have
%% -----File: 289.png---Folio 275-------
been able to add comparatively little. The general part of
the work was translated into German by Joh.\ Karl Burkhardt,
\index{Burkhardt, J.~K.}%
and appeared in Berlin, 1800--1802. Nathaniel Bowditch
\index{Bowditch}%
brought out an edition in English, with an extensive commentary,
in Boston, 1829--1839. The \textit{Mécanique Céleste} is not
easy reading. The difficulties lie, as a rule, not so much in the
subject itself as in the want of verbal explanation. A complicated
chain of reasoning receives often no explanation whatever.
Biot, who assisted Laplace in revising the work for
\index{Biot}%
the press, tells that he once asked Laplace some explanation
of a passage in the book which had been written not long
before, and that Laplace spent an hour endeavouring to recover
the reasoning which had been carelessly suppressed with the
remark, ``Il est facile de voir.'' Notwithstanding the important
researches in the work, which are due to Laplace himself,
it naturally contains a great deal that is drawn from his predecessors.
It is, in fact, the organised result of a century of
patient toil. But Laplace frequently neglects to properly
acknowledge the source from which he draws, and lets the
reader infer that theorems and formulæ due to a predecessor
are really his own.

We are told that when Laplace presented Napoleon with a
copy of the \textit{Mécanique Céleste}, the latter made the remark,
``M.~Laplace, they tell me you have written this large book on
the system of the universe, and have never even mentioned
its Creator.'' Laplace is said to have replied bluntly, ``Je
n'avais pas besoin de cette hypothèse-la.'' This assertion,
taken literally, is impious, but may it not have been intended
to convey a meaning somewhat different from its literal one?
Newton was not able to explain by his law of gravitation all
\index{Gravitation, theory of}%
questions arising in the mechanics of the heavens. Thus,
being unable to show that the solar system was stable, and
suspecting in fact that it was unstable, Newton expressed the
%% -----File: 290.png---Folio 276-------
opinion that the special intervention, from time to time, of a
powerful hand was necessary to preserve order. Now Laplace
was able to prove by the law of gravitation that the solar
system is stable, and in that sense may be said to have felt
no necessity for reference to the Almighty.

We now proceed to researches which belong more properly
to pure mathematics. Of these the most conspicuous are on
the theory of probability. Laplace has done more towards
\index{Probability}%
advancing this subject than any one other investigator. He
published a series of papers, the main results of which were
collected in his \textit{Théorie analytique des probabilités}, 1812. The
third edition (1820) consists of an introduction and two books.
The introduction was published separately under the title,
\textit{Essai philosophique sur les probabilités}, and is an admirable
and masterly exposition without the aid of analytical formulæ
of the principles and applications of the science. The first
book contains the theory of generating functions, which are
applied, in the second book, to the theory of probability.
Laplace gives in his work on probability his method of
approximation to the values of definite integrals. The solution
of linear differential equations was reduced by him to
definite integrals. One of the most important parts of the
work is the application of probability to the method of least
squares, which is shown to give the most probable as well as
the most convenient results.

The first printed statement of the principle of least squares
\index{Least squares}%
was made in 1806 by Legendre, without demonstration. Gauss
\index{Gauss!ref.\ to}%
\index{Legendre!ref.\ to}%
had used it still earlier, but did not publish it until 1809.
The first deduction of the law of probability of error that
appeared in print was given in 1808 by Robert Adrain in the
\index{Adrain}%
\textit{Analyst}, a journal published by himself in Philadelphia.\Reference{2}
Proofs of this law have since been given by Gauss, Ivory,
\index{Ivory!ref.\ to}%
Herschel, Hagen, and others; but all proofs contain some
\index{Hagen}%
\index{Herschel, J.~F.~W.!ref.\ to}%
%% -----File: 291.png---Folio 277-------
point of difficulty. Laplace's proof is perhaps the most satisfactory.

Laplace's work on probability is very difficult reading, particularly
the part on the method of least squares. The
analytical processes are by no means clearly established or
free from error. ``No one was more sure of giving the
result of analytical processes correctly, and no one ever took
so little care to point out the various small considerations on
which correctness depends'' (De~Morgan).
\index{Demorgan@{De Morgan}!ref.\ to}%

Of Laplace's papers on the attraction of ellipsoids, the most
\index{Attraction}%
important is the one published in 1785, and to a great extent
reprinted in the third volume of the \textit{Mécanique Céleste}. It
gives an exhaustive treatment of the general problem of
attraction of any ellipsoid upon a particle situated outside
\index{Ellipsoid!(attraction of)}%
or upon its surface. Spherical harmonics, or the so-called
``Laplace's coefficients,'' constitute a powerful analytic engine
\index{Laplace's coefficients}%
in the theory of attraction, in electricity, and magnetism.
The theory of spherical harmonics for two dimensions had
been previously given by Legendre. Laplace failed to make
due acknowledgment of this, and there existed, in consequence,
between the two great men, ``a feeling more
than coldness.'' The potential function,~$V$, is much used by
\index{Potential}%
Laplace, and is shown by him to satisfy the partial differential
equation $\dfrac{\partial^2V}{\partial x^2} + \dfrac{\partial^2V}{\partial y^2} + \dfrac{\partial^2V}{\partial z^2} = 0$. This is known as Laplace's
equation, and was first given by him in the more complicated
form which it assumes in polar co-ordinates. The notion
of potential was, however, not introduced into analysis by
Laplace. The honour of that achievement belongs to Lagrange.\Reference{49}
\index{Lagrange!ref.\ to}%

Among the minor discoveries of Laplace are his method of
solving equations of the second, third, and fourth degrees,
\index{Equations!solution of}%
his memoir on singular solutions of differential equations, his
\index{Singular solutions}%
%% -----File: 292.png---Folio 278-------
\index{Acoustics}%
\index{Differential equations}%
researches in finite differences and in determinants, the establishment
\index{Determinants}%
\index{Finite differences}%
of the expansion theorem in determinants which had
been previously given by Vandermonde for a special case, the
\index{Vandermonde}%
\index{Vandermonde!ref.\ to}%
determination of the complete integral of the linear differential
equation of the second order. In the \textit{Mécanique Céleste} he
made a generalisation of Lagrange's theorem on the development
\index{Lagrange!ref.\ to}%
of functions in series known as Laplace's theorem.

Laplace's investigations in physics were quite extensive.
We mention here his correction of Newton's formula on the
velocity of sound in gases by taking into account the changes
\index{Sound, velocity of}%
of elasticity due to the heat of compression and cold of rarefaction;
\index{Elasticity}%
his researches on the theory of tides; his mathematical
\index{Tides}%
theory of capillarity; his explanation of astronomical refraction;
\index{Capillarity}%
his formulæ for measuring heights by the barometer.

Laplace's writings stand out in bold contrast to those of
Lagrange in their lack of elegance and symmetry. Laplace
\index{Laplace|)}%
looked upon mathematics as the tool for the solution of physical
problems. The true result being once reached, he spent
little time in explaining the various steps of his analysis, or
in polishing his work. The last years of his life were spent
mostly at Arcueil in peaceful retirement on a country-place,
where he pursued his studies with his usual vigour until his
death. He was a great admirer of Euler, and would often
\index{Euler!ref.\ to}%
say, ``Lisez Euler, lisez Euler, c'est notre maître à tous.''

\textbf{Abnit-Théophile Vandermonde} (1735--1796) studied music
during his youth in Paris and advocated the theory that all
art rested upon one general law, through which any one could
become a composer with the aid of mathematics. He was the
first to give a connected and logical exposition of the theory
of determinants, and may, therefore, almost be regarded as
the founder of that theory. He and Lagrange originated the
method of combinations in solving equations.\Reference{20}

\textbf{Adrien Marie Legendre} (1752--1833) was educated at the
\index{Legendre|(}%
%% -----File: 293.png---Folio 279-------
\index{Elliptic functions}%
Collège Mazarin in Paris, where he began the study of mathematics
under Abbé Marie. His mathematical genius secured
\index{Marie, Abbé}%
for him the position of professor of mathematics at the military
school of Paris. While there he prepared an essay on
the curve described by projectiles thrown into resisting media
(ballistic curve), which captured a prize offered by the Royal
\index{Ballistic curve}%
Academy of Berlin. In 1780 he resigned his position in order
to reserve more time for the study of higher mathematics.
He was then made member of several public commissions.
In 1795 he was elected professor at the Normal School and
later was appointed to some minor government positions.
Owing to his timidity and to Laplace's unfriendliness toward
\index{Laplace!ref.\ to}%
him, but few important public offices commensurate with his
ability were tendered to him.

As an analyst, second only to Laplace and Lagrange, Legendre
\index{Lagrange!ref.\ to}%
enriched mathematics by important contributions, mainly
on elliptic integrals, theory of numbers, attraction of ellipsoids,
and least squares. The most important of Legendre's
works is his \textit{Fonctions elliptiques}, issued in two volumes in
1825 and~1826. He took up the subject where Euler, Landen,
\index{Euler!ref.\ to}%
\index{Landen!ref.\ to}%
and Lagrange had left it, and for forty years was the only one
to cultivate this new branch of analysis, until at last Jacobi and
\index{Jacobi!ref.\ to}%
Abel stepped in with admirable new discoveries.\Reference{52} Legendre
\index{Abel!ref.\ to}%
imparted to the subject that connection and arrangement
which belongs to an independent science. Starting with an
integral depending upon the square root of a polynomial of
the fourth degree in~$x$, he showed that such integrals can be
brought back to three canonical forms, designated by $F(\phi)$,
$E(\phi)$, and~$\Pi(\phi)$, the radical being expressed in the form
$\Delta(\phi) = \sqrt{1 - k^2 \sin^2\phi}$. He also undertook the prodigious task
of calculating tables of arcs of the ellipse for different degrees
of amplitude and eccentricity, which supply the means of
integrating a large number of differentials.
%% -----File: 294.png---Folio 280-------
\index{Ellipsoid!(attraction of)}%
\index{Eulerian integrals}%
\index{Legendre's function}%
\index{Spherical trigonometry}%
\index{Trigonometry!spherical}%

An earlier publication which contained part of his researches
on elliptic functions was his \textit{Calcul intégral} in three volumes
\index{Elliptic functions}%
(1811, 1816, 1817), in which he treats also at length of the
two classes of definite integrals named by him \emph{Eulerian}. He
tabulated the values of $\log \Gamma(p)$ for values of~$p$ between
$1$~and~$2$.

One of the earliest subjects of research was the attraction
of spheroids, which suggested to Legendre the function~$P_n$,
named after him. His memoir was presented to the Academy
of Sciences in 1783. The researches of Maclaurin and Lagrange
\index{Lagrange!ref.\ to}%
\index{Maclaurin!ref.\ to}%
suppose the point attracted by a spheroid to be at the surface
or within the spheroid, but Legendre showed that in order to
determine the attraction of a spheroid on any external point
it suffices to cause the surface of another spheroid described
upon the same foci to pass through that point. Other memoirs
on ellipsoids appeared later.

The two household gods to which Legendre sacrificed with
ever-renewed pleasure in the silence of his closet were the
elliptic functions and the theory of numbers. His researches
\index{Numbers!theory of numbers}%
\index{Theory of numbers}%
on the latter subject, together with the numerous scattered
fragments on the theory of numbers due to his predecessors
in this line, were arranged as far as possible into a systematic
whole, and published in two large quarto volumes, entitled
\textit{Théorie des nombres}, 1830. Before the publication of this
work Legendre had issued at divers times preliminary articles.
Its crowning pinnacle is the theorem of quadratic reciprocity,
\index{Quadratic reciprocity}%
previously indistinctly given by Euler without proof, but for the
\index{Euler!ref.\ to}%
first time clearly enunciated and partly proved by Legendre.\Reference{48}

While acting as one of the commissioners to connect Greenwich
and Paris geodetically, Legendre calculated all the triangles
in France. This furnished the occasion of establishing
formulæ and theorems on geodesics, on the treatment of the
spherical triangle as if it were a plane triangle, by applying
%% -----File: 295.png---Folio 281-------
\index{Axioms (of geometry)}%
\index{Least squares}%
certain corrections to the angles, and on the method of least
squares, published for the first time by him without demonstration
in~1806.

Legendre wrote an \textit{Éléments de Géométrie}, 1794, which
enjoyed great popularity, being generally adopted on the
Continent and in the United States as a substitute for Euclid.
\index{Euclid!ref.\ to}%
This great modern rival of Euclid passed through numerous
editions; the later ones containing the elements of trigonometry
and a proof of the irrationality of $\pi$ and~$\pi^2$. Much
\index{p@{$\pi$}: values for!proved to be irrational}%
attention was given by Legendre to the subject of parallel
lines. In the earlier editions of the \textit{Éléments}, he made direct
appeal to the senses for the correctness of the ``parallel-axiom.''
He then attempted to demonstrate that ``axiom,'' but his
proofs did not satisfy even himself. In Vol.~XII. of the
Memoirs of the Institute is a paper by Legendre, containing
\index{Legendre|)}%
his last attempt at a solution of the problem. Assuming
space to be infinite, he proved satisfactorily that it is impossible
for the sum of the three angles of a triangle to exceed two
right angles; and that if there be any triangle the sum of
whose angles is two right angles, then the same must be true
of all triangles. But in the next step, to show that this sum
cannot be less than two right angles, his demonstration necessarily
failed. If it could be granted that the sum of the three
angles is always equal to two right angles, then the theory of
parallels could be strictly deduced.
\index{Parallels}%

\textbf{Joseph Fourier} (1768--1830) was born at Auxerre, in central
\index{Fourier|(}%
France. He became an orphan in his eighth year. Through
the influence of friends he was admitted into the military
school in his native place, then conducted by the Benedictines
of the Convent of St.~Mark. He there prosecuted his studies,
particularly mathematics, with surprising success. He wished
to enter the artillery, but, being of low birth (the son of a
tailor), his application was answered thus: ``Fourier, not
%% -----File: 296.png---Folio 282-------
being noble, could not enter the artillery, although he were
a second Newton.''\Reference{53} He was soon appointed to the mathematical
\index{Newton!ref.\ to}%
chair in the military school. At the age of twenty-one
he went to Paris to read before the Academy of Sciences
a memoir on the resolution of numerical equations, which
\index{Equations!numerical}%
was an improvement on Newton's method of approximation.
This investigation of his early youth he never lost sight of.
He lectured upon it in the Polytechnic School; he developed
it on the banks of the Nile; it constituted a part of a work
entitled \textit{Analyse des equationes determines} (1831), which was
in press when death overtook him. This work contained
``Fourier's theorem'' on the number of real roots between
\index{Fourier's theorem}%
two chosen limits. Budan had published this result as early
\index{Budan}%
as 1807, but there is evidence to show that Fourier had established
it before Budan's publication. These brilliant results
were eclipsed by the theorem of Sturm, published in~1835.
\index{Sturm, J.~C.~F.!ref.\ to}%

Fourier took a prominent part at his home in promoting
the Revolution. Under the French Revolution the arts and
sciences seemed for a time to flourish. The reformation of
the weights and measures was planned with grandeur of conception.
The Normal School was created in 1795, of which
Fourier became at first pupil, then lecturer. His brilliant
success secured him a chair in the Polytechnic School, the
duties of which he afterwards quitted, along with Monge and
\index{Monge!ref.\ to}%
Berthollet, to accompany Napoleon on his campaign to Egypt.
Napoleon founded the Institute of Egypt, of which Fourier
became secretary. In Egypt he engaged not only in scientific
work, but discharged important political functions. After
his return to France he held for fourteen years the prefecture
of Grenoble. During this period he carried on his elaborate
investigations on the propagation of heat in solid bodies,
published in 1822 in his work entitled \textit{La Theorie Analytique
de la Chaleur}. This work marks an epoch in the history of
%% -----File: 297.png---Folio 283-------
\index{Functions!arbitrary}%
\index{Infinite series}%
\index{Kelvin, Lord!ref.\ to}%
mathematical physics. ``Fourier's series'' constitutes its
\index{Fourier's series}%
gem. By this research a long controversy was brought to a
close, and the fact established that any arbitrary function
can be represented by a trigonometric series. The first
\index{Trigonometric series}%
announcement of this great discovery was made by Fourier
in 1807, before the French Academy. The trigonometric
series $\smash[b]{\sum\limits_{n=0}^{n=\infty} (a_n \sin nx + b_n \cos nx)}$ represents the function~$\phi(x)$
for every value of~$x$, if the coefficients $a_n = \displaystyle\frac{1}{\pi} \int_{-\pi}^\pi \phi(x) \sin nx\, dx$,
and $b_n$ be equal to a similar integral. The weak point in
Fourier's analysis lies in his failure to prove generally that
the trigonometric series actually converges to the value of
the function. In 1827 Fourier succeeded Laplace as president
of the council of the Polytechnic School.

Before proceeding to the origin of modern geometry we shall
speak briefly of the introduction of higher analysis into Great
Britain. This took place during the first quarter of this century.
The British began to deplore the very small progress
that science was making in England as compared with its
racing progress on the Continent. In 1813 the ``Analytical
\index{Analytical Society (in Cambridge)}%
Society'' was formed at Cambridge. This was a small club
established by George Peacock, John Herschel, Charles Babbage,
\index{Babbage}%
\index{Herschel, J.~F.~W.!ref.\ to}%
\index{Peacock!ref.\ to, X}%
and a few other Cambridge students, to promote, as it
was humorously expressed, the principles of pure ``$D$-ism,''
that is, the Leibnizian notation in the calculus against those
\index{Notation!differential calculus}%
of ``dot-age,'' or of the Newtonian notation. This struggle
ended in the introduction into Cambridge of the notation
$\dfrac{dy}{dx}$, to the exclusion of the fluxional notation~$\dot{y}$. This
was a great step in advance, not on account of any great
superiority of the Leibnizian over the Newtonian notation,
but because the adoption of the former opened up to English
students the vast storehouses of continental discoveries. Sir
William Thomson, Tait, and some other modern writers find
\index{Tait}%
\index{Thomson, Sir William!ref.\ to}%
%% -----File: 298.png---Folio 284-------
\index{Arithmetical machine}%
\index{Calculating machines}%
\index{Fourier|)}%
\index{Machine, arithmetical}%
it frequently convenient to use both notations. Herschel,
\index{Herschel, J.~F.~W.!ref.\ to}%
Peacock, and Babbage translated, in 1816, from the French,
\index{Peacock}%
Lacroix's treatise on the differential and integral calculus, and
\index{Lacroix}%
added in 1820 two volumes of examples. Lacroix's was one
of the best and most extensive works on the calculus of that
time. Of the three founders of the ``Analytical Society,''
Peacock afterwards did most work in pure mathematics.
Babbage became famous for his invention of a calculating
engine superior to Pascal's. It was never finished, owing
\index{Pascal!ref.\ to}%
to a misunderstanding with the government, and a consequent
failure to secure funds. John Herschel, the eminent
astronomer, displayed his mastery over higher analysis in
memoirs communicated to the Royal Society on new applications
of mathematical analysis, and in articles contributed
to cyclopædias on light, on meteorology, and on the history
of mathematics.

\textbf{George Peacock} (1791--1858) was educated at Trinity College,
Cambridge, became Lowndean professor there, and later, dean
of Ely. His chief publications are his \textit{Algebra}, 1830 and~1842,
\index{Algebra!Peacock}%
and his \textit{Report on Recent Progress in Analysis}, which was the
first of several valuable summaries of scientific progress printed
in the volumes of the British Association. He was one of the
first to study seriously the fundamental principles of algebra,
and to fully recognise its purely symbolic character. He
advances, though somewhat imperfectly, the ``principle of the
permanence of equivalent forms.'' It assumes that the rules
applying to the symbols of arithmetical algebra apply also
in symbolical algebra. About this time D.~F. Gregory wrote
\index{Gregory, David F}%
a paper ``on the real nature of symbolical algebra,'' which
brought out clearly the commutative and distributive laws.
These laws had been noticed years before by the inventors
of symbolic methods in the calculus. It was Servois who
\index{Servois}%
introduced the names \emph{commutative} and \emph{distributive} in~1813.
%% -----File: 299.png---Folio 285-------
Peacock's investigations on the foundation of algebra were
considerably advanced by De~Morgan and Hankel.
\index{Demorgan@{De Morgan}!ref.\ to}%
\index{Hankel!ref.\ to}%

\textbf{James Ivory} (1765--1842) was a Scotch mathematician who
\index{Ivory}%
for twelve years, beginning in 1804, held the mathematical
chair in the Royal Military College at Marlow (now at Sandhurst).
He was essentially a self-trained mathematician, and
almost the only one in Great Britain previous to the organisation
of the Analytical Society who was well versed in continental
mathematics. Of importance is his memoir (\textit{Phil.\
Trans.}, 1809) in which the problem of the attraction of a
homogeneous ellipsoid upon an external point is reduced to
\index{Ellipsoid!(attraction of)}%
the simpler problem of the attraction of a related ellipsoid
upon a corresponding point interior to it. This is known as
``Ivory's theorem.'' He criticised with undue severity Laplace's
\index{Ivory's theorem}%
\index{Laplace!ref.\ to}%
solution of the method of least squares, and gave three proofs
\index{Least squares}%
of the principle without recourse to probability; but they are
\index{Probability}%
far from being satisfactory.

\Subsection{The Origin of Modern Geometry}
\index{Geometry!modern synthetic|(}%

By the researches of Descartes and the invention of the calculus,
the analytical treatment of geometry was brought into
great prominence for over a century. Notwithstanding the
efforts to revive synthetic methods made by Desargues, Pascal,
\index{Desargues!ref.\ to}%
\index{Pascal!ref.\ to}%
De~Lahire, Newton, and Maclaurin, the analytical method
\index{Delahire@{De Lahire}}%
\index{Maclaurin!ref.\ to}%
\index{Newton!ref.\ to}%
retained almost undisputed supremacy. It was reserved for
the genius of Monge to bring synthetic geometry in the
foreground, and to open up new avenues of progress. His
\textit{Géométrie descriptive} marks the beginning of a wonderful
development of modern geometry.

Of the two leading problems of descriptive geometry, the
one---to represent by drawings geometrical magnitudes---was
brought to a high degree of perfection before the time of
%% -----File: 300.png---Folio 286-------
Monge; the other---to solve problems on figures in space
\index{Monge|(}%
by constructions in a plane---had received considerable attention
before his time. His most noteworthy predecessor
in descriptive geometry was the Frenchman Frézier (1682--1773).
\index{Descriptive geometry|(}%
\index{Frezier@{Frézier}}%
\index{Geometry!descriptive|(}%
But it remained for Monge to create descriptive
geometry as a \emph{distinct} branch of science by imparting to it
geometric generality and elegance. All problems previously
treated in a special and uncertain manner were referred
back to a few general principles. He introduced the line
of intersection of the horizontal and the vertical plane as
the axis of projection. By revolving one plane into the
other around this axis or ground-line, many advantages were
gained.\Reference{54}

\textbf{Gaspard Monge} (1746--1818) was born at Beaune. The construction
of a plan of his native town brought the boy under
the notice of a colonel of engineers, who procured for him an
appointment in the college of engineers at Mézières. Being
of low birth, he could not receive a commission in the army,
but he was permitted to enter the annex of the school, where
surveying and drawing were taught. Observing that all the
operations connected with the construction of plans of fortification
were conducted by long arithmetical processes, he substituted
a geometrical method, which the commandant at first
refused even to look at, so short was the time in which it
could be practised; when once examined, it was received with
avidity. Monge developed these methods further and thus
created his descriptive geometry. Owing to the rivalry
between the French military schools of that time, he was not
permitted to divulge his new methods to any one outside of
this institution. In 1768 he was made professor of mathematics
at Mézières. In 1780, when conversing with two of his
pupils, S.~F. Lacroix and Gayvernon in Paris, he was obliged
\index{Lacroix}%
to say, ``All that I have here done by calculation, I could have
%% -----File: 301.png---Folio 287-------
\index{Imaginary quantities}%
done with the ruler and compass, but I am not allowed to
reveal these secrets to you.'' But Lacroix set himself to
\index{Lacroix}%
examine what the secret could be, discovered the processes, and
published them in 1795. The method was published by Monge
himself in the same year, first in the form in which the short-hand
writers took down his lessons given at the Normal School,
where he had been elected professor, and then again, in revised
form, in the \textit{Journal des écoles normales}. The next edition
occurred in 1798--1799. After an ephemeral existence of only
four months the Normal School was closed in~1795. In the
same year the Polytechnic School was opened, in the establishing
of which Monge took active part. He taught there
descriptive geometry until his departure from France to accompany
\index{Geometry!analytic}%
Napoleon on the Egyptian campaign. He was the first
president of the Institute of Egypt. Monge was a zealous
partisan of Napoleon and was, for that reason, deprived of all
his honours by Louis~XVIII\@. This and the destruction of the
Polytechnic School preyed heavily upon his mind. He did
not long survive this insult.

Monge's numerous papers were by no means confined to descriptive
geometry. His analytical discoveries are hardly less
remarkable. He introduced into analytic geometry the methodic
\index{Analytic geometry}%
use of the equation of a line. He made important
contributions to surfaces of the second degree (previously
\index{Surfaces, theory of}%
studied by Wren and Euler) and discovered between the
\index{Euler!ref.\ to}%
\index{Wren!ref.\ to}%
theory of surfaces and the integration of partial differential
\index{Partial differential equations}%
equations, a hidden relation which threw new light upon both
subjects. He gave the differential of curves of curvature,
established a general theory of curvature, and applied it to the
ellipsoid. He found that the validity of solutions was not
impaired when imaginaries are involved among subsidiary
quantities. Monge published the following books: \textit{Statics},
1786; \textit{Applications de l'algèbre à la géométrie}, 1805; \textit{Application
%% -----File: 302.png---Folio 288-------
\index{Brianchion}%
\index{Carnot, Lazare}%
de l'analyse à la géométrie}. The last two contain most of
his miscellaneous papers.

Monge was an inspiring teacher, and he gathered around
him a large circle of pupils, among which were Dupin, Servois,
\index{Dupin}%
\index{Servois}%
Brianchon, Hachette, Biot, and Poncelet.
\index{Biot}%
\index{Hachette}%
\index{Poncelet!ref.\ to}%

\textbf{Charles Dupin} (1784--1873), for many years professor of
mechanics in the Conservatoire des Arts et Métiers in Paris,
published in 1813 an important work on \textit{Développements de
géométrie}, in which is introduced the conception of conjugate
tangents of a point of a surface, and of the indicatrix.\Reference{53} It
contains also the theorem known as ``Dupin's theorem.''
Surfaces of the second degree and descriptive geometry were
successfully studied by \textit{Jean Nicolas Pierre Hachette} (1769--1834),
who became professor of descriptive geometry at the
Polytechnic School after the departure of Monge for Rome and
Egypt. In 1822 he published his \textit{Traité de géométrie descriptive}.

Descriptive geometry, which arose, as we have seen, in
technical schools in France, was transferred to Germany at
the foundation of technical schools there. G.~Schreiber,
\index{Schreiber}%
professor in Karlsruhe, was the first to spread Monge's
\index{Monge|)}%
geometry in Germany by the publication of a work thereon
in 1828--1829.\Reference{54} In the United States descriptive geometry was
\index{Descriptive geometry|)}%
\index{Geometry!descriptive|)}%
introduced in 1816 at the Military Academy in West Point
by Claude Crozet, once a pupil at the Polytechnic School in
\index{Crozet}%
Paris. Crozet wrote the first English work on the subject.\Reference{2}

\textbf{Lazare Nicholas Marguerite Carnot} (1753--1823) was born at
Nolay in Burgundy, and educated in his native province.
He entered the army, but continued his mathematical studies,
and wrote in 1784 a work on machines, containing the earliest
proof that kinetic energy is lost in collisions of bodies. With
the advent of the Revolution he threw himself into politics,
and when coalesced Europe, in 1793, launched against France
a million soldiers, the gigantic task of organising fourteen
%% -----File: 303.png---Folio 289-------
\index{Brianchion}%
\index{Carnot, Lazare}%
\index{Differential calculus!philosophy of}%
\index{Dupin}%
armies to meet the enemy was achieved by him. He was
banished in 1796 for opposing Napoleon's \textit{coup d'état}. The
refugee went to Geneva, where he issued, in 1797, a work
still frequently quoted, entitled, \textit{Réflexions sur la Métaphysique
du Calcul Infinitésimal}. He declared himself as an ``irreconcilable
enemy of kings.'' After the Russian campaign he
offered to fight for France, though not for the empire. On
the restoration he was exiled. He died in Magdeburg. His
\textit{Géométrie de position}, 1803, and his \textit{Essay on Transversals},
1806, are important contributions to modern geometry. While
Monge revelled mainly in three-dimensional geometry, Carnot
confined himself to that of two. By his effort to explain
the meaning of the negative sign in geometry he established
a ``geometry of position,'' which, however, is different from
the ``Geometrie der Lage'' of to-day. He invented a class
of general theorems on projective properties of figures, which
have since been pushed to great extent by Poncelet, Chasles,
\index{Chasles!ref.\ to}%
\index{Poncelet}%
and others.

\textbf{Jean Victor Poncelet} (1788--1867), a native of Metz, took
part in the Russian campaign, was abandoned as dead on the
bloody field of Krasnoi, and taken prisoner to Saratoff. Deprived
there of all books, and reduced to the remembrance
of what he had learned at the Lyceum at Metz and the Polytechnic
School, where he had studied with predilection the
works of Monge, Carnot, and Brianchon, he began to study
mathematics from its elements. He entered upon original
researches which afterwards made him illustrious. While
in prison he did for mathematics what Bunyan did for
literature,---produced a much-read work, which has remained
of great value down to the present time. He returned to
France in 1814, and in 1822 published the work in question,
entitled, \textit{Traité des Propriétés projectives des figures}. In it
he investigated the properties of figures which remain unaltered
%% -----File: 304.png---Folio 290-------
by projection of the figures. The projection is not
effected here by parallel rays of prescribed direction, as with
Monge, but by central projection. Thus perspective projection,
used before him by Desargues, Pascal, Newton, and Lambert,
\index{Desargues!ref.\ to}%
\index{Lambert!ref.\ to}%
\index{Newton!ref.\ to}%
\index{Pascal!ref.\ to}%
was elevated by him into a fruitful geometric method.
In the same way he elaborated some ideas of De~Lahire,
\index{Delahire@{De Lahire}}%
Servois, and Gergonne into a regular method---the method
\index{Gergonne!ref.\ to}%
\index{Servois}%
of ``reciprocal polars.'' To him we owe the Law of Duality
\index{Duality}%
\index{Reciprocal polars}%
as a consequence of reciprocal polars. As an independent
principle it is due to Gergonne. Poncelet wrote much on
\index{Poncelet}%
applied mechanics. In 1838 the Faculty of Sciences was
\index{Mechanics!more recent work}%
enlarged by his election to the chair of mechanics.

While in France the school of Monge was creating modern
geometry, efforts were made in England to revive Greek
geometry by \textbf{Robert Simson} (1687--1768) and \textbf{Matthew Stewart}
\index{Simson}%
\index{Stewart}%
(1717--1785). Stewart was a pupil of Simson and Maclaurin,
\index{Maclaurin!ref.\ to}%
and succeeded the latter in the chair at Edinburgh. During
the eighteenth century he and Maclaurin were the only prominent
mathematicians in Great Britain. His genius was ill-directed
by the fashion then prevalent in England to ignore
higher analysis. In his \textit{Four Tracts, Physical and Mathematical},
1761, he applied geometry to the solution of difficult
\index{Geometry!modern synthetic|)}%
astronomical problems, which on the Continent were approached
analytically with greater success. He published, in
1746, \textit{General Theorems}, and in 1763, his \textit{Propositiones geometricæ
more veterum demonstratæ}. The former work contains
sixty-nine theorems, of which only five are accompanied
by demonstrations. It gives many interesting new results
on the circle and the straight line. Stewart extended some
theorems on transversals due to Giovanni Ceva (1648--1737),
\index{Ceva}%
an Italian, who published in 1678 at Mediolani a work containing
the theorem now known by his name.
%% -----File: 305.png---Folio 291-------
\index{Bolyai, Johann!ref.\ to}%
\index{Bolyai, Wolfgang!ref.\ to}%


\Chapter{Recent Times}

\First{Never} more zealously and successfully has mathematics
been cultivated than in this century. Nor has progress, as
in previous periods, been confined to one or two countries.
While the French and Swiss, who alone during the preceding
epoch carried the torch of progress, have continued to develop
mathematics with great success, from other countries whole
armies of enthusiastic workers have wheeled into the front
rank. Germany awoke from her lethargy by bringing forward
Gauss, Jacobi, Dirichlet, and hosts of more recent men;
\index{Dirichlet!ref.\ to}%
\index{Gauss!ref.\ to}%
\index{Jacobi!ref.\ to}%
Great Britain produced her De~Morgan, Boole, Hamilton,
\index{Boole!ref.\ to}%
\index{Demorgan@{De Morgan}!ref.\ to}%
\index{Hamilton, W.~R.!ref.\ to}%
besides champions who are still living; Russia entered the
arena with her Lobatchewsky; Norway with Abel; Italy with
\index{Abel!ref.\ to}%
\index{Lobatchewsky!ref.\ to}%
Cremona; Hungary with her two Bolyais; the United States
\index{Cremona!ref.\ to}%
with Benjamin Peirce.
\index{Peirce, B.!ref.\ to}%

The productiveness of modern writers has been enormous.
``It is difficult,'' says Professor Cayley,\Reference[]{56} ``to give an idea of
\index{Cayley!ref.\ to}%
the vast extent of modern mathematics. This word `extent'
is not the right one: I mean extent crowded with beautiful
detail,---not an extent of mere uniformity such as an objectless
plain, but of a tract of beautiful country seen at first in
the distance, but which will bear to be rambled through and
studied in every detail of hillside and valley, stream, rock,
wood, and flower.'' It is pleasant to the mathematician to
think that in his, as in no other science, the achievements of
%% -----File: 306.png---Folio 292-------
every age remain possessions forever; new discoveries seldom
disprove older tenets; seldom is anything lost or wasted.

If it be asked wherein the utility of some modern extensions
of mathematics lies, it must be acknowledged that it is
at present difficult to see how they are ever to become applicable
to questions of common life or physical science. But
our inability to do this should not be urged as an argument
against the pursuit of such studies. In the first place, we
know neither the day nor the hour when these abstract
developments will find application in the mechanic arts, in
physical science, or in other branches of mathematics. For
example, the whole subject of graphical statics, so useful
\index{Graphical statics}%
to the practical engineer, was made to rest upon von Staudt's
\index{Von Staudt!ref.\ to}%
\textit{Geometrie der Lage}; Hamilton's ``principle of varying action''
\index{Action!varying}%
\index{Hamilton, W.~R.!ref.\ to}%
\index{Varying action, principle of}%
has its use in astronomy; complex quantities, general integrals,
\index{Complex quantities}%
and general theorems in integration offer advantages in
the study of electricity and magnetism. ``The utility of such
researches,'' says Spottiswoode,\Reference[]{57} ``can in no case be discounted,
\index{Spottiswoode!ref.\ to}%
or even imagined beforehand. Who, for instance, would have
supposed that the calculus of forms or the theory of substitutions
\index{Calculus!of operations}%
\index{Substitutions, theory of}%
would have thrown much light upon ordinary equations;
or that Abelian functions and hyperelliptic transcendents
\index{Abelian functions}%
\index{Hyperelliptic functions}%
would have told us anything about the properties of curves;
\index{Curves!theory of}%
or that the calculus of operations would have helped us in
\index{Operations, calculus of}%
any way towards the figure of the earth?'' A second reason
\index{Earth!figure of}%
\index{Figure of the earth}%
in favour of the pursuit of advanced mathematics, even when
there is no promise of practical application, is this, that mathematics,
like poetry and music, deserves cultivation for its
own sake.

The great characteristic of modern mathematics is its generalising
tendency. Nowadays little weight is given to isolated
theorems, ``except as affording hints of an unsuspected
new sphere of thought, like meteorites detached from some
%% -----File: 307.png---Folio 293-------
\index{Invariant}%
undiscovered planetary orb of speculation.'' In mathematics,
as in all true sciences, no subject is considered in itself alone,
but always as related to, or an outgrowth of, other things.
The development of the notion of continuity plays a leading
\index{Continuity}%
part in modern research. In geometry the principle of continuity,
\index{Geometry!modern synthetic|(}%
the idea of correspondence, and the theory of projection
\index{Correspondence, principle of}%
constitute the fundamental modern notions. Continuity
asserts itself in a most striking way in relation to the circular
points at infinity in a plane. In algebra the modern idea finds
\index{Infinity}%
expression in the theory of linear transformations and invariants,
and in the recognition of the value of homogeneity and
\index{Homogeneity}%
symmetry.


\Section{Synthetic Geometry}
\index{Synthetic geometry|(}%

The conflict between geometry and analysis which arose
near the close of the last century and the beginning of the
present has now come to an end. Neither side has come
out victorious. The greatest strength is found to lie, not in
the suppression of either, but in the friendly rivalry between
the two, and in the stimulating influence of the one upon the
other. Lagrange prided himself that in his \textit{\DPtypo{Mecanique}{Mécanique} Analytique}
\index{Lagrange!ref.\ to}%
he had succeeded in avoiding all figures; but since his
time mechanics has received much help from geometry.

Modern synthetic geometry was created by several investigators
about the same time. It seemed to be the outgrowth
of a desire for general methods which should serve as threads
of Ariadne to guide the student through the labyrinth of theorems,
corollaries, porisms, and problems. Synthetic geometry
was first cultivated by Monge, Carnot, and Poncelet in France;
\index{Carnot, Lazare!ref.\ to}%
\index{Monge!ref.\ to}%
\index{Poncelet!ref.\ to}%
it then bore rich fruits at the hands of Möbius and Steiner in
\index{Mobius@{Möbius}!ref.\ to}%
\index{Steiner!ref.\ to}%
Germany and Switzerland, and was finally developed to still
%% -----File: 308.png---Folio 294-------
\index{Anharmonic ratio}%
higher perfection by Chasles in France, von Staudt in Germany,
\index{Chasles!ref.\ to}%
\index{Von Staudt!ref.\ to}%
and Cremona in Italy.
\index{Cremona!ref.\ to|(}%

\textbf{Augustus Ferdinand Möbius} (1790--1868) was a native of
\index{Mobius@{Möbius}}%
Schulpforta in Prussia. He studied at Göttingen under
Gauss, also at Leipzig and Halle. In Leipzig he became, in
\index{Gauss!ref.\ to}%
1815, privat-docent, the next year extraordinary professor of
astronomy, and in 1844 ordinary professor. This position he
held till his death. The most important of his researches
are on geometry. They appeared in \textit{Crelle's Journal}, and in
his celebrated work entitled \textit{Der Barycentrische Calcul}, Leipzig,
1827. As the name indicates, this calculus is based upon
properties of the centre of gravity.\Reference{58} Thus, that the point~$S$
is the centre of gravity of weights $a$,~$b$, $c$,~$d$ placed at the
points $A$,~$B$, $C$,~$D$ respectively, is expressed by the equation
\[
(a + b + c + d)S = aA + bB + cC + dD.
\]
His calculus is the beginning of a quadruple algebra, and contains
the germs of Grassmann's marvellous system. In designating
\index{Grassmann!ref.\ to}%
segments of lines we find throughout this work for the
first time consistency in the distinction of positive and negative
by the order of letters $AB$,~$BA$. Similarly for triangles
and tetrahedra. The remark that it is always possible to give
three points $A$,~$B$,~$C$ such weights $\alpha$,~$\beta$,~$\gamma$ that any fourth point~$M$
in their plane will become a centre of mass, led Möbius
to a new system of co-ordinates in which the position of a
\index{Co-ordinates}%
point was indicated by an equation, and that of a line by
co-ordinates. By this algorithm he found by algebra many
geometric theorems expressing mainly invariantal properties,---for
example, the theorems on the anharmonic relation.
Möbius wrote also on statics and astronomy. He generalised
spherical trigonometry by letting the sides or angles of triangles
\index{Spherical trigonometry}%
\index{Trigonometry!spherical}%
exceed~$180°$.
%% -----File: 309.png---Folio 295-------
\index{Salmon}%

\textbf{Jacob Steiner} (1796--1863), ``the greatest geometrician since
\index{Steiner}%
the time of Euclid,'' was born in Utzendorf in the Canton of
Bern. He did not learn to write till he was fourteen. At
eighteen he became a pupil of Pestalozzi. Later he studied
at Heidelberg and Berlin. When Crelle started, in 1826, the
celebrated mathematical journal bearing his name, Steiner and
Abel became leading contributors. In 1832 Steiner published
his \textit{Systematische Entwickelung der Abhängigkeit geometrischer
Gestalten von einander}, ``in which is uncovered the organism
by which the most diverse phenomena (\textit{Erscheinungen}) in
the world of space are united to each other.'' Through the
influence of Jacobi and others, the chair of geometry was
\index{Jacobi!ref.\ to}%
founded for him at Berlin in~1834. This position he occupied
until his death, which occurred after years of bad health. In
his \textit{Systematische Entwickelungen}, for the first time, is the
principle of duality introduced at the outset. This book and
von Staudt's lay the foundation on which synthetic geometry
\index{Von Staudt!ref.\ to}%
in its present form rests. Not only did he fairly complete the
theory of curves and surfaces of the second degree, but he
\index{Surfaces, theory of}%
made great advances in the theory of those of higher degrees.
In his hands synthetic geometry made prodigious progress.
New discoveries followed each other so rapidly that he often
did not take time to record their demonstrations. In an
article in \textit{Crelle's Journal} on \textit{Allgemeine Eigenschaften Algebraischer
\index{Crelle's Journal}%
Curven} he gives without proof theorems which were
declared by Hesse to be ``like Fermat's theorems, riddles to
\index{Hesse!ref.\ to}%
the present and future generations.'' Analytical proofs of
some of them have been given since by others, but Cremona
finally proved them all by a synthetic method. Steiner discovered
synthetically the two prominent properties of a surface
of the third order; viz.~that it contains twenty-seven
straight lines and a pentahedron which has the double points
for its vertices and the lines of the Hessian of the given surface
\index{Hessian}%
%% -----File: 310.png---Folio 296-------
for its edges.\Reference{55} The first property was discovered analytically
somewhat earlier in England by Cayley and Salmon,
\index{Cayley!ref.\ to}%
and the second by Sylvester. Steiner's work on this subject
\index{Steiner}%
\index{Sylvester!ref.\ to}%
was the starting-point of important researches by H.~Schröter,
\index{SchroterH@{Schröter, H.}!ref.\ to}%
F.~August, L.~Cremona, and R.~Sturm. Steiner made investigations
\index{August}%
\index{Cremona!ref.\ to|)}%
\index{Sturm, R.}%
by synthetic methods on maxima and minima, and
arrived at the solution of problems which at that time altogether
surpassed the analytic power of the calculus of variations.
\index{Calculus!of variations}%
He generalised the \textit{hexagrammum mysticum} and also
\index{Hexagrammum mysticum}%
Malfatti's problem.\Reference{59} Malfatti, in 1803, proposed the problem,
\index{Malfatti}%
\index{Malfatti's problem}%
to cut three cylindrical holes out of a three-sided prism in
such a way that the cylinders and the prism have the same
altitude and that the volume of the cylinders be a maximum.
This problem was reduced to another, now generally known
as Malfatti's problem: to inscribe three circles in a triangle
that each circle will be tangent to two sides of a triangle and
to the other two circles. Malfatti gave an analytical solution,
but Steiner gave without proof a construction, remarked that
there were thirty-two solutions, generalised the problem by
replacing the three lines by three circles, and solved the
analogous problem for three dimensions. This general problem
was solved analytically by C.~H. Schellbach (1809--1892)
\index{Schellbach}%
and Cayley, and by Clebsch with the aid of the addition
\index{Clebsch!ref.\ to}%
theorem of elliptic functions.\Reference{60}
\index{Elliptic functions}%

Steiner's researches are confined to synthetic geometry. He
hated analysis as thoroughly as Lagrange disliked geometry.
\index{Lagrange!ref.\ to}%
Steiner's \textit{Gesammelte Werke} were published in Berlin in 1881
and 1882.

\textbf{Michel Chasles} (1793--1880) was born at Epernon, entered
\index{Chasles|(}%
the Polytechnic School of Paris in 1812, engaged afterwards
in business, which he later gave up that he might devote all
his time to scientific pursuits. In 1841 he became professor of
geodesy and mechanics at the Polytechnic School; later,
%% -----File: 311.png---Folio 297-------
\index{Cubic curves}%
``Professeur de Géométrie supérieure à la Faculté des Sciences
de Paris.'' He was a voluminous writer on geometrical subjects.
In 1837 he published his admirable \textit{Aperçu historique
sur l'origine et le développement des méthodes en géométrie}, containing
a history of geometry and, as an appendix, a treatise
``sur deux principes généraux de la Science.'' The \textit{Aperçu
historique} is still a standard historical work; the appendix
contains the general theory of Homography (Collineation) and
of duality (Reciprocity). The name \emph{duality} is due to Joseph
\index{Duality}%
Diaz Gergonne (1771--1859). Chasles introduced the term
\index{Gergonne}%
\emph{anharmonic ratio}, corresponding to the German \textit{Doppelverhältniss}
\index{Anharmonic ratio}%
and to Clifford's \emph{cross-ratio}. Chasles and Steiner
\index{Clifford!ref.\ to}%
\index{Steiner!ref.\ to}%
elaborated independently the modern synthetic or projective
geometry. Numerous original memoirs of Chasles were published
later in the \textit{Journal de l'École Polytechnique}. He gave
a reduction of cubics, different from Newton's in this, that the
\index{Newton!ref.\ to}%
five curves from which all others can be projected are symmetrical
with respect to a centre. In 1864 he began the publication,
in the \textit{Comptes rendus}, of articles in which he solves
by his ``method of characteristics'' and the ``principle of correspondence''
\index{Characteristics, method of}%
\index{Correspondence, principle of}%
\index{Method of characteristics}%
an immense number of problems. He determined,
for instance, the number of intersections of two curves
in a plane. The method of characteristics contains the basis
of enumerative geometry. The application of the principle of
\index{Enumerative geometry}%
correspondence was extended by Cayley, A.~Brill, H.~G. Zeuthen,
\index{Brill, A.}%
\index{Cayley!ref.\ to}%
\index{Zeuthen!ref.\ to}%
H.~A. Schwarz, G.~H. Halphen (1844--1889), and others.
\index{Halphen!ref.\ to}%
\index{Schwarz!ref.\ to}%
The full value of these principles of Chasles was not brought
out until the appearance, in 1879, of the \textit{Kalkül der Abzählenden
Geometrie} by Hermann Schubert of Hamburg. This
\index{Schubert}%
work contains a masterly discussion of the problem of enumerative
geometry, viz.~to determine how many geometric figures
of given definition satisfy a sufficient number of conditions.
Schubert extended his enumerative geometry to $n$-dimensional
space.\Reference{55}
%% -----File: 312.png---Folio 298-------

To Chasles we owe the introduction into projective geometry
of non-projective properties of figures by means of the infinitely
distant imaginary sphero-circle.\Reference{61} Remarkable is his
\index{Imaginary points, lines, etc}%
complete solution, in 1846, by synthetic geometry, of the
difficult question of the attraction of an ellipsoid on an external
\index{Ellipsoid!(attraction of)}%
point. This was accomplished analytically by Poisson in
\index{Poisson!ref.\ to}%
1835. The labours of Chasles and Steiner raised synthetic
\index{Steiner!ref.\ to}%
geometry to an honoured and respected position by the side
of analysis.

\textbf{Karl Georg Christian von Staudt} (1798--1867) was born in
\index{Von Staudt}%
Rothenburg on the Tauber, and, at his death, was professor
in Erlangen. His great works are the \textit{Geometrie der Lage},
Nürnberg, 1847, and his \textit{Beiträge zur Geometrie der Lage}, 1856--1860.
The author cut loose from algebraic formulæ and from
metrical relations, particularly the anharmonic ratio of Steiner
and Chasles, and then created a geometry of position, which
\index{Chasles|)}%
is a complete science in itself, independent of all measurements.
He shows that projective properties of figures have
no dependence whatever on measurements, and can be established
without any mention of them. In his theory of what
he calls ``Würfe,'' he even gives a geometrical definition of
a number in its relation to geometry as determining the position
of a point. The \textit{Beiträge} contains the first complete and
general theory of imaginary points, lines, and planes in projective
geometry. Representation of an imaginary point is
sought in the combination of an involution with a determinate
direction, both on the real line through the point.
While purely projective, von Staudt's method is intimately
related to the problem of representing by actual points and
lines the imaginaries of analytical geometry. This was systematically
undertaken by C.~F. Maximilien Marie, who worked,
\index{Marie, C. F. M.}%
however, on entirely different lines. An independent attempt
has been made recently (1893) by F.~H. Loud of Colorado
\index{Loud}%
%% -----File: 313.png---Folio 299-------
\index{Funicular polygons}%
College. Von~Staudt's geometry of position was for a long
\index{Von Staudt}%
time disregarded, mainly, no doubt, because his book is
extremely condensed. An impulse to the study of this subject
was given by Culmann, who rests his graphical statics upon
\index{Culmann}%
\index{Graphical statics}%
the work of von~Staudt. An interpreter of von~Staudt was
at last found in Theodor Reye of Strassburg, who wrote a
\index{Reye}%
\textit{Geometrie der Lage} in 1868.

Synthetic geometry has been studied with much success by
\textbf{Luigi Cremona}, professor in the University of Rome. In
\index{Cremona}%
his \textit{Introduzione ad una teoria geometrica delle curve piane}
he developed by a uniform method many new results and
proved synthetically all important results reached before that
time by analysis. His writings have been translated into
German by M.~Curtze, professor at the gymnasium in Thorn.
\index{Curtze, M}%
The theory of the transformation of curves and of the correspondence
of points on curves was extended by him to three
dimensions. Ruled surfaces, surfaces of the second order,
\index{Surfaces, theory of}%
space-curves of the third order, and the general theory of
surfaces have received much attention at his hands.

\textbf{Karl Culmann}, professor at the Polytechnicum in Zürich,
published an epoch-making work on \textit{Die graphische Statik},
Zürich, 1864, which has rendered graphical statics a great
rival of analytical statics. Before Culmann, \textit{B.~E. Cousinery}
\index{Cousinery}%
had turned his attention to the graphical calculus, but he
made use of perspective, and not of modern geometry.\Reference{62} Culmann
is the first to undertake to present the graphical calculus
as a symmetrical whole, holding the same relation to the new
geometry that analytical mechanics does to higher analysis.
He makes use of the polar theory of reciprocal figures as
expressing the relation between the force and the funicular
polygons. He deduces this relation without leaving the plane
of the two figures. But if the polygons be regarded as projections
of lines in space, these lines may be treated as reciprocal
%% -----File: 314.png---Folio 300-------
\index{Culmann}%
\index{Non-Euclidean geometry|(}%
\index{Parallels}%
elements of a ``Nullsystem.'' This was done by \textit{Clerk
Maxwell} in 1864, and elaborated further by \textit{Cremona}.\Reference{63} The
\index{Cremona!ref.\ to}%
\index{Maxwell!ref.\ to}%
graphical calculus has been applied by \textit{O.~Mohr} of Dresden
\index{Mohr}%
to the elastic line for continuous spans. \textit{Henry~T. Eddy}, of
\index{Eddy}%
the Rose Polytechnic Institute, gives graphical solutions of
problems on the maximum stresses in bridges under concentrated
loads, with aid of what he calls ``reaction polygons.''
\index{Reaction polygons}%
A standard work, \textit{La Statique graphique}, 1874, was issued by
Maurice Levy of Paris.
\index{Levy}%

Descriptive geometry (reduced to a science by Monge in
\index{Descriptive geometry}%
\index{Geometry!descriptive}%
\index{Monge!ref.\ to}%
France, and elaborated further by his successors, \textit{Hachette},
\index{Hachette}%
\textit{Dupin}, \textit{Olivier}, \textit{J.~de la Gournerie}) was soon studied also in
\index{Dupin!ref.\ to}%
\index{Gournerie}%
\index{Olivier}%
other countries. The French directed their attention mainly
to the theory of surfaces and their curvature; the Germans
and Swiss, through Schreiber, Pohlke, Schlessinger, and particularly
\index{Pohlke}%
\index{Schlessinger}%
\index{Schreiber}%
Fiedler, interwove projective and descriptive geometry.
\index{Fiedler}%
Bellavitis in Italy worked along the same line. The
\index{Bellavitis!ref.\ to}%
theory of shades and shadows was first investigated by the
French writers just quoted, and in Germany treated most
exhaustively by Burmester.\Reference{62}
\index{Burmester}%

During the present century very remarkable generalisations
have been made, which reach to the very root of two of the
oldest branches of mathematics,---elementary algebra and
geometry. In algebra the laws of operation have been extended;
in geometry the axioms have been searched to the
\index{Axioms (of geometry)}%
bottom, and the conclusion has been reached that the space
defined by Euclid's axioms is not the only possible non-contradictory
space. Euclid proved (I.~27) that ``if a straight
line falling on two other straight lines make the alternate
angles equal to one another, the two straight lines shall be
parallel to one another.'' Being unable to prove that in every
other case the two lines are not parallel, he \emph{assumed} this to
be true in what is generally called the 12th ``axiom,'' by some
%% -----File: 315.png---Folio 301-------
the 11th ``axiom.'' But this so-called axiom is far from
axiomatic. After centuries of desperate but fruitless attempts
to prove Euclid's assumption, the bold idea dawned upon
the minds of several mathematicians that a geometry might
be built up without assuming the parallel-axiom. While
Legendre still endeavoured to establish the axiom by rigid
\index{Legendre!ref.\ to}%
proof, Lobatchewsky brought out a publication which assumed
\index{Lobatchewsky}%
the contradictory of that axiom, and which was the first of
a series of articles destined to clear up obscurities in the
fundamental concepts, and to greatly extend the field of
geometry.

\textbf{Nicholaus Ivanovitch Lobatchewsky} (1793--1856) was born at
Makarief, in Nischni-Nowgorod, Russia, studied at Kasan, and
from 1827 to 1846 was professor and rector of the University
of Kasan. His views on the foundation of geometry were
first made public in a discourse before the physical and mathematical
faculty at Kasan, and first printed in the Kasan
\textit{Messenger} for 1829, and then in the \textit{Gelehrte Schriften der
Universität Kasan}, 1836--1838, under the title, ``New Elements
of Geometry, with a complete theory of Parallels.'' Being
\index{Parallels}%
in the Russian language, the work remained unknown to
foreigners, but even at home it attracted no notice. In 1840
he published a brief statement of his researches in Berlin.
Lobatchewsky constructed an ``imaginary geometry,'' as he
\index{Imaginary geometry}%
called it, which has been described by Clifford as ``quite
simple, merely Euclid without the vicious assumption.'' A
remarkable part of this geometry is this, that through a
point an indefinite number of lines can be drawn in a plane,
none of which cut a given line in the same plane. A similar
system of geometry was deduced independently by the Bolyais
in Hungary, who called it ``absolute geometry.''
\index{Absolute geometry}%

\textbf{Wolfgang Bolyai de~Bolya} (1775--1856) was born in Szekler-Land,
\index{Bolyai, Wolfgang}%
Transylvania. After studying at Jena, he went to
%% -----File: 316.png---Folio 302-------
Göttingen, where he became intimate with Gauss, then nineteen
\index{Gauss!ref.\ to}%
years old. Gauss used to say that Bolyai was the only
\index{Bolyai, Johann}%
\index{Bolyai, Wolfgang}%
man who fully understood his views on the metaphysics of
mathematics. Bolyai became professor at the Reformed College
of Maros-Vásárhely, where for forty-seven years he had
for his pupils most of the present professors of Transylvania.
The first publications of this remarkable genius were dramas
and poetry. Clad in old-time planter's garb, he was truly
original in his private life as well as in his mode of thinking.
He was extremely modest. No monument, said he, should
stand over his grave, only an apple-tree, in memory of the
three apples; the two of Eve and Paris, which made hell out
of earth, and that of Newton, which elevated the earth again
\index{Newton!ref.\ to}%
into the circle of heavenly bodies.\Reference{64} His son, \textbf{Johann Bolyai}
(1802--1860), was educated for the army, and distinguished
himself as a profound mathematician, an impassioned violin-player,
and an expert fencer. He once accepted the challenge
of thirteen officers on condition that after each duel he might
play a piece on his violin, and he vanquished them all.

The chief mathematical work of Wolfgang Bolyai appeared
in two volumes, 1832--1833, entitled \textit{Tentamen juventutem
studiosam in elementa matheseos puræ\ldots introducendi}. It
is followed by an appendix composed by his son Johann on
\textit{The Science Absolute of Space}. Its twenty-six pages make the
name of Johann Bolyai immortal. He published nothing else,
but he left behind one thousand pages of manuscript which
have never been read by a competent mathematician! His
father seems to have been the only person in Hungary who
really appreciated the merits of his son's work. For thirty-five
years this appendix, as also Lobatchewsky's researches,
remained in almost entire oblivion. Finally Richard Baltzer
\index{Baltzer, R.!ref.\ to}%
of the University of Giessen, in 1867, called attention to the
wonderful researches. Johann Bolyai's \textit{Science Absolute of}
%% -----File: 317.png---Folio 303-------
\textit{Space} and Lobatchewsky's \textit{Geometrical Researches on the
\index{Lobatchewsky!ref.\ to}%
Theory of Parallels} (1840) were rendered easily accessible to
\index{Parallels}%
American readers by translations into English made in 1891
by George Bruce Halsted of the University of Texas.
\index{Halsted}%

The Russian and Hungarian mathematicians were not the
only ones to whom pangeometry suggested itself. A copy of
the \textit{Tentamen} reached Gauss, the elder Bolyai's former room-mate %[** PP: Only instance]
\index{Gauss!ref.\ to}%
at Göttingen, and this Nestor of German mathematicians
was surprised to discover in it worked out what he himself
had begun long before, only to leave it after him in his
papers. As early as 1792 he had started on researches of that
character. His letters show that in 1799 he was trying to
prove \textit{a~priori} the reality of Euclid's system; but some time
\index{Euclid!ref.\ to}%
within the next thirty years he arrived at the conclusion
reached by Lobatchewsky and Bolyai. In 1829 he wrote to
Bessel, stating that his ``conviction that we cannot found
\index{Bessel!ref.\ to}%
geometry completely a~priori has become, if possible, still
firmer,'' and that ``if number is merely a product of our
mind, space has also a \emph{reality beyond} our mind of which we
cannot fully foreordain the laws \textit{a~priori}.'' The term \emph{non-Euclidean
geometry} is due to Gauss. It has recently been
brought to notice that \textit{Geronimo Saccheri}, a Jesuit father of
\index{Saccheri}%
Milan, in 1733 anticipated Lobatchewsky's doctrine of the
parallel angle. Moreover, G.~B. Halsted has pointed out
that in 1766 Lambert wrote a paper ``Zur Theorie der Parallellinien,''
\index{Lambert!ref.\ to}%
published in the \textit{Leipziger Magazin für reine und
angewandte Mathematik}, 1786, in which: (1)~The failure of
the parallel-axiom in surface-spherics gives a geometry with
angle-sum $> 2$ right angles; (2)~In order to make intuitive
a geometry with angle-sum $< 2$ right angles we need the aid
of an ``imaginary sphere" (pseudo-sphere); (3)~In a space
with the angle-sum differing from 2 right angles, there is
an absolute measure (Bolyai's natural unit for length).
%% -----File: 318.png---Folio 304-------

In 1854, nearly twenty years later, Gauss heard from his
\index{Gauss!ref.\ to}%
pupil, \textit{Riemann}, a marvellous dissertation carrying the discussion
\index{Riemann!ref.\ to}%
one step further by developing the notion of \nnobrkdash-ply
extended magnitude, and the measure-relations of which a
manifoldness of $n$ dimensions is capable, on the assumption
that every line may be measured by every other. Riemann
applied his ideas to space. He taught us to distinguish
between ``unboundedness'' and ``infinite extent.'' According
\index{Infinity}%
to him we have in our mind a more general notion of space,
\ie~a notion of non-Euclidean space; but we learn \emph{by experience}
that our physical space is, if not exactly, at least to high
degree of approximation, Euclidean space. Riemann's profound
dissertation was not published until 1867, when it
appeared in the \textit{Göttingen Abhandlungen}. Before this the
idea of \nnobrkdash-dimensions had suggested itself under various
\index{Hyperspace}%
aspects to Lagrange, Plücker, and H.~Grassmann. About the
\index{Grassmann!ref.\ to}%
\index{Lagrange!ref.\ to}%
\index{Plücker!ref.\ to}%
same time with Riemann's paper, others were published from
the pens of \textit{Helmholtz} and \textit{Beltrami}. These contributed powerfully
\index{Beltrami}%
\index{Helmholtz!ref.\ to}%
to the victory of logic over excessive empiricism. This
period marks the beginning of lively discussions upon this subject.
Some writers---Bellavitis, for example---were able to
\index{Bellavitis!ref.\ to}%
see in non-Euclidean geometry and \nnobrkdash-dimensional space nothing
but huge caricatures, or diseased outgrowths of mathematics.
Helmholtz's article was entitled \textit{Thatsachen, welche
der Geometrie zu Grunde liegen}, 1868, and contained many of
the ideas of Riemann. Helmholtz popularised the subject in
lectures, and in articles for various magazines.

\textbf{Eugenio Beltrami}, born at Cremona, Italy, in 1835, and now
professor at Rome, wrote the classical paper \textit{Saggio di interpretazione
della geometria non-euclidea} (Giorn.\ di Matem.,~6),
which is analytical (and, like several other papers, should be
mentioned elsewhere were we to adhere to a strict separation
between synthesis and analysis). He reached the brilliant
%% -----File: 319.png---Folio 305-------
and surprising conclusion that the theorems of non-Euclidean
geometry find their realisation upon surfaces of constant negative
curvature. He studied, also, surfaces of constant positive
curvature, and ended with the interesting theorem that the
space of constant positive curvature is contained in the space
of constant negative curvature. These researches of Beltrami,
\index{Beltrami}%
Helmholtz, and Riemann culminated in the conclusion that
\index{Helmholtz!ref.\ to}%
\index{Riemann!ref.\ to}%
on surfaces of constant curvature we may have three geometries,---the
non-Euclidean on a surface of constant negative
curvature, the spherical on a surface of constant positive curvature,
and the Euclidean geometry on a surface of zero curvature.
The three geometries do not contradict each other, but
are members of a system,---a geometrical trinity. The ideas
of \DPtypo{hyper-space}{hyperspace} were brilliantly expounded and popularised in
\index{Hyperspace}%
England by Clifford.
\index{Clifford}%

\textbf{William Kingdon Clifford} (1845--1879) was born at Exeter,
educated at Trinity College, Cambridge, and from 1871 until
his death professor of applied mathematics in University College,
London. His premature death left incomplete several
brilliant researches which he had entered upon. Among these
are his paper \textit{On Classification of Loci} and his \textit{Theory of
Graphs}. He wrote articles \textit{On the Canonical Form and
Dissection of a Riemann's Surface}, on \textit{Biquaternions}, and
an incomplete work on the \textit{Elements of Dynamic}. The
theory of polars of curves and surfaces was generalised by
him and by Reye. His classification of loci, 1878, being a
\index{Reye!ref.\ to}%
general study of curves, was an introduction to the study
of \nnobrkdash-dimensional space in a direction mainly projective.
This study has been continued since chiefly by G.~Veronese
\index{Veronese}%
of Padua, C.~Segre of Turin, E.~Bertini, F.~Aschieri, P.~Del~Pezzo
\index{Aschieri}%
\index{Bertini}%
\index{Delpezzo@{Del Pezzo}}%
\index{Segre}%
of Naples.

Beltrami's researches on non-Euclidean geometry were followed,
in 1871, by important investigations of Felix Klein,
\index{Klein!ref.\ to}%
%% -----File: 320.png---Folio 306-------
\index{Clifford}%
\index{Parallels}%
resting upon Cayley's \textit{Sixth Memoir on Quantics}, 1859. The
\index{Cayley!ref.\ to}%
question whether it is not possible to so express the metrical
properties of figures that they will not vary by projection (or
linear transformation) had been solved for special projections
by Chasles, Poncelet, and E.~Laguerre (1834--1886) of Paris,
\index{Chasles!ref.\ to}%
\index{Laguerre}%
\index{Poncelet!ref.\ to}%
but it remained for Cayley to give a general solution by defining
the distance between two points as an arbitrary constant
multiplied by the logarithm of the anharmonic ratio in which
\index{Anharmonic ratio}%
the line joining the two points is divided by the fundamental
quadric. Enlarging upon this notion, Klein showed the independence
\index{Klein!ref.\ to}%
of projective geometry from the parallel-axiom, and
by properly choosing the law of the measurement of distance
deduced from projective geometry the spherical, Euclidean,
and pseudospherical geometries, named by him respectively
the elliptic, parabolic, and hyperbolic geometries. This suggestive
investigation was followed up by numerous writers,
particularly by G.~Battaglini of Naples, E.~d'Ovidio of Turin,
\index{Battaglini}%
\index{Dovidio@{D'Ovidio}}%
R.~de~Paolis of Pisa, F.~Aschieri, A.~Cayley, F.~Lindemann
\index{Aschieri}%
\index{Depaolis@{De Paolis}}%
\index{Lindemann!ref.\ to}%
of Munich, E.~Schering of Göttingen, W.~Story of Clark
\index{Schering}%
\index{Story}%
University, H.~Stahl of Tübingen, A.~Voss of Würzburg,
\index{Stahl}%
\index{Voss}%
Homersham Cox, A.~Buchheim.\Reference{55} The geometry of $n$ dimensions
\index{Buchheim!ref.\ to}%
\index{Cox}%
was studied along a line mainly metrical by a host of
writers, among whom may be mentioned Simon Newcomb of
\index{Newcomb!ref.\ to}%
the Johns Hopkins University, L.~Schläfli of Bern, W.~I.
\index{Schlafli@{Schläfli}}%
Stringham of the University of California, W.~Killing of
\index{Killing}%
\index{Stringham}%
Münster, T.~Craig of the Johns Hopkins, R.~Lipschitz of
\index{Craig, T.}%
\index{Lipschitz}%
Bonn. R.~S. Heath and Killing investigated the kinematics
\index{Heath}%
and mechanics of such a space. Regular solids in \nnobrkdash-dimensional
space were studied by Stringham, Ellery~W. Davis
\index{Davis, E.~W.}%
of the University of Nebraska, R.~Hoppe of Berlin, and
\index{Hoppe}%
others. Stringham gave pictures of projections upon our
space of regular solids in four dimensions, and Schlegel at
\index{Schlegel!ref.\ to, XII}%
Hagen constructed models of such projections. These are
%% -----File: 321.png---Folio 307-------
\index{Non-Euclidean geometry|)}%
among the most curious of a series of models published by
L.~Brill in Darmstadt. It has been pointed out that if a
\index{Brill, L.}%
fourth dimension existed, certain motions could take place
which we hold to be impossible. Thus Newcomb showed the
\index{Newcomb!ref.\ to}%
possibility of turning a closed material shell inside out by simple
flexure without either stretching or tearing; Klein pointed
\index{Klein!ref.\ to}%
out that knots could not be tied; Veronese showed that a
\index{Veronese!ref.\ to}%
body could be removed from a closed room without breaking
the walls; C.~S. Peirce proved that a body in four-fold space
\index{Peirce, C.~S.!ref.\ to}%
either rotates about two axes at once, or cannot rotate without
losing one of its dimensions.

\Section{Analytic Geometry}
\index{Analytic geometry|(}%
\index{Geometry!analytic|(}%

In the preceding chapter we endeavoured to give a flash-light %[** PP: Only instance]
view of the rapid advance of synthetic geometry. In
connection with hyperspace we also mentioned analytical
treatises. Modern synthetic and modern analytical geometry
have much in common, and may be grouped together
under the common name ``projective geometry.'' Each has
\index{Projective geometry}%
advantages over the other. The continual direct viewing of
figures as existing in space adds exceptional charm to the
study of the former, but the latter has the advantage in this,
that a well-established routine in a certain degree may outrun
thought itself, and thereby aid original research. While in
Germany Steiner and von Staudt developed synthetic geometry,
\index{Synthetic geometry|)}%
Plücker laid the foundation of modern analytic geometry.
\index{Geometry!modern synthetic|)}%
\index{Plücker|(}%

\textbf{Julius Plücker} (1801--1868) was born at Elberfeld, in Prussia.
After studying at Bonn, Berlin, and Heidelberg, he spent
a short time in Paris attending lectures of Monge and his
pupils. Between 1826 and 1836 he held positions successively
at Bonn, Berlin, and Halle. He then became professor of
%% -----File: 322.png---Folio 308-------
physics at Bonn. Until 1846 his original researches were on
geometry. In 1828 and in 1831 he published his \textit{Analytisch-Geometrische
Entwicklungen} in two volumes. Therein he
adopted the abbreviated notation (used before him in a more
restricted way by Bobillier), and avoided the tedious process
\index{Bobillier}%
of algebraic elimination by a geometric consideration. In the
\index{Elimination}%
second volume the principle of duality is formulated analytically.
\index{Duality}%
With him duality and homogeneity found expression
\index{Homogeneity}%
already in his system of co-ordinates. The homogenous or
\index{Co-ordinates}%
tri-linear system used by him is much the same as the co-ordinates
of Möbius. In the identity of analytical operation
and geometric construction Plücker looked for the source of
\index{Plücker!ref.\ to}%
his proofs. The \textit{System der Analytischen Geometrie}, 1835, contains
a complete classification of plane curves of the third
order, based on the nature of the points at infinity. The
\index{Infinity}%
\textit{Theorie der Algebraischen Curven}, 1839, contains, besides an
enumeration of curves of the fourth order, the analytic relations
between the ordinary singularities of plane curves
known as ``Plücker's equations,'' by which he was able to
explain ``Poncelet's paradox.'' The discovery of these relations
\index{Poncelet!ref.\ to}%
\index{Poncelet's paradox}%
is, says Cayley, ``the most important one beyond all
\index{Cayley!ref.\ to}%
comparison in the entire subject of modern geometry.'' But
in Germany Plücker's researches met with no favour. His
method was declared to be unproductive as compared with
the synthetic method of Steiner and Poncelet! His relations
\index{Steiner!ref.\ to}%
with Jacobi were not altogether friendly. Steiner once
\index{Jacobi!ref.\ to}%
declared that he would stop writing for \textit{Crelle's Journal} if
Plücker continued to contribute to it.\Reference{66} The result was that
many of Plücker's researches were published in foreign journals,
and that his work came to be better known in France
and England than in his native country. The charge was
also brought against Plücker that, though occupying the chair
of physics, he was no physicist. This induced him to relinquish
%% -----File: 323.png---Folio 309-------
\index{Congruency of lines}%
mathematics, and for nearly twenty years to devote
his energies to physics. Important discoveries on Fresnel's
wave-surface, magnetism, spectrum-analysis were made by
him. But towards the close of his life he returned to his
first love,---mathematics,---and enriched it with new discoveries.
By considering space as made up of lines he created
a ``new geometry of space.'' Regarding a right line as a
curve involving four arbitrary parameters, one has the whole
system of lines in space. By connecting them by a single
relation, he got a ``complex'' of lines; by connecting them
\index{Complex of lines}%
with a twofold relation, he got a ``congruency'' of lines. His
first researches on this subject were laid before the Royal
Society in~1865. His further investigations thereon appeared
in 1868 in a posthumous work entitled \textit{Neue Geometrie des
Raumes gegründet auf die Betrachtung der geraden Linie als
Raumelement}, edited by Felix Klein. Plücker's analysis lacks
\index{Klein!ref.\ to}%
the elegance found in Lagrange, Jacobi, Hesse, and Clebsch.
\index{Clebsch!ref.\ to}%
\index{Hesse|(}%
\index{Hesse!ref.\ to}%
\index{Jacobi!ref.\ to}%
\index{Lagrange!ref.\ to}%
For many years he had not kept up with the progress of
geometry, so that many investigations in his last work had
already received more general treatment on the part of others.
The work contained, nevertheless, much that was fresh and
original. The theory of complexes of the second degree, left
unfinished by Plücker, was continued by Felix Klein, who
\index{Plücker|)}%
greatly extended and supplemented the ideas of his master.

\textbf{Ludwig Otto Hesse} (1811--1874) was born at Königsberg, and
studied at the university of his native place under Bessel,
\index{Bessel!ref.\ to}%
Jacobi, Richelot, and F.~Neumann. Having taken the doctor's
\index{Neumann, C.!ref.\ to}%
\index{Neumann, F.~E.!ref.\ to}%
\index{Richelot!ref.\ to}%
degree in 1840, he became docent at Königsberg, and in 1845
extraordinary professor there. Among his pupils at that time
were Durège, Carl Neumann, Clebsch, Kirchhoff. The Königsberg
\index{Durège!ref.\ to}%
\index{Kirchhoff!ref.\ to}%
period was one of great activity for Hesse. Every new
discovery increased his zeal for still greater achievement.
His earliest researches were on surfaces of the second order,
\index{Surfaces, theory of}%
%% -----File: 324.png---Folio 310-------
\index{Invariant}%
and were partly synthetic. He solved the problem to construct
any tenth point of such a surface when nine points are given.
The analogous problem for a conic had been solved by Pascal
\index{Pascal!ref.\ to}%
by means of the hexagram. A difficult problem confronting
mathematicians of this time was that of elimination. Plücker
\index{Elimination}%
had seen that the main advantage of his special method in
analytic geometry lay in the avoidance of algebraic elimination.
Hesse, however, showed how by determinants to make
algebraic elimination easy. In his earlier results he was
anticipated by Sylvester, who published his dialytic method
\index{Sylvester!ref.\ to}%
of elimination in 1840. These advances in algebra Hesse
applied to the analytic study of curves of the third order. By
linear substitutions, he reduced a form of the third degree in
three variables to one of only four terms, and was led to an
important determinant involving the second differential coefficient
of a form of the third degree, called the ``Hessian.''
\index{Hessian}%
The ``Hessian'' plays a leading part in the theory of invariants,
a subject first studied by Cayley. Hesse showed that
his determinant gives for every curve another curve, such that
the double points of the first are points on the second, or
``Hessian." Similarly for surfaces (Crelle, 1844). Many of
\index{Surfaces, theory of}%
the most important theorems on curves of the third order are
due to Hesse. He determined the curve of the 14th order,
which passes through the 56~points of contact of the 28~bi-tangents
of a curve of the fourth order. His great memoir on
this subject (Crelle, 1855) was published at the same time as
was a paper by Steiner treating of the same subject.

Hesse's income at Königsberg had not kept pace with his
growing reputation. Hardly was he able to support himself
and family. In 1855 he accepted a more lucrative position at
Halle, and in 1856 one at Heidelberg. Here he remained until
1868, when he accepted a position at a technic school in
Munich.\Reference{67} At Heidelberg he revised and enlarged upon his
%% -----File: 325.png---Folio 311-------
\index{Gournerie}%
\index{McCullagh}%
previous researches, and published in 1861 his \textit{Vorlesungen
über die Analytische Geometrie des Raumes, insbesondere über
Flächen 2.~Ordnung}. More elementary works soon followed.
While in Heidelberg he elaborated a principle, his ``Uebertragungsprincip."
According to this, there corresponds to every
point in a plane a pair of points in a line, and the projective
geometry of the plane can be carried back to the geometry of
points in a line.

The researches of Plücker and Hesse were continued in England
\index{Hesse|)}%
by Cayley, Salmon, and Sylvester. It may be premised here
\index{Cayley!ref.\ to}%
\index{Salmon|(}%
\index{Sylvester!ref.\ to}%
that among the early writers on analytical geometry in England
was \textbf{James Booth} (1806--1878), whose chief results are embodied
\index{Booth}%
in his \textit{Treatise on Some New Geometrical Methods}; and \textbf{James
MacCullagh} (1809--1846), who was professor of natural philosophy
\index{MacCullagh}%
at Dublin, and made some valuable discoveries on the
theory of quadrics. The influence of these men on the
progress of geometry was insignificant, for the interchange of
scientific results between different nations was not so complete
at that time as might have been desired. In further illustration
of this, we mention that Chasles in France elaborated
\index{Chasles!ref.\ to}%
subjects which had previously been disposed of by Steiner in
\index{Steiner!ref.\ to}%
Germany, and Steiner published researches which had been
given by Cayley, Sylvester, and Salmon nearly five years
earlier. Cayley and Salmon in 1849 determined the straight
lines in a cubic surface, and studied its principal properties,
while Sylvester in 1851 discovered the pentahedron of such a
surface. Cayley extended Plücker's equations to curves of
higher singularities. Cayley's own investigations, and those
of M.~Nöther of Erlangen, G.~H. Halphen (1844--1889) of the
\index{Halphen}%
\index{Nother@{Nöther}}%
Polytechnic School in Paris, De~La~\DPtypo{Gournérie}{Gournerie} of Paris, A.~Brill
\index{Brill, A.}%
of Tübingen, lead to the conclusion that each higher singularity
of a curve is equivalent to a certain number of simple
singularities,---the node, the ordinary cusp, the double tangent,
%% -----File: 326.png---Folio 312-------
\index{Deficiency of curves}%
and the inflection. Sylvester studied the ``twisted Cartesian,''
\index{Sylvester!ref.\ to}%
\index{Twisted Cartesian}%
a curve of the fourth order. Salmon helped powerfully
towards the spreading of a knowledge of the new algebraic and
geometric methods by the publication of an excellent series of
text-books (\textit{Conic Sections}, \textit{Modern Higher Algebra}, \textit{Higher
Plane Curves}, \textit{Geometry of Three Dimensions}), which have
been placed within easy reach of German readers by a free
translation, with additions, made by Wilhelm Fiedler of the
\index{Fiedler}%
Polytechnicum in Zürich. The next great worker in the field
of analytic geometry was Clebsch.
\index{Clebsch}%

\textbf{Rudolf Friedrich Alfred Clebsch} (1833--1872) was born at
Königsberg in Prussia, studied at the university of that place
under Hesse, Richelot, F.~Neumann. From 1858 to 1863 he
\index{Hesse!ref.\ to}%
\index{Neumann, F.~E.!ref.\ to}%
\index{Richelot!ref.\ to}%
held the chair of theoretical mechanics at the Polytechnicum
in Carlsruhe. The study of Salmon's works led him into
algebra and geometry. In 1863 he accepted a position at the
University of \DPtypo{Giesen}{Giessen}, where he worked in conjunction with
Paul Gordan (now of Erlangen). In 1868 Clebsch went to
\index{Gordan}%
Göttingen, and remained there until his death. He worked
successively at the following subjects: Mathematical physics,
the calculus of variations and partial differential equations of
the first order, the general theory of curves and surfaces,
Abelian functions and their use in geometry, the theory of
\index{Abelian functions}%
invariants, and ``Flächenabbildung.''\Reference{68} He proved theorems
on the pentahedron enunciated by Sylvester and Steiner; he
\index{Steiner!ref.\ to}%
made systematic use of ``deficiency'' (\textit{Geschlecht}) as a fundamental
principle in the classification of algebraic curves. The
notion of deficiency was known before him to Abel and Riemann.
\index{Abel!ref.\ to}%
\index{Riemann!ref.\ to}%
At the beginning of his career, Clebsch had shown
how elliptic functions could be advantageously applied to
Malfatti's problem. The idea involved therein, viz.\ the use
\index{Malfatti's problem}%
of higher transcendentals in the study of geometry, led him
to his greatest discoveries. Not only did he apply Abelian
%% -----File: 327.png---Folio 313-------
\index{Armemante}%
\index{Ptolemy!ref.\ to}%
functions to geometry, but conversely, he drew geometry into
the service of Abelian functions.

Clebsch made liberal use of determinants. His study of
\index{Clebsch}%
\index{Determinants}%
curves and surfaces began with the determination of the points
of contact of lines which meet a surface in four consecutive
points. Salmon had proved that these points lie on the intersection
\index{Salmon|)}%
of the surface with a derived surface of the degree
$11n - 24$, but his solution was given in inconvenient form.
Clebsch's investigation thereon is a most beautiful piece of
analysis.

The representation of one surface upon another (\textit{Flächenabbildung}),
\index{Flächenabbildung}%
so that they have a $(1, 1)$ correspondence, was
thoroughly studied for the first time by Clebsch. The representation
of a sphere on a plane is an old problem which
drew the attention of Ptolemæus, Gerard Mercator, Lambert,
\index{Lambert!ref.\ to}%
\index{Mercator, G.}%
Gauss, Lagrange. Its importance in the construction of maps
\index{Gauss!ref.\ to}%
\index{Lagrange!ref.\ to}%
is obvious. Gauss was the first to represent a surface upon
another with a view of more easily arriving at its properties.
Plücker, Chasles, Cayley, thus represented on a plane the
\index{Cayley!ref.\ to}%
\index{Chasles!ref.\ to}%
\index{Plücker!ref.\ to}%
geometry of quadric surfaces; Clebsch and Cremona, that of
\index{Cremona!ref.\ to}%
cubic surfaces. Other surfaces have been studied in the same
way by recent writers, particularly M.~Nöther of Erlangen,
\index{Nother@{Nöther}}%
Armenante, Felix Klein, Korndörfer, Caporali, H.~G. Zeuthen
\index{Caporali}%
\index{Klein!ref.\ to}%
\index{Korndörfer}%
\index{Zeuthen}%
of Copenhagen. A fundamental question which has as yet
received only a partial answer is this: What surfaces can be
represented by a $(1, 1)$ correspondence upon a given surface?
This and the analogous question for curves was studied by
Clebsch. Higher correspondences between surfaces have been
investigated by Cayley and Nöther. The theory of surfaces
has been studied also by \textbf{Joseph Alfred Serret} (1819--1885), professor
\index{Serret}%
at the Sorbonne in Paris, \textbf{Jean Gaston Darboux} of Paris,
\index{Darboux}%
\textit{John Casey} of Dublin (died 1891), \textit{W.~R.~W. Roberts} of Dublin,
\index{Casey}%
\index{Roberts}%
\textit{H.~Schröter} (1829--1892) of Breslau. Surfaces of the
\index{SchroterH@{Schröter, H.}}%
%% -----File: 328.png---Folio 314-------
\index{Differential equations}%
\index{Fresnel's wave-surface}%
fourth order were investigated by Kummer, and Fresnel's
\index{Kummer!ref.\ to}%
wave-surface, studied by Hamilton, is a particular case of
\index{Hamilton, W.~R.!ref.\ to}%
Kummer's quartic surface, with sixteen canonical points and
sixteen singular tangent planes.\Reference{56}

The infinitesimal calculus was first applied to the determination
of the measure of curvature of surfaces by Lagrange,
\index{Curvature, measure of}%
\index{Lagrange!ref.\ to}%
\index{Surfaces, theory of}%
Euler, and \DPtypo{Meunier}{Meusnier} (1754--1793) of Paris. Then followed the
\index{Euler!ref.\ to}%
\index{Meunier}%
researches of Monge and Dupin, but they were eclipsed by
\index{Dupin!ref.\ to}%
\index{Monge!ref.\ to}%
the work of Gauss, who disposed of this difficult subject in a
\index{Gauss!ref.\ to}%
way that opened new vistas to geometricians. His treatment
is embodied in the \textit{Disquisitiones generales circa superficies
curvas} (1827) and \textit{Untersuchungen über gegenstände\DPnote{** Non-capitalization seems to be a legitimate variant} der
höheren Geodäsie} of 1843 and~1846. He defined the measure
of curvature at a point to be the reciprocal of the product
of the two principal radii of curvature at that point. From
this flows the theorem of \textit{Johann August Grunert} (1797--1872;
\index{Grunert}%
professor in Greifswald), that the arithmetical mean of the
radii of curvature of all normal sections through a point is the
radius of a sphere which has the same measure of curvature
as has the surface at that point. Gauss's deduction of the
formula of curvature was simplified through the use of determinants
by \textit{Heinrich Richard Baltzer} (1818--1887) of Giessen.\Reference{69}
\index{Baltzer, R.}%
Gauss obtained an interesting theorem that if one surface be
developed (\textit{abgewickelt}) upon another, the measure of curvature
remains unaltered at each point. The question whether
two surfaces having the same curvature in corresponding
points can be unwound, one upon the other, was answered
by F.~Minding in the affirmative only when the curvature is
\index{Minding}%
constant. The case of variable curvature is difficult, and was
studied by Minding, J.~Liouville (1806--1882) of the Polytechnic
\index{Liouville!ref.\ to}%
School in Paris, Ossian Bonnet of Paris (died 1892).
\index{Bonnet, O.}%
Gauss's measure of curvature, expressed as a function of curvilinear
co-ordinates, gave an impetus to the study of differential-invariants,
\index{Co-ordinates}%
%% -----File: 329.png---Folio 315-------
\index{Analytic geometry|)}%
\index{Kelvin, Lord!ref.\ to}%
or differential-parameters, which have been
investigated by Jacobi, C.~Neumann, Sir James Cockle,
\index{Cockle}%
\index{Jacobi!ref.\ to}%
\index{Neumann, C.!ref.\ to}%
Halphen, and elaborated into a general theory by Beltrami,
\index{Beltrami!ref.\ to}%
\index{Halphen!ref.\ to}%
S.~Lie, and others. Beltrami showed also the connection
between the measure of curvature and the geometric axioms.
\index{Axioms (of geometry)}%

Various researches have been brought under the head of
``analysis situs.'' The subject was first investigated by
\index{Analysis situs}%
Leibniz, and was later treated by Gauss, whose theory of
\index{Gauss!ref.\ to}%
\index{Leibniz!ref.\ to}%
knots (\textit{Verschlingungen}) has been employed recently by J.~B.
Listing, O.~Simony, F.~Dingeldey, and others in their ``topologic
\index{Dingeldey}%
\index{Listing}%
\index{Simony}%
studies.'' Tait was led to the study of knots by Sir
William Thomson's theory of vortex atoms. In the hands
\index{Thomson, Sir William!ref.\ to}%
of Riemann the analysis situs had for its object the determination
\index{Riemann!ref.\ to}%
of what remains unchanged under transformations
brought about by a combination of infinitesimal distortions.
In continuation of his work, Walter Dyck of Munich wrote on
\index{Dyck}%
the analysis situs of three-dimensional spaces.

Of geometrical text-books not yet mentioned, reference
should be made to \textbf{Alfred Clebsch's} \textit{Vorlesungen über Geometrie},
\index{Clebsch!ref.\ to}%
edited by Ferdinand Lindemann, now of Munich; \textbf{Frost's}
\index{Frost}%
\index{Lindemann}%
\textit{Solid Geometry}; \textbf{Durège's} \textit{Ebene Curven dritter Ordnung}.
\index{Durège!ref.\ to}%
\index{Geometry!analytic|)}%


\Section{Algebra}
\index{Algebra!recent|(}%

The progress of algebra in recent times may be considered
under three principal heads: the study of fundamental laws
and the birth of new algebras, the growth of the theory of
equations, and the development of what is called modern
higher algebra.

We have already spoken of George Peacock and D.~F.
\index{Peacock!ref.\ to, X}%
Gregory in connection with the fundamental laws of algebra.
\index{Gregory, David F}%
Much was done in this line by De~Morgan.
%% -----File: 330.png---Folio 316-------

\textbf{Augustus De~Morgan} (1806--1871) was born at Madura (Madras),
\index{Demorgan@{De Morgan}}%
and educated at Trinity College, Cambridge. His scruples
about the doctrines of the established church prevented
him from proceeding to the M.A.~degree, and from sitting
for a fellowship. In 1828 he became professor at the newly
established University of London, and taught there until
1867, except for five years, from 1831--1835. De~Morgan was
a unique, manly character, and pre-eminent as a teacher. The
value of his original work lies not so much in increasing our
stock of mathematical knowledge as in putting it all upon a
thoroughly logical basis. He felt keenly the lack of close
reasoning in mathematics as he received it. He said once:
``We know that mathematicians care no more for logic than
\index{Logic}%
logicians for mathematics. The two eyes of exact science are
mathematics and logic: the mathematical sect puts out the
logical eye, the logical sect puts out the mathematical eye;
each believing that it can see better with one eye than with
two.'' De~Morgan saw with both eyes. He analysed logic
mathematically, and studied the logical analysis of the laws,
symbols, and operations of mathematics; he wrote a \textit{Formal
Logic} as well as a \textit{Double Algebra}, and corresponded both with
Sir William Hamilton, the metaphysician, and Sir William
\index{Hamilton, W.}%
\index{Hamilton, W.~R.!ref.\ to}%
Rowan Hamilton, the mathematician. Few contemporaries
were as profoundly read in the history of mathematics as
was De~Morgan. No subject was too insignificant to receive
his attention. The authorship of ``Cocker's Arithmetic'' and
the work of circle-squarers was investigated as minutely as was
\index{Circle-squarers}%
the history of the invention of the calculus. Numerous articles
of his lie scattered in the volumes of the \textit{Penny} and \textit{English
Cyclopædias}. His \textit{Differential Calculus}, 1842, is still a
standard work, and contains much that is original with the
author. For the \textit{Encyclopædia Metropolitana} he wrote on the
calculus of functions (giving principles of symbolic reasoning)
%% -----File: 331.png---Folio 317-------
and on the theory of probability. Celebrated is his \textit{Budget of
Paradoxes}, 1872. He published memoirs ``On the Foundation
of Algebra'' (\textit{Trans.\ of Cam.\ Phil.\ Soc.}, 1841, 1842, 1844, and~1847).

In Germany symbolical algebra was studied by Martin Ohm,
\index{Ohm, M}%
who wrote a \textit{System der Mathematik} in~1822. The ideas of
Peacock and De~Morgan recognise the possibility of algebras
which differ from ordinary algebra. Such algebras were
indeed not slow in forthcoming, but, like non-Euclidean
geometry, some of them were slow in finding recognition.
This is true of Grassmann's, Bellavitis's, and Peirce's discoveries,
\index{Bellavitis!ref.\ to}%
\index{Grassmann!ref.\ to}%
\index{Peirce, B.!ref.\ to}%
but Hamilton's quaternions met with immediate
\index{Hamilton, W.~R.!ref.\ to}%
\index{Quaternions!ref.\ to}%
appreciation in England. These algebras offer a geometrical
interpretation of imaginaries. During the times of Descartes,
\index{Descartes!ref.\ to}%
Newton, and Euler, we have seen the negative and the imaginary,
\index{Euler!ref.\ to}%
\index{Newton!ref.\ to}%
$\sqrt{-1}$, accepted as numbers, but the latter was still
regarded as an algebraic fiction. The first to give it a geometric
picture, analogous to the geometric interpretation of the
negative, was \textit{H.~Kühn}, a teacher in Danzig, in a publication of
\index{Kuhn@{Kühn, H.}}%
1750--1751. He represented $a\sqrt{-1}$ by a line perpendicular
to the line~$a$, and equal to~$a$ in length, and construed $\sqrt{-1}$ as
the mean proportional between $+1$~and~$-1$. This same idea
was developed further, so as to give a geometric interpretation
of $a + \sqrt{-b}$, by \textit{Jean-Robert Argand} (1768--?) of Geneva,
\index{Argand}%
in a remarkable \textit{Essai} (1806).\Reference{70} The writings of Kühn and
Argand were little noticed, and it remained for Gauss to break
\index{Gauss!ref.\ to}%
down the last opposition to the imaginary. He introduced~$i$ as
an independent unit co-ordinate to~$1$, and $a + ib$ as a ``complex
\index{Complex quantities}%
number.'' The connection between complex numbers and
points on a plane, though artificial, constituted a powerful
aid in the further study of symbolic algebra. The mind
required a visual representation to aid it. The notion of
what we now call vectors was growing upon mathematicians,
%% -----File: 332.png---Folio 318-------
and the geometric addition of vectors in space was discovered
independently by Hamilton, Grassmann, and others, about the
\index{Grassmann!ref.\ to}%
\index{Hamilton, W.~R.}%
same time.

\textbf{William Rowan Hamilton} (1805--1865) was born of Scotch
parents in Dublin. His early education, carried on at home,
was mainly in languages. At the age of thirteen he is said to
have been familiar with as many languages as he had lived
years. About this time he came across a copy of Newton's
\textit{Universal Arithmetic}. After reading that, he took up successively
analytical geometry, the calculus, Newton's \textit{Principia},
Laplace's \textit{Mécanique Céleste}. At the age of eighteen he
published a paper correcting a mistake in Laplace's work. In
1824 he entered Trinity College, Dublin, and in 1827, while he
was still an undergraduate, he was appointed to the chair of
astronomy. His early papers were on optics. In 1832 he
predicted conical refraction, a discovery by aid of mathematics
which ranks with the discovery of Neptune by
Le~Verrier and Adams. Then followed papers on the \textit{Principle
of Varying Action} (1827) and a general method of
\index{Action!varying}%
\index{Varying action, principle of}%
dynamics (1834--1835). He wrote also on the solution of
\index{Dynamics}%
equations of the fifth degree, the hodograph, fluctuating
functions, the numerical solution of differential equations.
\index{Differential equations}%

The capital discovery of Hamilton is his quaternions, in
\index{Quaternions}%
which his study of algebra culminated. In 1835 he published
in the \textit{Transactions of the Royal Irish Academy} his Theory of
Algebraic Couples. He regarded algebra ``as being no mere
art, nor language, nor primarily a science of quantity, but
rather as the science of order of progression.'' Time appeared
to him as the picture of such a progression. Hence his definition
of algebra as ``the science of pure time.'' It was the
subject of years' meditation for him to determine what he
should regard as the product of each pair of a system of perpendicular
directed lines. At last, on the 16th~of October,
%% -----File: 333.png---Folio 319-------
1843, while walking with his wife one evening, along the
Royal Canal in Dublin, the discovery of quaternions flashed
\index{Quaternions}%
upon him, and he then engraved with his knife on a stone in
Brougham Bridge the fundamental formula $i^2 = j^2 = k^2 = ijk =
-1$. At the general meeting of the Irish Academy, a month
later, he made the first communication on quaternions. An
account of the discovery was given the following year in the
\textit{Philosophical Magazine}. Hamilton displayed wonderful fertility
\index{Hamilton, W.~R.}%
in their development. His \textit{Lectures on Quaternions},
delivered in Dublin, were printed in 1852. His \textit{Elements of
Quaternions} appeared in 1866. Quaternions were greatly
admired in England from the start, but on the Continent they
received less \DPtypo{attenttion}{attention}. P.~G. Tait's \textit{Elementary Treatise}
\index{Tait}%
helped powerfully to spread a knowledge of them in England.
Cayley, Clifford, and Tait advanced the subject somewhat by
\index{Cayley!ref.\ to}%
\index{Clifford!ref.\ to}%
original contributions. But there has been little progress in
recent years, except that made by Sylvester in the solution of
\index{Sylvester!ref.\ to}%
quaternion equations, nor has the application of quaternions
to physics been as extended as was predicted. The change
in notation made in France by Hoüel and by Laisant has been
\index{Houel@{Hoüel}}%
\index{Laisant}%
considered in England as a wrong step, but the true cause for
the lack of progress is perhaps more deep-seated. There is
indeed great doubt as to whether the quaternionic product can
claim a necessary and fundamental place in a system of vector
analysis. Physicists claim that there is a loss of naturalness
in taking the square of a vector to be negative. In order to
meet more adequately their wants, \textit{J.~W. Gibbs} of Yale University
\index{Gibbs!ref.\ to}%
and \textit{A.~Macfarlane} of the University of Texas, have
\index{Macfarlane}%
each suggested an algebra of vectors with a new notation.
Each gives a definition of his own for the product of two
vectors, but in such a way that the square of a vector is
positive. A third system of vector analysis has been used by
\textit{Oliver Heaviside} in his electrical researches.
\index{Heaviside}%
%% -----File: 334.png---Folio 320-------

\textbf{Hermann Grassmann} (1809--1877) was born at Stettin,
\index{Grassmann|(}%
attended a gymnasium at his native place (where his father
was teacher of mathematics and physics), and studied theology
in Berlin for three years. In 1834 he succeeded Steiner as
\index{Steiner!ref.\ to}%
teacher of mathematics in an industrial school in Berlin, but
returned to Stettin in 1836 to assume the duties of teacher of
mathematics, the sciences, and of religion in a school there.\Reference{71}
Up to this time his knowledge of mathematics was pretty
much confined to what he had learned from his father, who
had written two books on ``Raumlehre'' and ``Grössenlehre.''
But now he made his acquaintance with the works of Lacroix,
\index{Lacroix}%
Lagrange, and Laplace. He noticed that Laplace's results
\index{Laplace!ref.\ to}%
could be reached in a shorter way by some new ideas advanced
in his father's books, and he proceeded to elaborate this
abridged method, and to apply it in the study of tides. He
was thus led to a new geometric analysis. In 1840 he had
made considerable progress in its development, but a new
book of Schleiermacher drew him again to theology. In
1842 he resumed mathematical research, and becoming thoroughly
convinced of the importance of his new analysis,
decided to devote himself to it. It now became his ambition
to secure a mathematical chair at a university, but in this he
never succeeded. In 1844 appeared his great classical work,
the \textit{Lineale Ausdehnungslehre}, which was full of new and
\index{Ausdehnungslehre}%
strange matter, and so general, abstract, and out of fashion in
its mode of exposition, that it could hardly have had less
influence on European mathematics during its first twenty
years, had it been published in China.\DPnote{ouch} Gauss, Grunert, and
\index{Gauss!ref.\ to}%
\index{Grunert!ref.\ to}%
Möbius glanced over it, praised it, but complained of the
\index{Mobius@{Möbius}!ref.\ to}%
strange terminology and its ``philosophische Allgemeinheit.''
Eight years afterwards, Bretschneider of Gotha was said to be
\index{Bretschneider}%
the only man who had read it through. An article in \textit{Crelle's
Journal}, in which Grassmann eclipsed the geometers of that
%% -----File: 335.png---Folio 321-------
\index{Curves!theory of}%
\index{Differential equations}%
\index{Matrices}%
time by constructing, with aid of his method, geometrically
any algebraic curve, remained again unnoticed. Need we marvel
if Grassmann turned his attention to other subjects,---to
Schleiermacher's philosophy, to politics, to philology? Still,
articles by him continued to appear in \textit{Crelle's Journal}, and in
1862 came out the second part of his \textit{Ausdehnungslehre}. It
\index{Ausdehnungslehre}%
was intended to show better than the first part the broad
scope of the Ausdehnungslehre, by considering not only geometric
applications, but by treating also of algebraic functions,
infinite series, and the differential and integral calculus. But
the second part was no more appreciated than the first. At the
age of fifty-three, this wonderful man, with heavy heart, gave
up mathematics, and directed his energies to the study of Sanskrit,
achieving in philology results which were better appreciated,
and which vie in splendour with those in mathematics.

Common to the Ausdehnungslehre and to quaternions are geometric
addition, the function of two vectors represented in quaternions
by $S\alpha\beta$~and~$V\alpha\beta$, and the linear vector functions. The
quaternion is peculiar to Hamilton, while with Grassmann we
\index{Hamilton, W.~R.!ref.\ to}%
find in addition to the algebra of vectors a geometrical algebra
of wide application, and resembling Möbius's \textit{Barycentrische
\index{Mobius@{Möbius}!ref.\ to}%
Calcul}, in which the point is the fundamental element. Grassmann
developed the idea of the ``external product,'' the ``internal
product,'' and the ``open product.'' The last we now call
a matrix. His Ausdehnungslehre has very great extension,
having no limitation to any particular number of dimensions.
Only in recent years has the wonderful richness of
his discoveries begun to be appreciated. A second edition of
the \textit{Ausdehnungslehre} of 1844 was printed in 1877. C.~S.
Peirce gave a representation of Grassmann's system in the
\index{Peirce, C.~S.!ref.\ to}%
logical notation, and E.~W. Hyde of the University of Cincinnati
\index{Hyde}%
wrote the first text-book on Grassmann's calculus in
\index{Grassmann|)}%
the English language.
%% -----File: 336.png---Folio 322-------
\index{Aequipollences@{Æquipollences}}%

Discoveries of less value, which in part covered those of
Grassmann and Hamilton, were made by \textit{Saint-Venant} (1797--1886),
\index{Saint-Venant!ref.\ to}%
who described the multiplication of vectors, and the
addition of vectors and oriented areas; by \textit{Cauchy}, whose
\index{Cauchy!ref.\ to}%
``clefs algébriques'' were units subject to combinatorial multiplication,
and were applied by the author to the theory of
elimination in the same way as had been done earlier by
Grassmann; by \textbf{Justus Bellavitis} (1803--1880), who published
\index{Bellavitis}%
in 1835 and 1837 in the \textit{Annali delle Scienze} his calculus of
æquipollences. Bellavitis, for many years professor at Padua,
was a self-taught mathematician of much power, who in his
thirty-eighth year laid down a city office in his native place,
Bassano, that he might give his time to science.\Reference{65}

The first impression of Grassmann's ideas is marked in the
writings of \textbf{Hermann Hankel} (1839--1873), who published in
\index{Hankel}%
1867 his \textit{Vorlesungen über die Complexen Zahlen}. Hankel,
then docent in Leipzig, had been in correspondence with
Grassmann. The ``alternate numbers'' of Hankel are subject
\index{Alternate numbers}%
to his law of combinatorial multiplication. In considering
the foundations of algebra Hankel affirms the principle of
the permanence of formal laws previously enunciated incompletely
by Peacock. Hankel was a close student of mathematical
history, and left behind an unfinished work thereon.
Before his death he was professor at Tübingen. His \textit{\DPtypo{Complexe}{Complexen}
Zahlen} was at first little read, and we must turn to
\textbf{Victor Schlegel} of Hagen as the successful interpreter of Grassmann.
\index{Schlegel}%
Schlegel was at one time a young colleague of Grassmann
at the Marienstifts-Gymnasium in Stettin. Encouraged
by Clebsch, Schlegel wrote a \textit{System der Raumlehre} which
\index{Clebsch!ref.\ to}%
explained the essential conceptions and operations of the
Ausdehnungslehre.

Multiple algebra was powerfully advanced by Peirce, whose
theory is not geometrical, as are those of Hamilton and Grassmann.
%% -----File: 337.png---Folio 323-------
\textbf{Benjamin Peirce} (1809--1880) was born at Salem, Mass.,
\index{Peirce, B.}%
\index{Peirce, C.~S.}%
and graduated at Harvard College, having as undergraduate
carried the study of mathematics far beyond the limits of the
college course.\Reference{2} When Bowditch was preparing his translation
\index{Bowditch}%
and commentary of the \textit{Mécanique Céleste}, young Peirce
helped in reading the proof-sheets. He was made professor at
Harvard in 1833, a position which he retained until his death.
For some years he was in charge of the \textit{Nautical Almanac}
and superintendent of the United States Coast Survey. He
published a series of college text-books on mathematics, an
\textit{Analytical Mechanics}, 1855, and calculated, together with Sears~C.
Walker of Washington, the orbit of Neptune. Profound
\index{Walker}%
are his researches on \textit{Linear Associative Algebra}. The first of
\index{Linear associative algebra}%
several papers thereon was read at the first meeting of the
American Association for the Advancement of Science in 1864.
Lithographed copies of a memoir were distributed among friends
in 1870, but so small seemed to be the interest taken in this
subject that the memoir was not printed until 1881 (\textit{Am.\ Jour.\
Math.}, Vol.~IV., No.~2). Peirce works out the multiplication
tables, first of \emph{single} algebras, then of \emph{double} algebras, and so
on up to sextuple, making in all 162~algebras, which he shows
to be possible on the consideration of symbols $A$,~$B$,~etc.,
which are linear functions of a determinate number of letters
or units $i$,~$j$, $k$,~$l$,~etc., with coefficients which are ordinary analytical
magnitudes, real or imaginary,---the letters $i$,~$j$,~etc.,
being such that every binary combination $i^2$,~$ij$, $ji$,~etc., is equal
to a linear function of the letters, but under the restriction of
satisfying the associative law.\Reference{56} \textit{Charles~S. Peirce}, a son of
Benjamin Peirce, and one of the foremost writers on mathematical
logic, showed that these algebras were all defective
\index{Logic}%
forms of quadrate algebras which he had previously discovered
by logical analysis, and for which he had devised a simple
notation. Of these quadrate algebras quaternions is a simple
%% -----File: 338.png---Folio 324-------
example; nonions is another. C.~S. Peirce showed that of all
linear associative algebras there are only three in which division
is unambiguous. These are ordinary single algebra,
ordinary double algebra, and quaternions, from which the
imaginary scalar is excluded. He showed that his father's
algebras are operational and matricular. Lectures on multiple
algebra were delivered by J.~J. Sylvester at the Johns Hopkins
\index{Sylvester!ref.\ to}%
University, and published in various journals. They treat
largely of the algebra of matrices. The theory of matrices
\index{Matrices}%
was developed as early as 1858 by Cayley in an important
\index{Cayley!ref.\ to}%
memoir which, in the opinion of Sylvester, ushered in the
reign of Algebra the Second. Clifford, Sylvester, H.~Taber,
\index{Clifford!ref.\ to}%
\index{Taber}%
C.~H. Chapman, carried the investigations much further. The
\index{Chapman}%
originator of matrices is really Hamilton, but his theory, published
\index{Hamilton, W.~R.!ref.\ to}%
in his \textit{Lectures on Quaternions}, is less general than that
of Cayley. The latter makes no reference to Hamilton.

The theory of determinants\Reference[]{73} was studied by Hoëné Wronski
\index{Determinants}%
\index{Wronski}%
in Italy and J.~Binet in France; but they were forestalled by
\index{Binet}%
the great master of this subject, Cauchy. In a paper (\textit{Jour.\
\index{Cauchy!ref.\ to}%
de l'ecole Polyt.}, IX.,~16) Cauchy developed several general
theorems. He introduced the name \emph{determinant}, a term
previously used by Gauss in the functions considered by him.
\index{Gauss!ref.\ to}%
In 1826 Jacobi began using this calculus, and he gave brilliant
\index{Jacobi!ref.\ to}%
proof of its power. In 1841 he wrote extended memoirs on
determinants in \textit{Crelle's Journal}, which rendered the theory
easily accessible. In England the study of linear transformations
of quantics gave a powerful impulse. Cayley developed
skew-determinants and Pfaffians, and introduced the use of
determinant brackets, or the familiar pair of upright lines.
More recent researches on determinants appertain to special
forms. ``Continuants'' are due to Sylvester; ``alternants,''
originated by Cauchy, have been developed by Jacobi, N.~Trudi,
\index{Trudi}%
H.~Nägelbach, and G.~Garbieri; ``axisymmetric determinants,''
\index{Garbieri}%
\index{Nagelbach@{Nägelbach}}%
%% -----File: 339.png---Folio 325-------
\index{Invariant}%
first used by Jacobi, have been studied by V.~A. Lebesgue,
\index{Lebesgue}%
Sylvester, and Hesse; ``circulants'' are due to E.~Catalan of
\index{Catalan, E.}%
\index{Hesse!ref.\ to}%
\index{Sylvester!ref.\ to}%
Liège, W.~Spottiswoode (1825--1883), J.~W.~L. Glaisher, and
\index{Glaisher!ref.\ to}%
\index{Spottiswoode}%
R.~F. Scott; for ``centro-symmetric determinants'' we are
\index{Determinants}%
\index{Scott}%
indebted to G.~Zehfuss. E.~B. Christoffel of Strassburg and
\index{Christoffel}%
\index{Zehfuss}%
G.~Frobenius discovered the properties of ``Wronskians,'' first
\index{Frobenius}%
used by Wronski. V.~Nachreiner and S.~Günther, both of
\index{Nachreiner}%
Munich, pointed out relations between determinants and continued
fractions; Scott uses Hankel's alternate numbers in his
\index{Hankel!ref.\ to}%
treatise. Text-books on determinants were written by Spottiswoode
(1851), Brioschi (1854), Baltzer (1857), Günther
\index{Baltzer, R.!ref.\ to}%
\index{Brioschi!ref.\ to}%
\index{Gunther@{Günther, S.}}%
(1875), Dostor (1877), Scott (1880), Muir (1882), Hanus
\index{Dostor}%
\index{Hanus}%
\index{Muir}%
(1886).

Modern higher algebra is especially occupied with the
theory of linear transformations. Its development is mainly
the work of Cayley and Sylvester.
\index{Cayley}%

\textbf{Arthur Cayley}, born at Richmond, in Surrey, in 1821, was
educated at Trinity College, Cambridge.\Reference{74} He came out Senior
Wrangler in 1842. He then devoted some years to the study
and practice of law. On the foundation of the Sadlerian professorship
at Cambridge, he accepted the offer of that chair,
thus giving up a profession promising wealth for a very
modest provision, but which would enable him to give all his
time to mathematics. Cayley began his mathematical publications
in the \textit{Cambridge Mathematical Journal} while he was
still an undergraduate. Some of his most brilliant discoveries
were made during the time of his legal practice. There is
hardly any subject in pure mathematics which the genius of
Cayley has not enriched, but most important is his creation
of a new branch of analysis by his theory of invariants.
Germs of the principle of invariants are found in the writings
of Lagrange, Gauss, and particularly of Boole, who
\index{Boole!ref.\ to}%
\index{Gauss!ref.\ to}%
\index{Lagrange!ref.\ to}%
showed, in 1841, that invariance is a property of discriminants
%% -----File: 340.png---Folio 326-------
generally, and who applied it to the theory of orthogonal
substitution. Cayley set himself the problem to determine
\index{Cayley}%
\textit{a~priori} what functions of the coefficients of a given equation
possess this property of invariance, and found, to begin with,
in 1845, that the so-called ``hyper-determinants'' possessed it.
Boole made a number of additional discoveries. Then Sylvester
\index{Sylvester}%
began his papers in the \textit{Cambridge and Dublin Mathematical
Journal} on the Calculus of Forms. After this,
discoveries followed in rapid succession. At that time Cayley
and Sylvester were both residents of London, and they
stimulated each other by frequent oral communications. It
has often been difficult to determine how much really belongs
to each.

\textbf{James Joseph Sylvester} was born in London in 1814, and
educated at St.~Johns College, Cambridge. He came out
Second Wrangler in 1837. His Jewish origin incapacitated
him from taking a degree. In 1846 he became a student at
the Inner Temple, and was called to the bar in 1850. He
became professor of natural philosophy at University College,
London; then, successively, professor of mathematics at the
University of Virginia, at the Royal Military Academy in
Woolwich, at the Johns Hopkins University in Baltimore,
and is, since 1883, professor of geometry at Oxford. His
first printed paper was on Fresnel's optic theory, 1837. Then
followed his researches on invariants, the theory of equations,
theory of partitions, multiple algebra, the theory of numbers,
and other subjects mentioned elsewhere. About 1874 he took
part in the development of the geometrical theory of link-work %[** PP: Only instance]
movements, originated by the beautiful discovery of
A.~Peaucellier, Capitaine du Génie à Nice (published in
\index{Peaucellier}%
\textit{Nouvelles Annales}, 1864 and 1873), and made the subject
of close study by A.~B. Kempe. To Sylvester is ascribed the
\index{Kempe}%
general statement of the theory of contravariants, the discovery
\index{Contravariants}%
%% -----File: 341.png---Folio 327-------
\index{Macmahon}%
of the partial differential equations satisfied by the
invariants and covariants of binary quantics, and the subject
\index{Covariants}%
of mixed concomitants. In the \textit{American Journal of Mathematics}
are memoirs on binary and ternary quantics, elaborated
partly with aid of \textit{F.~Franklin}, now professor at the Johns
\index{Franklin}%
Hopkins University. At Oxford, Sylvester has opened up a
new subject, the theory of reciprocants, treating of the functions
\index{Reciprocants}%
of a dependent variable~$y$ and the functions of its differential
coefficients in regard to~$x$, which remain unaltered by
the interchange of $x$~and~$y$. This theory is more general than
one on differential invariants by Halphen (1878), and has
\index{Differential invariants}%
\index{Halphen!ref.\ to}%
been developed further by J.~Hammond of Oxford, McMahon
\index{Hammond, J}%
of Woolwich, A.~R. Forsyth of Cambridge, and others. Sylvester
\index{Forsyth}%
playfully lays claim to the appellation of the Mathematical
Adam, for the many names he has introduced into
mathematics. Thus the terms \emph{invariant}, \emph{discriminant}, \emph{Hessian},
\index{Hessian}%
\emph{Jacobian}, are his.

The great theory of invariants, developed in England mainly
by Cayley and Sylvester, came to be studied earnestly in Germany,
France, and Italy. One of the earliest in the field was
\textbf{Siegfried Heinrich Aronhold} (1819--1884), who demonstrated
\index{Aronhold}%
the existence of invariants, $S$~and~$T$, of the ternary cubic.
Hermite discovered evectants and the theorem of reciprocity
named after him. Paul Gordan showed, with the aid of
\index{Gordan}%
symbolic methods, that the number of distinct forms for a
binary quantic is finite. Clebsch proved this to be true for
\index{Clebsch!ref.\ to}%
quantics with any number of variables. A very much simpler
proof of this was given in 1891, by David Hilbert of Königsberg.
\index{Hilbert}%
In Italy, F.~Brioschi of Milan and \textit{Faà de Bruno}
\index{Brioschi}%
\index{Bruno, Faà de}%
(1825--1888) contributed to the theory of invariants, the
latter writing a text-book on binary forms, which ranks by
the side of Salmon's treatise and those of Clebsch and
Gordan. Among other writers on invariants are E.~B. Christoffel,
\index{Christoffel}%
%% -----File: 342.png---Folio 328-------
\index{Burkhardt, H.}%
\index{Invariant}%
\index{Sylvester!ref.\ to}%
\index{Tchirnhausen!ref.\ to}%
Wilhelm Fiedler, P.~A. McMahon, J.~W.~L. Glaisher of
\index{Fiedler}%
\index{Glaisher!ref.\ to}%
\index{McMahon}%
Cambridge, Emory McClintock of New York. McMahon discovered
\index{McClintock}%
that the theory of semi-invariants is a part of that of
\index{Semi-invariants}%
symmetric functions. The modern higher algebra has reached
\index{Symmetric functions}%
out and indissolubly connected itself with several other
branches of mathematics---geometry, calculus of variations,
\index{Calculus!of variations}%
mechanics. Clebsch extended the theory of binary forms to
\index{Clebsch!ref.\ to}%
\index{Mechanics!more recent work}%
ternary, and applied the results to geometry. Clebsch, Klein,
\index{Klein!ref.\ to}%
Weierstrass, Burckhardt, and Bianchi have used the theory of
\index{Bianchi}%
\index{Weierstrass!ref.\ to}%
invariants in hyperelliptic and Abelian functions.
\index{Abelian functions}%
\index{Hyperelliptic functions}%

In the theory of equations Lagrange, Argand, and Gauss
\index{Equations!theory of|(}%
furnished proof to the important theorem that every algebraic
equation has a real or a complex root. Abel proved rigorously
\index{Abel!ref.\ to}%
that the general algebraic equation of the fifth or of higher
degrees cannot be solved by radicals (\textit{Crelle},~I., 1826). A modification
of Abel's proof was given by Wantzel. Before Abel,
\index{Wantzel}%
an Italian physician, \textit{Paolo Ruffini} (1765--1822), had printed
\index{Ruffini}%
proofs of the insolvability, which were criticised by his
countryman Malfatti. Though inconclusive, Ruffini's papers
\index{Malfatti}%
are remarkable as containing anticipations of Cauchy's theory
\index{Cauchy!ref.\ to}%
of groups.\Reference{76} A transcendental solution of the quintic involving
\index{Groups, theory of|(}%
elliptic integrals was given by Hermite (\textit{Compt.\ Rend.}, 1858,
\index{Elliptic integrals}%
\index{Hermite!ref.\ to}%
1865, 1866). After Hermite's first publication, Kronecker, in
\index{Kronecker!ref.\ to}%
1858, in a letter to Hermite, gave a second solution in which
was obtained a simple resolvent of the sixth degree. \textit{Jerrard}, in
\index{Jerrard}%
his \textit{Mathematical Researches} (1832--1835), reduced the quintic
to the trinomial form by an extension of the method of
Tschirnhausen. This important reduction had been effected
as early as 1786 by \textit{E.~S. Bring}, a Swede, and brought out
\index{Bring}%
in a publication of the University of Lund. Jerrard, like
Tschirnhausen, believed that his method furnished a general
algebraic solution of equations of any degree. In 1836 William~R.
Hamilton made a report on the validity of Jerrard's
\index{Hamilton, W.~R.!ref.\ to}%
%% -----File: 343.png---Folio 329-------
\index{Elliptic functions}%
\index{Hamilton's numbers}%
method, and showed that by his process the quintic could be
transformed to any one of the four trinomial forms. Hamilton
defined the limits of its applicability to higher equations.
Sylvester investigated this question, What is the lowest degree
an equation can have in order that it may admit of being
deprived of $i$~consecutive terms by aid of equations not higher
than $i$th~degree. He carried the investigation as far as $i = 8$,
and was led to a series of numbers which he named ``Hamilton's
numbers.'' A transformation of equal importance to Jerrard's
is that of Sylvester, who expressed the quintic as the sum of
three fifth-powers. The covariants and invariants of higher
equations have been studied much in recent years.

Abel's proof that higher equations cannot always be solved
algebraically led to the inquiry as to what equations of a given
degree can be solved by radicals. Such equations are the
ones discussed by Gauss in considering the division of the
\index{Division of the circle}%
\index{Gauss!ref.\ to}%
circle. Abel advanced one step further by proving that an
\index{Circle!division of}%
irreducible equation can always be solved in radicals, if, of
two of its roots, the one can be expressed rationally in terms
of the other, provided that the degree of the equation is
prime; if it is not prime, then the solution depends upon
that of equations of lower degree. Through geometrical considerations,
Hesse came upon algebraically solvable equations
\index{Hesse!ref.\ to}%
of the ninth degree, not included in the previous groups.
The subject was powerfully advanced in Paris by the youthful
\textit{Evariste Galois} (born, 1811; killed in a duel, 1832), who
\index{Galois}%
introduced the notion of a group of substitutions. To him are
\index{Substitutions, theory of}%
due also some valuable results in relation to another set of
equations, presenting themselves in the theory of elliptic
functions, viz.\ the modular equations. Galois's labours gave
\index{Modular equations}%
birth to the important theory of substitutions, which has been
\index{Theory of substitutions}%
greatly advanced by \textit{C.~Jordan} of Paris, \textit{J.~A. Serret} (1819--1885)
\index{Jordan}%
of the Sorbonne in Paris, L.~Kronecker (1823--1891) of
\index{Kronecker}%
%% -----File: 344.png---Folio 330-------
\index{Sturm's theorem}%
\index{Warring}%
Berlin, Klein of Göttingen, M.~Nöther of Erlangen, C.~Hermite
\index{Hermite!ref.\ to}%
\index{Klein!ref.\ to}%
\index{Nother@{Nöther}}%
of Paris, A.~Capelli of Naples, L.~Sylow of Friedrichshald,
\index{Capelli}%
\index{Sylow}%
E.~Netto of Giessen. Netto's book, the \textit{Substitutionstheorie},
\index{Netto}%
has been translated into English by F.~N. Cole of
\index{Cole}%
the University of Michigan, who contributed to the theory.
A simple group of 504~substitutions of nine letters, discovered
by Cole, has been shown by E.~H. Moore of the University of
\index{Moore}%
Chicago to belong to a doubly-infinite system of simple groups.
\index{Groups, theory of|)}%
The theory of substitutions has important applications in the
theory of differential equations. Kronecker published, in 1882,
\index{Kronecker!ref.\ to}%
his \textit{Grundzüge einer Arithmetischen Theorie der Algebraischen
Grössen}.

Since Fourier and Budan, the solution of numerical equations
has been advanced by \textit{W.~G. Horner} of Bath, who gave
\index{Horner}%
an improved method of approximation (\textit{Philosophical Transactions},
1819). \textbf{Jacques Charles François Sturm} (1803--1855), a
\index{Sturm, J.~C.~F.}%
native of Geneva, Switzerland, and the successor of Poisson
\index{Poisson!ref.\ to}%
in the chair of mechanics at the Sorbonne, published in 1829
his celebrated theorem determining the number and situation
of roots of an equation comprised between given limits. Sturm
tells us that his theorem stared him in the face in the midst
of some mechanical investigations connected with the motion
of a compound pendulum.\Reference{77} This theorem, and Horner's
method, offer together sure and ready means of finding the
real roots of a numerical equation.

The symmetric functions of the sums of powers of the roots
\index{Symmetric functions}%
of an equation, studied by Newton and Waring, was considered
\index{Newton!ref.\ to}%
more recently by Gauss, Cayley, Sylvester, Brioschi. Cayley
\index{Brioschi!ref.\ to}%
\index{Cayley!ref.\ to}%
\index{Gauss!ref.\ to}%
\index{Sylvester!ref.\ to}%
gives rules for the ``weight'' and ``order'' of symmetric functions.

The theory of elimination was greatly advanced by Sylvester,
\index{Elimination}%
Cayley, Salmon, Jacobi, Hesse, Cauchy, Brioschi, and
\index{Cauchy!ref.\ to}%
\index{Hesse!ref.\ to}%
\index{Jacobi!ref.\ to}%
\index{Salmon}%
Gordan. Sylvester gave the dialytic method (\textit{Philosophical
\index{Dialytic method of elimination}%
\index{Gordan}%
%% -----File: 345.png---Folio 331-------
\index{Algebra!recent|)}%
Magazine}, 1840), and in 1852 established a theorem relating
to the expression of an eliminant as a determinant. Cayley
made a new statement of Bézout's method of elimination and
\index{Bezout's@{Bézout's method of elimination}}%
\index{Elimination}%
established a general theory of elimination (1852).

\Section{Analysis}
\index{Analysis!modern|(}%

Under this head we find it convenient to consider the subjects
of the differential and integral calculus, the calculus of
variations, infinite series, probability, and differential equations.
\index{Equations!theory of|)}%
Prominent in the development of these subjects was
Cauchy.
\index{Cauchy|(}%

\textbf{Augustin-Louis Cauchy}\Reference[]{78} (1789--1857) was born in Paris,
and received his early education from his father. Lagrange
and Laplace, with whom the father came in frequent contact,
foretold the future greatness of the young boy. At the
École Centrale du Panthéon he excelled in ancient classical
studies. In 1805 he entered the Polytechnic School, and two
years later the École des Ponts et Chaussées. Cauchy left for
Cherbourg in 1810, in the capacity of engineer. Laplace's
\textit{Mécanique Céleste and} Lagrange's \textit{Fonctions Analytiques} were
among his book companions there. Considerations of health
induced him to return to Paris after three years. Yielding to
the persuasions of Lagrange and Laplace, he renounced engineering
in favour of pure science. We find him next holding
a professorship at the Polytechnic School. On the expulsion
of Charles~X., and the accession to the throne of Louis Philippe
in 1830, Cauchy, being exceedingly conscientious, found himself
unable to take the oath demanded of him. Being, in
consequence, deprived of his positions, he went into voluntary
exile. At Fribourg in Switzerland, Cauchy resumed his
studies, and in 1831 was induced by the king of Piedmont to
%% -----File: 346.png---Folio 332-------
accept the chair of mathematical physics, especially created
for him at the university of Turin. In 1833 he obeyed the
call of his exiled king, Charles~X., to undertake the education
of a grandson, the Duke of Bordeaux. This gave Cauchy an
opportunity to visit various parts of Europe, and to learn how
extensively his works were being read. Charles~X. bestowed
upon him the title of Baron. On his return to Paris in 1838,
a chair in the College de France was offered to him, but the
oath demanded of him prevented his acceptance. He was
nominated member of the Bureau of Longitude, but declared
ineligible by the ruling power. During the political events of
1848 the oath was suspended, and Cauchy at last became
professor at the Polytechnic School. On the establishment of
the second empire, the oath was re-instated, but Cauchy and
Arago were exempt from it. Cauchy was a man of great
\index{Arago}%
piety, and in two of his publications staunchly defended the
Jesuits.

Cauchy was a prolific and profound mathematician. By a
prompt publication of his results, and the preparation of
standard text-books, he exercised a more immediate and
beneficial influence upon the great mass of mathematicians
than any contemporary writer. He was one of the leaders
in infusing rigour into analysis. His researches extended
over the field of series, of imaginaries, theory of numbers,
differential equations, theory of substitutions, theory of functions,
determinants, mathematical astronomy, light, elasticity,
etc.,---covering pretty much the whole realm of mathematics,
pure and applied.

Encouraged by Laplace and Poisson, Cauchy published in
1821 his \textit{Cours d'Analyse de l'École Royale Polytechnique}, a
work of great merit. Had it been studied more diligently by
writers of text-books in England and the United States, many
a lax and loose method of analysis hardly as yet eradicated
%% -----File: 347.png---Folio 333-------
\index{Differential equations}%
from elementary text-books would have been discarded over
half a century ago. Cauchy was the first to publish a
rigorous proof of Taylor's theorem. He greatly improved
\index{Taylor's theorem}%
the exposition of fundamental principles of the differential
\index{Differential calculus!philosophy of}%
calculus by his mode of considering limits and his new theory
\index{Calculus!of variations|(}%
on the continuity of functions. The method of Cauchy and
\index{Continuity}%
Duhamel was accepted with favour by Hoüel and others. In
\index{Duhamel}%
England special attention to the clear exposition of fundamental
principles was given by De~Morgan. Recent American
\index{Demorgan@{De Morgan}!ref.\ to}%
treatises on the calculus introduce time as an independent variable,
and the allied notions of velocity and acceleration---thus
virtually returning to the method of fluxions.
\index{Fluxions}%

Cauchy made some researches on the calculus of variations.
\index{Cauchy|)}%
This subject is now in its essential principles the same as
when it came from the hands of Lagrange. Recent studies
pertain to the variation of a double integral when the limits
are also variable, and to variations of multiple integrals in
general. Memoirs were published by Gauss in 1829, Poisson
\index{Gauss!ref.\ to}%
\index{Poisson!ref.\ to}%
in 1831, and Ostrogradsky of St.~Petersburg in 1834, without,
\index{Ostrogradsky}%
however, determining in a general manner the number and
form of the equations which must subsist at the limits in case
of a double or triple integral. In 1837 Jacobi published a
\index{Jacobi!ref.\ to}%
memoir, showing that the difficult integrations demanded by
the discussion of the second variation, by which the existence
of a maximum or minimum can be ascertained, are included
in the integrations of the first variation, and thus are superfluous.
This important theorem, presented with great brevity
by Jacobi, was elucidated and extended by V.~A. Lebesgue,
\index{Lebesgue}%
C.~E. Delaunay, Eisenlohr, S.~Spitzer, Hesse, and Clebsch. An
\index{Clebsch!ref.\ to}%
\index{Delaunay!ref.\ to}%
\index{Eisenlohr}%
\index{Hesse!ref.\ to}%
\index{Spitzer}%
important memoir by Sarrus on the question of determining
\index{Sarrus}%
the limiting equations which must be combined with the
indefinite equations in order to determine completely the
maxima and minima of multiple integrals, was awarded a
\index{Maxima and minima}%
%% -----File: 348.png---Folio 334-------
\index{Criteria of convergence|(}%
\index{Jellet}%
prize by the French Academy in 1845, honourable mention
being made of a paper by Delaunay. Sarrus's method was
\index{Delaunay!ref.\ to}%
simplified by Cauchy. In 1852 G.~Mainardi attempted to
\index{Mainardi}%
exhibit a new method of discriminating maxima and minima,
\index{Maxima and minima}%
and extended Jacobi's theorem to double integrals. Mainardi
and F.~Brioschi showed the value of determinants in exhibiting
\index{Brioschi!ref.\ to}%
\index{Determinants}%
the terms of the second variation. In 1861 \textit{Isaac Todhunter}
\index{Todhunter}%
(1820--1884) of St.~John's College, Cambridge, published
his valuable work on the \textit{History of the Progress of the Calculus
of Variations}, which contains researches of his own. In 1866
he published a most important research, developing the theory
of discontinuous solutions (discussed in particular cases by
Legendre), and doing for this subject what Sarrus had done
for multiple integrals.

The following are the more important authors of systematic
treatises on the calculus of variations, and the dates of publication:
\index{Calculus!of variations|)}%
Robert Woodhouse, Fellow of Caius College, Cambridge,
\index{Woodhouse}%
1810; Richard Abbatt in London, 1837; John Hewitt
\index{Abbatt}%
Jellett (1817--1888), once Provost of Trinity College, Dublin,
1850; G.~W. Strauch in \DPtypo{Zurich}{Zürich}, 1849; Moigno and Lindelöf,
\index{Lindelöf}%
\index{Moigno}%
\index{Strauch}%
1861; Lewis Buffett Carll of Flushing in New York, 1881.
\index{Carll}%

The lectures on definite integrals, delivered by Dirichlet in
\index{Definite integrals}%
\index{Dirichlet!ref.\ to}%
1858, have been elaborated into a standard work by G.~F.
Meyer. The subject has been treated most exhaustively by
\index{Meyer, G.~F.}%
D.~Bierens de Haan of Leiden in his \textit{Exposé de la théorie des
\index{Haan}%
intégrals définies}, Amsterdam, 1862.

The history of infinite series illustrates vividly the salient
\index{Infinite series|(}%
feature of the new era which analysis entered upon during the
\index{Analysis!modern|)}%
first quarter of this century. Newton and Leibniz felt the
\index{Leibniz!ref.\ to}%
\index{Newton!ref.\ to}%
necessity of inquiring into the convergence of infinite series,
\index{Convergence of series|(}%
but they had no proper criteria, excepting the test advanced
by Leibniz for alternating series. By Euler and his contemporaries
\index{Euler!ref.\ to}%
the \emph{formal} treatment of series was greatly extended,
%% -----File: 349.png---Folio 335-------
\index{Multiplication of series}%
while the necessity for determining the convergence was generally
lost sight of. Euler reached some very pretty results
on infinite series, now well known, and also some very
\index{Absolutely convergent series}%
absurd results, now quite forgotten. The faults of his time
found their culmination in the Combinatorial School in Germany,
\index{Combinatorial School}%
which has now passed into deserved oblivion. At the
beginning of the period now under consideration, the doubtful,
or plainly absurd, results obtained from infinite series stimulated
profounder inquiries into the validity of operations with
them. Their \emph{actual contents} came to be the primary, \emph{form} a
secondary, consideration. The first important and strictly
rigorous investigation of series was made by Gauss in connection
\index{Gauss!ref.\ to}%
with the hypergeometric series. The criterion developed
\index{Hypergeometric series}%
by him settles the question of convergence in every case
which it is intended to cover, and thus bears the stamp of
generality so characteristic of Gauss's writings. Owing to the
strangeness of treatment and unusual rigour, Gauss's paper
excited little interest among the mathematicians of that time.

More fortunate in reaching the public was Cauchy, whose
\index{Cauchy!ref.\ to}%
\textit{Analyse Algébrique} of 1821 contains a rigorous treatment of
series. All series whose sum does not approach a fixed limit
as the number of terms increases indefinitely are called divergent.
Like Gauss, he institutes comparisons with geometric
series, and finds that series with positive terms are convergent
or not, according as the $n$th root of the $n$th term, or the ratio
of the $(n + 1)$th term and the $n$th term, is ultimately less
or greater than unity. To reach some of the cases where
these expressions become ultimately unity and fail, Cauchy
established two other tests. He showed that series with negative
\Addendlabel{335}%
terms converge when the absolute values of the terms
converge, and then deduces Leibniz's test for alternating
series. The product of two convergent series was not found
to be necessarily convergent. Cauchy's theorem that the
%% -----File: 350.png---Folio 336-------
product of two absolutely convergent series converges to the
product of the sums of the two series was shown half a century
later by F.~Mertens of Graz to be still true if, of the two
\index{Mertens}%
convergent series to be multiplied together, only one is absolutely
convergent.

The most outspoken critic of the old methods in series
was Abel. His letter to his friend Holmboe (1826) contains
\index{Abel!ref.\ to}%
\index{Holmboe}%
severe criticisms. It is very interesting reading, even to
modern students. In his demonstration of the binomial theorem
he established the theorem that if two series and their
product series are all convergent, then the product series will
converge towards the product of the sums of the two given
series. This remarkable result would dispose of the whole
problem of multiplication of series if we had a universal
\index{Multiplication of series}%
practical criterion of convergency for semi-convergent series.
\index{Semi-convergent series}%
Since we do not possess such a criterion, theorems have been
recently established by A.~Pringsheim of Munich and A.~Voss
\index{Pringsheim|(}%
\index{Voss!ref.\ to}%
of Würzburg which remove in certain cases the necessity of
applying tests of convergency to the product series by the
application of tests to easier related expressions. Pringsheim
reaches the following interesting conclusions: The product
of two semi-convergent series can never converge absolutely,
but a semi-convergent series, or even a divergent series, multiplied
by an absolutely convergent series, \emph{may} yield an absolutely
convergent product.

The researches of Abel and Cauchy caused a considerable
stir. We are told that after a scientific meeting in which
Cauchy had presented his first researches on series, Laplace
\index{Laplace!ref.\ to}%
hastened home and remained there in seclusion until he had
examined the series in his \textit{Mécanique Céleste}. Luckily, every
one was found to be convergent! We must not conclude,
however, that the new ideas at once displaced the old. On the
contrary, the new views were generally accepted only after a
%% -----File: 351.png---Folio 337-------
severe and long struggle. As late as 1844 De~Morgan began a
\index{Demorgan@{De Morgan}!ref.\ to}%
paper on ``divergent series'' in this style: ``I believe it will
\index{Absolutely convergent series}%
\index{Divergent series}%
be generally admitted that the heading of this paper describes
the only subject yet remaining, of an elementary character, on
which a serious schism exists among mathematicians as to the
absolute correctness or incorrectness of results.''

First in time in the evolution of more delicate criteria of
convergence and divergence come the researches of Josef Ludwig
Raabe (\textit{Crelle}, Vol.~IX.); then follow those of De~Morgan
\index{Raabe}%
as given in his calculus. De~Morgan established the logarithmic
criteria which were discovered in part independently
by J.~Bertrand. The forms of these criteria, as given by
\index{Bertrand}%
Bertrand and by Ossian Bonnet, are more convenient than
\index{Bonnet, O.!ref.\ to}%
De~Morgan's. It appears from Abel's posthumous papers
\index{Abel!ref.\ to}%
that he had anticipated the above-named writers in establishing
logarithmic criteria. It was the opinion of Bonnet
\index{Logarithmic criteria of convergence}%
that the logarithmic criteria never fail; but Du~Bois-Reymond
\index{Bois-Reymond, P.~du|(}%
and Pringsheim have each discovered series demonstrably
convergent in which these criteria fail to determine
the convergence. The criteria thus far alluded to have
been called by Pringsheim \emph{special} criteria, because they all
depend upon a comparison of the $n$th term of the series
with special functions $a^n$, $n^x$, $n(\log n)^x$, etc. Among the
first to suggest \emph{general} criteria, and to consider the subject
from a still wider point of view, culminating in a regular
mathematical theory, was Kummer. He established a theorem
\index{Kummer!ref.\ to}%
yielding a test consisting of two parts, the first part of
which was afterwards found to be superfluous. The study
of general criteria was continued by U.~Dini of Pisa, Paul
\index{Dini}%
Du~Bois-Reymond, G.~Kohn of Minden, and Pringsheim.
\index{Kohn}%
Du~Bois-Reymond divides criteria into two classes: criteria
of the \emph{first kind} and criteria of the \emph{second kind}, according as
the general $n$th term, or the ratio of the $(n + 1)$th term and
%% -----File: 352.png---Folio 338-------
the $n$th~term, is made the basis of research. Kummer's is a
\index{Kummer!ref.\ to}%
criterion of the second kind. A criterion of the first kind,
analogous to this, was invented by Pringsheim. From the
general criteria established by Du~Bois-Reymond and Pringsheim
respectively, all the special criteria can be derived. The
theory of Pringsheim is very complete, and offers, in addition
\index{Pringsheim|)}%
to the criteria of the first kind and second kind, entirely new
criteria of a \emph{third kind}, and also generalised criteria of the
second kind, which apply, however, only to series with never
\index{Absolutely convergent series}%
increasing terms. Those of the third kind rest mainly on the
consideration of the limit of the difference either of consecutive
terms or of their reciprocals. In the generalised criteria
of the second kind he does not consider the ratio of two consecutive
terms, but the ratio of any two terms however far
apart, and deduces, among others, two criteria previously given
by Kohn and Ermakoff respectively.

Difficult questions arose in the study of Fourier's series.\Reference{79}
\index{Fourier's series}%
Cauchy was the first who felt the necessity of inquiring into
\index{Cauchy!ref.\ to}%
its convergence. But his mode of proceeding was found
by Dirichlet to be unsatisfactory. Dirichlet made the first
\index{Dirichlet!ref.\ to}%
thorough researches on this subject (\textit{Crelle}, Vol.~IV.). They
culminate in the result that whenever the function does not
become infinite, does not have an infinite number of discontinuities,
and does not possess an infinite number of
maxima and minima, then Fourier's series converges toward
the value of that function at all places, except points of
discontinuity, and there it converges toward the mean of
the two boundary values. Schläfli of Bern and Du~Bois-Reymond
\index{Schlafli@{Schläfli}!ref.\ to}%
expressed doubts as to the correctness of the mean
value, which were, however, not well founded. Dirichlet's
conditions are sufficient, but not necessary. Lipschitz, of
\index{Lipschitz!ref.\ to}%
Bonn, proved that Fourier's series still represents the function
when the number of discontinuities is infinite, and
%% -----File: 353.png---Folio 339-------
\index{Criteria of convergence|)}%
\index{Definite integrals}%
\index{Undulatory theory of light}%
established a condition on which it represents a function
having an infinite number of maxima and minima. Dirichlet's
\index{Dirichlet!ref.\ to}%
\index{Maxima and minima}%
belief that all continuous functions can be represented
by Fourier's series at all points was shared by Riemann and
\index{Fourier's series}%
\index{Riemann!ref.\ to}%
H.~Hankel, but was proved to be false by Du~Bois-Reymond
\index{Hankel!ref.\ to}%
and H.~A. Schwarz.
\index{Schwarz!ref.\ to}%

Riemann inquired what properties a function must have,
so that there may be a trigonometric series which, whenever
\index{Trigonometric series}%
it is convergent, converges toward the value of the function.
He found necessary and sufficient conditions for this. They
do not decide, however, whether such a series actually represents
the function or not. Riemann rejected Cauchy's definition
\index{Cauchy!ref.\ to}%
of a definite integral on account of its arbitrariness, gave
a new definition, and then inquired when a function has an
integral. His researches brought to light the fact that continuous
functions need not always have a differential coefficient.
But this property, which was shown by Weierstrass to
\index{Weierstrass!ref.\ to}%
belong to large classes of functions, was not found necessarily
to exclude them from being represented by Fourier's series.
Doubts on some of the conclusions about Fourier's series were
thrown by the observation, made by Weierstrass, that the
integral of an infinite series can be shown to be equal to the
\index{Infinite series|)}%
sum of the integrals of the separate terms only when the series
converges \emph{uniformly} within the region in question. The subject
of uniform convergence was investigated by Philipp Ludwig
\index{Convergence of series|)}%
Seidel (1848) and G.~G. Stokes (1847), and has assumed
\index{Seidel}%
\index{Stokes!ref.\ to}%
great importance in Weierstrass' theory of functions. It
became necessary to prove that a trigonometric series representing
a continuous function converges uniformly. This was
done by Heinrich Eduard Heine (1821--1881), of Halle. Later
\index{Heine}%
researches on Fourier's series were made by G.~Cantor and
\index{Cantor, G.}%
Du~Bois-Reymond.
\index{Bois-Reymond, P.~du|)}%

As compared with the vast development of other mathematical
%% -----File: 354.png---Folio 340-------
branches, the theory of probability has made very
\index{Probability}%
insignificant progress since the time of Laplace. Improvements
\index{Laplace!ref.\ to}%
and \DPtypo{simplications}{simplifications} in the mode of exposition have been
made by A.~De~Morgan, G.~Boole, A.~Meyer (edited by E.~Czuber),
\index{Boole!ref.\ to}%
\index{Czuber}%
\index{Demorgan@{De Morgan}!ref.\ to}%
\index{Meyer, A.}%
J.~Bertrand. Cournot's and Westergaard's treatment
\index{Bertrand}%
\index{Cournot}%
\index{Westergaard}%
of insurance and the theory of life-tables are classical. Applications
\index{Insurance}%
of the calculus to statistics have been made by L.~A.~J.
\index{Statistics}%
Quetelet (1796--1874), director of the observatory at Brussels;
\index{Quetelet}%
by Lexis; Harald Westergaard, of Copenhagen; and Düsing.
\index{Dusing@{Düsing}}%
\index{Lexis}%

Worthy of note is the rejection of inverse probability by the
\index{Inverse probability}%
best authorities of our time. This branch of probability had
been worked out by Thomas Bayes (died 1761) and by Laplace
\index{Bayes}%
(Bk.~II., Ch.~VI. of his \textit{Théorie Analytique}). By it some
logicians have explained induction. For example, if a man,
\index{Induction}%
who has never heard of the tides, were to go to the shore of the
Atlantic Ocean and witness on $m$~successive days the rise of the
sea, then, says Quetelet, he would be entitled to conclude that
there was a probability equal to $\dfrac{m+1}{m+2}$ that the sea would rise
next day. Putting $m = 0$, it is seen that this view rests upon
the unwarrantable assumption that the probability of a totally
unknown event is~$\frac{1}{2}$, or that of all theories proposed for investigation
one-half are true. W.~S. Jevons in his \textit{Principles of
\index{Jevons}%
Science} founds induction upon the theory of inverse probability,
and F.~Y. Edgeworth also accepts it in his \textit{Mathematical
\index{Edgeworth}%
Psychics}.

The only noteworthy recent addition to probability is the
subject of ``local probability,'' developed by several English
\index{Local probability}%
and a few American and French mathematicians. The earliest
problem on this subject dates back to the time of Buffon, the
\index{Buffon}%
naturalist, who proposed the problem, solved by himself and
Laplace, to determine the probability that a short needle,
thrown at random upon a floor ruled with equidistant parallel
%% -----File: 355.png---Folio 341-------
lines, will fall on one of the lines. Then came Sylvester's
\index{Sylvester!ref.\ to}%
four-point problem: to find the probability that four points,
\index{Four-point problem}%
\index{Probability}%
taken at random within a given boundary, shall form a re-entrant %[** PP: Only instance]
quadrilateral. Local probability has been studied in
England by A.~R. Clarke, H.~McColl, S.~Watson, J.~Wolstenholme,
\index{Clarke}%
\index{McColl}%
\index{Watson, S.}%
\index{Wolstenholme}%
but with greatest success by \textit{M.~W. Crofton} of the
\index{Crofton}%
military school at Woolwich. It was pursued in America by
E.~B. Seitz; in France by C.~Jordan, E.~Lemoine, E.~Barbier,
\index{Barbier}%
\index{Jordan!ref.\ to}%
\index{Lemoine}%
\index{Seitz}%
and others. Through considerations of local probability,
Crofton was led to the evaluation of certain definite integrals.
\index{Definite integrals}%

The first full scientific treatment of differential equations
\index{Differential equations|(}%
was given by Lagrange and Laplace. This remark is especially
true of partial differential equations. The latter were investigated
\index{Partial differential equations|etseq}%
in more recent time by Monge, Pfaff, Jacobi, Émile Bour
\index{Bour}%
\index{Jacobi!ref.\ to}%
\index{Monge!ref.\ to}%
\index{Pfaff}%
(1831--1866) of Paris, A.~Weiler, Clebsch, A.~N. Korkine of St.~Petersburg,
\index{Clebsch!ref.\ to}%
\index{Korkine!ref.\ to}%
\index{Weiler}%
G.~Boole, A.~Meyer, Cauchy, Serret, Sophus Lie,
\index{Boole!ref.\ to}%
\index{Cauchy!ref.\ to}%
\index{Lie!ref.\ to}%
\index{Meyer, A.}%
\index{Serret!ref.\ to}%
and others. In 1873 their \DPtypo{reseaches}{researches}, on partial differential
equations of the first order, were presented in text-book form
by Paul Mansion, of the University of Gand. The keen
\index{Mansion}%
researches of \textbf{Johann Friedrich Pfaff} (1795--1825) marked a
decided advance. He was an intimate friend of young Gauss
at Göttingen. Afterwards he was with the astronomer Bode.
\index{Bode}%
Later he became professor at Helmstädt, then at Halle. By a
peculiar method, Pfaff found the general integration of partial
differential equations of the first order for any number
of variables. Starting from the theory of ordinary differential
equations of the first order in $n$~variables, he gives first their
general integration, and then considers the integration of the
partial differential equations as a particular case of the former,
assuming, however, as known, the general integration of differential
equations of any order between two variables. His
researches led Jacobi to introduce the name ``Pfaffian problem.''
\index{Pfaffian problem}%
From the connection, observed by Hamilton, between
\index{Hamilton, W.~R.!ref.\ to}%
%% -----File: 356.png---Folio 342-------
\index{Maxima and minima}%
\index{Pfaffian problem}%
a system of ordinary differential equations (in analytical
mechanics) and a partial differential equation, Jacobi drew
the conclusion that, of the series of systems whose successive
integration Pfaff's method demanded, all but the first system
\index{Pfaff}%
were entirely superfluous. Clebsch considered Pfaff's problem
\index{Clebsch!ref.\ to}%
from a new point of view, and reduced it to systems of simultaneous
linear partial differential equations, which can be
established independently of each other without any integration.
Jacobi materially advanced the theory of differential
equations of the first order. The problem to determine unknown
functions in such a way that an integral containing
these functions and their differential coefficients, in a prescribed
manner, shall reach a maximum or minimum value,
demands, in the first place, the vanishing of the first variation
of the integral. This condition leads to differential equations,
the integration of which determines the functions. To ascertain
whether the value is a maximum or a minimum, the
second variation must be examined. This leads to new and
difficult differential equations, the integration of which, for
the simpler cases, was ingeniously deduced by Jacobi from
the integration of the differential equations of the first variation.
Jacobi's solution was perfected by Hesse, while Clebsch
\index{Hesse!ref.\ to}%
extended to the general case Jacobi's results on the second
variation. Cauchy gave a method of solving partial differential
\index{Cauchy!ref.\ to}%
equations of the first order having any number of
variables, which was corrected and extended by Serret, J.~Bertrand,
\index{Bertrand}%
\index{Serret!ref.\ to}%
O.~Bonnet in France, and Imschenetzky in Russia.
\index{Bonnet, O.!ref.\ to}%
\index{Imschenetzky}%
Fundamental is the proposition of Cauchy that every ordinary
differential equation admits in the vicinity of any non-singular
point of an integral, which is synectic within a certain circle
of convergence, and is developable by Taylor's theorem.
\index{Taylor's theorem}%
Allied to the point of view indicated by this theorem is that
of Riemann, who regards a function of a single variable as
\index{Riemann!ref.\ to}%
%% -----File: 357.png---Folio 343-------
\index{Logic}%
defined by the position and nature of its singularities, and
who has applied this conception to that linear differential
equation of the second order, which is satisfied by the hypergeometric
series. This equation was studied also by Gauss
\index{Gauss!ref.\ to}%
and Kummer. Its general theory, when no restriction is
\index{Kummer!ref.\ to}%
imposed upon the value of the variable, has been considered
by J.~Tannery, of Paris, who employed Fuchs' method of
\index{Fuchs}%
\index{Tannery}%
linear differential equations and found all of Kummer's
twenty-four integrals of this equation. This study has been
continued by Édouard Goursat of Paris.
\index{Goursat}%

A standard text-book on \textit{Differential Equations}, including
original matter on integrating factors, singular solutions, and
especially on symbolical methods, was prepared in 1859 by
\textbf{George Boole} (1815--1864), at one time professor in Queen's
\index{Boole}%
University, Cork, Ireland. He was a native of Lincoln, and a
self-educated mathematician of great power. His treatise on
\textit{Finite Differences} (1860) and his \textit{Laws of Thought} (1854) are
\index{Finite differences}%
works of high merit.

The fertility of the conceptions of Cauchy and Riemann
with regard to differential equations is attested by the
researches to which they have given rise on the part of
\textbf{Lazarus Fuchs} of Berlin (born 1835), \textbf{Felix Klein} of Göttingen
\index{Klein}%
(born 1849), \textbf{Henri Poincaré} of Paris (born 1854), and others.
\index{Poincaré}%
The study of linear differential equations entered a new
period with the publication of Fuchs' memoirs of 1866
and~1868. Before this, linear equations with \emph{constant} coefficients
were almost the only ones for which general methods
of integration were known. While the general theory of
these equations has recently been presented in a new light
by Hermite, Darboux, and Jordan, Fuchs began the study
\index{Darboux}%
\index{Hermite!ref.\ to}%
\index{Jordan!ref.\ to}%
from the more general standpoint of the linear differential
equations whose coefficients are not constant. He directed
his attention mainly to those whose integrals are all regular.
%% -----File: 358.png---Folio 344-------
\index{Invariant}%
If the variable be made to describe all possible paths enclosing
one or more of the critical points of the equation, we
have a certain substitution corresponding to each of the
paths; the aggregate of all these substitutions being called
a \emph{group}. The forms of integrals of such equations were
examined by Fuchs and by G.~Frobenius by independent
\index{Frobenius}%
\index{Fuchs!ref.\ to}%
methods. Logarithms generally appear in the integrals of
a group, and Fuchs and Frobenius investigated the conditions
under which no logarithms shall appear. Through the study
of groups the reducibility or irreducibility of linear differential
\index{Groups, theory of|(}%
equations has been examined by Frobenius and Leo
Königsberger. The subject of linear differential equations,
\index{Konigsberger@{Königsberger}!ref.\ to}%
not all of whose integrals are regular, has been attacked by
G.~Frobenius of Berlin, W.~Thomé of Greifswald (born 1841),
\index{Thome@{Thomé}}%
and Poincaré, but the resulting theory of irregular integrals
\index{Irregular integrals}%
is as yet in very incomplete form.

The theory of invariants associated with linear differential
equations has been developed by Halphen and by A.~R. Forsyth.
\index{Forsyth}%
\index{Halphen!ref.\ to}%

The researches above referred to are closely connected with
the theory of functions and of groups. Endeavours have thus
\index{Theory of functions}%
been made to determine the nature of the function defined by
a differential equation from the differential equation itself,
and not from any analytical expression of the function, obtained
first by solving the differential equation. Instead of studying
the properties of the integrals of a differential equation for all
the values of the variable, investigators at first contented themselves
with the study of the properties in the vicinity of a
given point. The nature of the integrals at singular points
and at ordinary points is entirely different. \textit{Albert Briot}
\index{Briot}%
(1817--1882) and \textit{Jean Claude Bouquet} (1819--1885), both of
\index{Bouquet}%
Paris, studied the case when, near a singular point, the differential
equations take the form $(x-x_0) \dfrac{dy}{dx} = \displaystyle\int (xy)$. Fuchs
%% -----File: 359.png---Folio 345-------
\index{Kleinian groups}%
\index{Theory of functions}%
gave the development in series of the integrals for the particular
case of linear equations. Poincaré did the same for the
\index{Poincaré!ref.\ to}%
case when the equations are not linear, as also for partial
differential equations of the first order. The developments for
ordinary points were given by Cauchy and Madame Kowalevsky.
\index{Cauchy!ref.\ to}%
\index{Kowalevsky!ref.\ to}%

The attempt to express the integrals by developments that
are always convergent and not limited to particular points in
a plane necessitates the introduction of new transcendents, for
the old functions permit the integration of only a small number
\index{Functions!theory of}%
of differential equations. Poincaré tried this plan with
linear equations, which were then the best known, having
been studied in the vicinity of given points by Fuchs, Thomé,
\index{Fuchs!ref.\ to}%
\index{Thome@{Thomé}!ref.\ to}%
Frobenius, Schwarz, Klein, and Halphen. Confining himself to
\index{Frobenius}%
\index{Halphen!ref.\ to}%
\index{Klein!ref.\ to}%
\index{Schwarz!ref.\ to}%
those with rational algebraical coefficients, Poincaré was able
to integrate them by the use of functions named by him \emph{Fuchsians}.\Reference{81}
He divided these equations into ``families.'' If the
integral of such an equation be subjected to a certain transformation,
the result will be the integral of an equation
belonging to the same family. The new transcendents have a
great analogy to elliptic functions; while the region of the
\index{Elliptic functions}%
latter may be divided into parallelograms, each representing a
group, the former may be divided into curvilinear polygons,
so that the knowledge of the function inside of one polygon
carries with it the knowledge of it inside the others. Thus
Poincaré arrives at what he calls \emph{Fuchsian groups}. He found,
\index{Fuchsian groups}%
moreover, that Fuchsian functions can be expressed as the
\index{Fuchsian functions}%
ratio of two transcendents (theta-fuchsians) in the same way
\index{Theta-fuchsians}%
that elliptic functions can be. If, instead of linear substitutions
with real coefficients, as employed in the above groups,
imaginary coefficients be used, then discontinuous groups are
obtained, which he called \emph{Kleinians}. The extension to non-linear %[** PP: Only instance]
equations of the method thus applied to linear equations
has been begun by Fuchs and Poincaré.
%% -----File: 360.png---Folio 346-------
\index{Abelian functions}%
\index{Algebraic functions}%
\index{Elliptic functions}%

We have seen that among the earliest of the several kinds
of ``groups'' are the finite discontinuous groups (groups in
the theory of substitution), which since the time of Galois
have become the leading concept in the theory of algebraic
equations; that since 1876 Felix Klein, H.~Poincaré, and
\index{Klein!ref.\ to}%
\index{Poincaré!ref.\ to}%
others have applied the theory of finite and infinite discontinuous
groups to the theory of functions and of differential
\index{Theory of functions}%
equations. The finite continuous groups were first made
\index{Groups, theory of|)}%
the subject of general research in 1873 by Sophus Lie, now
\index{Lie}%
of Leipzig, and applied by him to the integration of ordinary
linear partial differential equations.

Much interest attaches to the determination of those linear
differential equations which can be integrated by simpler
functions, such as algebraic, elliptic, or Abelian. This has
been studied by C.~Jordan, P.~Appel of Paris (born 1858), and
\index{Appel}%
\index{Jordan!ref.\ to}%
Poincaré.

The mode of integration above referred to, which makes
known the properties of equations from the standpoint of the
theory of functions, does not suffice in the application of
differential equations to questions of mechanics. If we consider
\index{Mechanics!more recent work}%
the function as defining a plane curve, then the general
form of the curve does not appear from the above mode of
investigation. It is, however, often desirable to construct
the curves defined by differential equations. Studies having
this end in view have been carried on by Briot and Bouquet,
\index{Bouquet!ref.\ to}%
\index{Briot!ref.\ to}%
and by Poincaré.\Reference{81}

The subject of singular solutions of differential equations
has been materially advanced since the time of Boole by G.~Darboux
\index{Boole!ref.\ to}%
\index{Darboux}%
and Cayley. The papers prepared by these mathematicians
\index{Cayley!ref.\ to}%
point out a difficulty as yet unsurmounted: whereas
a singular solution, from the point of view of the integrated
equation, ought to be a phenomenon of universal, or at least of
general occurrence, it is, on the other hand, a very special and
%% -----File: 361.png---Folio 347-------
exceptional phenomenon from the point of view of the differential
equation.\Reference{89} A geometrical theory of singular solutions
resembling the one used by Cayley was previously employed
by W.~W. Johnson of Annapolis.
\index{Johnson}%

An advanced \textit{Treatise on Linear Differential Equations}
(1889) was brought out by Thomas Craig of the Johns Hopkins
\index{Craig, T.}%
University. He chose the algebraic method of presentation
followed by Hermite and Poincaré, instead of the geometric
\index{Hermite!ref.\ to}%
\index{Poincaré!ref.\ to}%
method preferred by Klein and Schwarz. A notable work, the
\index{Klein!ref.\ to}%
\index{Schwarz!ref.\ to}%
\textit{Traité d'Analyse}, is now being published by Émile Picard of
\index{Picard, E.}%
Paris, the interest of which is made to centre in the subject of
differential equations.
\index{Differential equations|)}%


\Section{Theory of Functions}
\index{Theory of functions|(}%

We begin our sketch of the vast progress in the theory of
functions by considering the special class called elliptic functions.
\index{Elliptic functions|(}%
These were richly developed by Abel and Jacobi.
\index{Abel}%
\index{Jacobi!ref.\ to}%

\textbf{Niels Henrick Abel} (1802--1829) was born at Findoë in Norway,
and was prepared for the university at the cathedral
school in Christiania. He exhibited no interest in mathematics
until 1818, when B.~Holmboe became lecturer there,
\index{Holmboe}%
and aroused Abel's interest by assigning original problems
to the class. Like Jacobi and many other young men who
became eminent mathematicians, Abel found the first exercise
of his talent in the attempt to solve by algebra the general
equation of the fifth degree. In 1821 he entered the University
in Christiania. The works of Euler, Lagrange, and
Legendre were closely studied by him. The idea of the inversion
of elliptic functions dates back to this time. His extraordinary
success in mathematical study led to the offer of a
stipend by the government, that he might continue his studies
%% -----File: 362.png---Folio 348-------
\index{Abelian functions}%
\index{Binomial formula}%
\index{Integral calculus}%
in Germany and France. Leaving Norway in 1825, Abel visited
\index{Abel}%
the astronomer, Schumacher, in Hamburg, and spent six
\index{Schumacher!ref.\ to}%
months in Berlin, where he became intimate with \textbf{August
Leopold Crelle} (1780--1855), and met Steiner. Encouraged by
\index{Crelle}%
\index{Steiner!ref.\ to}%
Abel and Steiner, Crelle started his journal in 1826. Abel
began to put some of his work in shape for print. His proof
of the impossibility of solving the general equation of the fifth
degree by radicals,---first printed in 1824 in a very concise
form, and difficult of apprehension,---was elaborated in greater
detail, and published in the first volume. He entered also
upon the subject of infinite series (particularly the binomial
\index{Infinite series}%
theorem, of which he gave in \textit{Crelle's Journal} a rigid general
investigation), the study of functions, and of the integral
calculus. The obscurities everywhere encountered by him
owing to the prevailing loose methods of analysis he endeavoured
to clear up. For a short time he left Berlin for Freiberg,
where he had fewer interruptions to work, and it was
there that he made researches on hyperelliptic and Abelian
\index{Hyperelliptic functions}%
functions. In July, 1826, Abel left Germany for Paris without
having met Gauss! Abel had sent to Gauss his proof of
\index{Gauss!ref.\ to}%
1824 of the impossibility of solving equations of the fifth
\index{Equations!solution of}%
degree, to which Gauss never paid any attention. This slight,
and a haughtiness of spirit which he associated with Gauss,
prevented the genial Abel from going to Göttingen. A similar
feeling was entertained by him later against Cauchy. Abel
\index{Cauchy!ref.\ to}%
remained ten months in Paris. He met there Dirichlet,
\index{Dirichlet!ref.\ to}%
Legendre, Cauchy, and others; but was little appreciated.
He had already published several important memoirs in
\textit{Crelle's Journal}, but by the French this new periodical was
as yet hardly known to exist, and Abel was too modest to
speak of his own work. Pecuniary embarrassments induced
him to return home after a second short stay in Berlin. At
Christiania he for some time gave private lessons, and served
%% -----File: 363.png---Folio 349-------
\index{Elliptic integrals}%
\index{Periodicity of functions}%
as docent. Crelle secured at last an appointment for him at
\index{Crelle!ref.\ to}%
Berlin; but the news of it did not reach Norway until after
the death of Abel at Froland.\Reference{82}

At nearly the same time with Abel, Jacobi published articles
\index{Jacobi!ref.\ to}%
on elliptic functions. Legendre's favourite subject, so long
\index{Legendre!ref.\ to}%
neglected, was at last to be enriched by some extraordinary
discoveries. The advantage to be derived by inverting the
elliptic integral of the first kind and treating it as a function
of its amplitude (now called elliptic function) was recognised
by Abel, and a few months later also by Jacobi. A second
fruitful idea, also arrived at independently by both, is the
introduction of imaginaries leading to the observation that
the new functions simulated at once trigonometric and exponential
functions. For it was shown that while trigonometric
functions had only a real period, and exponential only an imaginary,
\index{Imaginary quantities}%
elliptic functions had both sorts of periods. These two
discoveries were the foundations upon which Abel and Jacobi,
each in his own way, erected beautiful new structures. Abel
developed the curious expressions representing elliptic functions
by infinite series or quotients of infinite products.
\index{Infinite products}%
\index{Infinite series}%
Great as were the achievements of Abel in elliptic functions,
they were eclipsed by his researches on what are now called
Abelian functions. Abel's theorem on these functions was
\index{Abelian functions}%
given by him in several forms, the most general of these
being that in his \textit{Mémoire sur une propriété générale d'une
classe très-étendue de fonctions transcendentes} (1826). The history
of this memoir is interesting. A few months after his
arrival in Paris, Abel submitted it to the French Academy.
Cauchy and Legendre were appointed to examine it; but said
\index{Cauchy!ref.\ to}%
nothing about it until after Abel's death. In a brief statement
of the discoveries in question, published by Abel in \textit{Crelle's
Journal}, 1829, reference is made to that memoir. This led
Jacobi to inquire of Legendre what had become of it. Legendre
%% -----File: 364.png---Folio 350-------
says that the manuscript was so badly written as to be
illegible, and that Abel was asked to hand in a better copy,
\index{Abel!ref.\ to}%
which he neglected to do. The memoir remained in Cauchy's
\index{Cauchy!ref.\ to}%
hands. It was not published until 1841. By a singular mishap,
the manuscript was lost before the proof-sheets were
read.

In its form, the contents of the memoir belongs to the integral
calculus. Abelian integrals depend upon an irrational
\index{Abelian integrals}%
function $y$ which is connected with $x$ by an algebraic equation
$F(x, y) = 0$. Abel's theorem asserts that a sum of such
\index{Addition theorem of elliptic integrals}%
integrals can be expressed by a definite number $p$ of similar
integrals, where $p$ depends merely on the properties of the
equation $F(x, y) = 0$. It was shown later that $p$ is the deficiency
of the curve $F(x, y) = 0$. The addition theorems of
elliptic integrals are deducible from Abel's theorem. The
\index{Elliptic integrals}%
hyperelliptic integrals introduced by Abel, and proved by him
to possess multiple periodicity, are special cases of Abelian
\index{Periodicity of functions}%
integrals whenever $p= \text{or} > 3$. The reduction of Abelian to
elliptic integrals has been studied mainly by Jacobi, Hermite,
\index{Hermite!ref.\ to}%
\index{Jacobi!ref.\ to}%
Königsberger, Brioschi, Goursat, E.~Picard, and O.~Bolza of
\index{Bolza}%
\index{Brioschi!ref.\ to}%
\index{Goursat!ref.\ to}%
\index{Konigsberger@{Königsberger}!ref.\ to}%
\index{Picard, E.}%
the University of Chicago.

Two editions of Abel's works have been published: the first
by Holmboe in 1839, and the second by Sylow and Lie in
\index{Holmboe}%
\index{Lie!ref.\ to}%
\index{Sylow!ref.\ to}%
1881.

Abel's theorem was pronounced by Jacobi the greatest discovery
of our century on the integral calculus. The aged
\index{Integral calculus}%
Legendre, who greatly admired Abel's genius, called it ``\textit{monumentum
\index{Legendre!ref.\ to}%
aere perennius}.'' During the few years of work
allotted to the young Norwegian, he penetrated new fields of
research, the development of which has kept mathematicians
busy for over half a century.

Some of the discoveries of Abel and Jacobi were anticipated
by Gauss. In the \textit{Disquisitiones Arithmeticæ} he observed
\index{Gauss!ref.\ to}%
%% -----File: 365.png---Folio 351-------
that the principles which he used in the division of the circle
were applicable to many other functions, besides the circular,
and particularly to the transcendents dependent on the integral
$\displaystyle\int \frac{dx}{\sqrt{1-x^4}}$. From this Jacobi\Reference[]{83} concluded that Gauss had
\index{Gauss!ref.\ to}%
\index{Jacobi|(}%
thirty years earlier considered the nature and properties of
elliptic functions and had discovered their double periodicity.
The papers in the collected works of Gauss confirm this conclusion.

\textbf{Carl Gustav Jacob Jacobi}\Reference[]{84} (1804--1851) was born of Jewish
parents at Potsdam. Like many other mathematicians he was
initiated into mathematics by reading Euler. At the University
of Berlin, where he pursued his mathematical studies
independently of the lecture courses, he took the degree of
Ph.D. in 1825. After giving lectures in Berlin for two years,
he was elected extraordinary professor at Königsberg, and two
years later to the ordinary professorship there. After the
publication of his \textit{Fundamenta Nova} he spent some time in
travel, meeting Gauss in Göttingen, and Legendre, Fourier,
\index{Fourier!ref.\ to}%
\index{Legendre!ref.\ to}%
Poisson, in Paris. In 1842 he and his colleague, Bessel, attended
\index{Bessel!ref.\ to}%
\index{Poisson!ref.\ to}%
the meetings of the British Association, where they
made the acquaintance of English mathematicians.

His early researches were on Gauss' approximation to the
value of definite integrals, partial differential equations, Legendre's
\index{Definite integrals}%
coefficients, and cubic residues. He read Legendre's
\textit{Exercises}, which give an account of elliptic integrals. When
he returned the book to the library, he was depressed in spirits
and said that important books generally excited in him new
ideas, but that this time he had not been led to a single original
thought. Though slow at first, his ideas flowed all the
richer afterwards. Many of his discoveries in elliptic functions
were made independently by Abel. Jacobi communicated
his first researches to \textit{Crelle's Journal}. In 1829, at the age
%% -----File: 366.png---Folio 352-------
\index{Kowalevsky!ref.\ to}%
of twenty-five, he published his \textit{Fundamenta Nova Theoriæ
Functionum Ellipticarum}, which contains in condensed form
the main results in elliptic functions. This work at once
secured for him a wide reputation. He then made a closer
study of theta-functions and lectured to his pupils on a new
\index{Theta-functions}%
theory of elliptic functions based on the theta-functions. He
developed a theory of transformation which led him to a multitude
of formulæ containing~$q$, a transcendental function of
the modulus, defined by the equation $q = e^{-\pi k'/k}$. He was also
led by it to consider the two new functions $\Eta$~and~$\Theta$, which
taken each separately with two different arguments are the
four (single) theta-functions designated by the $\Theta_1$, $\Theta_2$, $\Theta_3$, $\Theta_4$.\Reference{56}
In a short but very important memoir of 1832, he shows that
for the hyperelliptic integral of any class the direct functions
\index{Hyperelliptic integrals}%
to which Abel's theorem has reference are not functions of a
\index{Abel's theorem}%
single variable, such as the elliptic $\sn$,~$\cn$,~$\dn$, but functions of
$p$~variables.\Reference{56} Thus in the case $p = 2$, which Jacobi especially
considers, it is shown that Abel's theorem has reference to
two functions $\lambda(u, v)$, $\lambda_1(u, v)$, each of two variables, and
gives in effect an addition-theorem for the expression of the
functions $\lambda(u + u', v + v')$, $\lambda_1(u + u', v + v')$ algebraically in
terms of the functions $\lambda(u, v)$, $\lambda_1(u, v)$, $\lambda(u', v')$, $\lambda_1(u', v')$. By
the memoirs of Abel and Jacobi it may be considered that the
notion of the Abelian function of $p$~variables was established
and the addition-theorem for these functions given. Recent
studies touching Abelian functions have been made by Weierstrass,
\index{Abelian functions}%
\index{Weierstrass!ref.\ to}%
E.~Picard, Madame Kowalevski, and Poincaré. Jacobi's
\index{Poincaré!ref.\ to}%
work on differential equations, determinants, dynamics, and
the theory of numbers is mentioned elsewhere.

In 1842 Jacobi visited Italy for a few months to recuperate
\index{Jacobi|)}%
his health. At this time the Prussian government gave him
a pension, and he moved to Berlin, where the last years of his
life were spent.
%% -----File: 367.png---Folio 353-------

The researches on functions mentioned thus far have been
greatly extended. In 1858 \textbf{Charles Hermite} of Paris (born 1822),
\index{Hermite}%
introduced in place of the variable $q$ of Jacobi a new variable $\omega$
\index{Jacobi!ref.\ to}%
connected with it by the equation $q = e^{i\pi\omega}$, so that $\omega = ik'/k$, and
was led to consider the functions $\phi(\omega)$, $\psi(\omega)$, $\chi(\omega)$.\Reference{56} Henry
Smith regarded a theta-function with the argument equal to
\index{Smith, H.!ref.\ to}%
zero, as a function of $\omega$. This he called an omega-function,
\index{Omega-function}%
while the three functions $\phi(\omega)$, $\psi(\omega)$, $\chi(\omega)$, are his modular
functions. Researches on theta-functions with respect to real
\index{Theta-functions}%
and imaginary arguments have been made by Meissel of Kiel,
\index{Meissel}%
J.~Thomae of Jena, Alfred Enneper of Göttingen (1830--1885).
\index{Enneper}%
\index{Thomae}%
A general formula for the product of two theta-functions was
given in 1854 by H.~Schröter of Breslau (1829--1892). These
\index{SchroterH@{Schröter, H.}!ref.\ to}%
functions have been studied also by Cauchy, Königsberger of
\index{Cauchy!ref.\ to}%
\index{Konigsberger@{Königsberger}}%
Heidelberg (born 1837), F.~S. Richelot of Königsberg (1808--1875),
\index{Richelot}%
Johann Georg Rosenhain of Königsberg (1816--1887),
\index{Rosenhain!ref.\ to}%
L.~Schläfli of Bern (born 1818).\Reference{85}
\index{Schlafli@{Schläfli}!ref.\ to}%

Legendre's method of reducing an elliptic differential to its
\index{Legendre!ref.\ to}%
normal form has called forth many investigations, most important
of which are those of Richelot and of Weierstrass of
\index{Weierstrass!ref.\ to}%
Berlin.

The algebraic transformations of elliptic functions involve
a relation between the old modulus and the new one which
Jacobi expressed by a differential equation of the third order,
and also by an algebraic equation, called by him ``modular
equation.'' The notion of modular equations was familiar to
\index{Modular equations}%
Abel, but the development of this subject devolved upon later
\index{Abel!ref.\ to}%
investigators. These equations have become of importance in
the theory of algebraic equations, and have been studied by
Sohnke, E.~Mathieu, L.~Königsberger, E.~Betti of Pisa (died
\index{Betti}%
\index{Mathieu!ref.\ to}%
\index{Sohnke}%
1892), C.~Hermite of Paris, Joubert of Angers, Francesco
\index{Joubert}%
Brioschi of Milan, Schläfli, H.~Schröter, M.~Gudermann of
\index{Brioschi!ref.\ to}%
\index{Gudermann}%
Cleve, Gützlaff.
\index{Gutzlaff@{Gützlaff}}%
%% -----File: 368.png---Folio 354-------
\index{Invariant}%
\index{Primary factors, Weierstrass' theory of}%

Felix Klein of Göttingen has made an extensive study of
\index{Klein!ref.\ to}%
modular functions, dealing with a type of operations lying
\index{Modular functions}%
between the two extreme types, known as the theory of substitutions
\index{Theory of substitutions}%
and the theory of invariants and covariants. Klein's
\index{Covariants}%
theory has been presented in book-form by his pupil, Robert
Fricke. The bolder features of it were first published in his
\index{Fricke}%
\textit{Ikosaeder}, 1884. His researches embrace the theory of modular
functions as a specific class of elliptic functions, the
statement of a more general problem as based on the doctrine
of groups of operations, and the further development of the
subject in connection with a class of Riemann's surfaces.
\index{Riemann!ref.\ to}%

The elliptic functions were expressed by Abel as quotients
of doubly infinite products. He did not, however, inquire
\index{Infinite products}%
rigorously into the convergency of the products. In 1845
Cayley studied these products, and found for them a complete
\index{Cayley!ref.\ to}%
theory, based in part upon geometrical interpretation, which
he made the basis of the whole theory of elliptic functions.
Eisenstein discussed by purely analytical methods the general
\index{Eisenstein!ref.\ to}%
doubly infinite product, and arrived at results which have
been greatly simplified in form by the theory of primary
factors, due to Weierstrass. A certain function involving a
\index{Weierstrass!ref.\ to}%
doubly infinite product has been called by Weierstrass the
sigma-function, and is the basis of his beautiful theory of
\index{Sigma-function}%
elliptic functions. The first systematic presentation of Weierstrass'
theory of elliptic functions was published in 1886 by
G.~H. Halphen in his \textit{Théorie des fonctions elliptiques et des
\index{Halphen!ref.\ to}%
leurs applications}. Applications of these functions have been
given also by A.~G. Greenhill. Generalisations analogous to
\index{Greenhill}%
those of Weierstrass on elliptic functions have been made
by Felix Klein on hyperelliptic functions.
\index{Hyperelliptic functions}%

Standard works on elliptic functions have been published by
\index{Elliptic functions|)}%
\textit{Briot and Bouquet} (1859), by \textit{Königsberger}, \textit{Cayley}, \textit{Heinrich
\index{Bouquet!ref.\ to}%
\index{Briot!ref.\ to}%
\index{Konigsberger@{Königsberger}!ref.\ to}%
Durège} of Prague (1821--1893), and others.
\index{Durège}%
%% -----File: 369.png---Folio 355-------

Jacobi's work on Abelian and theta-functions was greatly
\index{Theta-functions}%
extended by \textbf{Adolph Göpel} (1812--1847), professor in a gymnasium
\index{Gopel@{Göpel}}%
near Potsdam, and \textbf{Johann Georg Rosenhain} of Königsberg
\index{Rosenhain}%
(1816--1887). Göpel in his \Lat{\textit{Theoriæ transcendentium primi
ordinis adumbratio levis}} (\textit{Crelle},~35, 1847) and Rosenhain in
several memoirs established each independently, on the analogy
of the single theta-functions, the functions of two variables,
called double theta-functions, and worked out in connection
with them the theory of the Abelian functions of two variables.
\index{Abelian functions|(}%
The theta-relations established by Göpel and Rosenhain received
for thirty years no further development, notwithstanding
the fact that the double theta series came to be of increasing
importance in analytical, geometrical, and mechanical problems,
and that Hermite and Königsberger had considered the
\index{Hermite!ref.\ to}%
\index{Konigsberger@{Königsberger}!ref.\ to}%
subject of transformation. Finally, the investigations of C.~W.
Borchardt of Berlin (1817--1880), treating of the representation
\index{Borchardt}%
of Kummer's surface by Göpel's biquadratic relation between
\index{Kummer!ref.\ to}%
four theta-functions of two variables, and researches of H.~H.
Weber of Marburg, F.~Prym of Würzburg, Adolf Krazer, and
\index{Krazer}%
\index{Prym}%
\index{Weber, H.~H.}%
Martin Krause of Dresden led to broader views. Researches
\index{Krause}%
on double theta-functions, made by Cayley, were extended to
\index{Cayley!ref.\ to}%
quadruple theta-functions by Thomas Craig of the Johns
\index{Craig, T.}%
Hopkins University.

Starting with the integrals of the most general form and
considering the inverse functions corresponding to these integrals
(the Abelian functions of $p$~variables), \textit{Riemann}
\index{Riemann!ref.\ to}%
defined the theta-functions of $p$~variables as the sum of a
$p$\nobreakdash-tuply infinite series of exponentials, the general term depending
on $p$~variables. Riemann shows that the Abelian
functions are algebraically connected with theta-functions of
the proper arguments, and presents the theory in the broadest
form.\Reference{56} He rests the theory of the multiple theta-functions
upon the general principles of the theory of functions of a
complex variable.
%% -----File: 370.png---Folio 356-------
\index{Bernoulli, James (born 1758)}%
\index{Bernoulli, John (born 1667)!ref.\ to}%
\index{Riemann's surfaces!ref.\ to}%

Through the researches of A.~Brill of Tübingen, M.~Nöther
\index{Brill, A.}%
\index{Nother@{Nöther}}%
of Erlangen, and Ferdinand Lindemann of Munich, made
\index{Lindemann!ref.\ to}%
in connection with Riemann-Roch's theorem and the theory
\index{Riemann!ref.\ to}%
of residuation, there has grown out of the theory of Abelian
functions a theory of algebraic functions and point-groups on
\index{Functions!definition of}%
\index{Functions!theory of|(}%
algebraic curves.

Before proceeding to the general theory of functions, we
make mention of the ``calculus of functions,'' studied chiefly
\index{Calculus!of variations}%
by C.~Babbage, J.~F.~W. Herschel, and De~Morgan, which was
\index{Babbage}%
\index{Demorgan@{De Morgan}!ref.\ to}%
\index{Herschel, J.~F.~W.!ref.\ to}%
not so much a theory of functions as a theory of the solution
of functional equations by means of known functions or
symbols.

The history of the general theory of functions begins with
the adoption of new definitions of a function. With the
Bernoullis and Leibniz, $y$ was called a function of~$x$, if there
\index{Leibniz!ref.\ to}%
existed an equation between these variables which made it
possible to calculate~$y$ for any given value of~$x$ lying anywhere
between $-\infty$~and~$+\infty$. The study of Fourier's theory
\index{Fourier!ref.\ to}%
of heat led Dirichlet to a new definition: $y$ is called a function
\index{Dirichlet!ref.\ to}%
of~$x$, if $y$ possess one or more definite values for each of certain
values that $x$ is assumed to take in an interval $x_0$~to~$x_1$. In
functions thus defined, there need be no analytical connection
between $y$~and~$x$, and it becomes necessary to look for possible
discontinuities. A great revolution in the ideas of a function
was brought about by Cauchy when, in a function as defined
\index{Cauchy!ref.\ to}%
by Dirichlet, he gave the variables imaginary values, and when
he extended the notion of a definite integral by letting the
variable pass from one limit to the other by a succession of
imaginary values along arbitrary paths. Cauchy established
several fundamental theorems, and gave the first great impulse
to the study of the general theory of functions. His researches
were continued in France by Puiseux and Liouville. But more
\index{Liouville!ref.\ to}%
\index{Puiseux}%
profound investigations were made in Germany by Riemann.
%% -----File: 371.png---Folio 357-------

\textbf{Georg Friedrich Bernhard Riemann} (1826--1866) was born at
\index{Riemann|(}%
Breselenz in Hanover. His father wished him to study
theology, and he accordingly entered upon philological and
theological studies at Göttingen. He attended also some lectures
on mathematics. Such was his predilection for this
science that he abandoned theology. After studying for a
time under Gauss and Stern, he was drawn, in 1847, to Berlin
\index{Gauss!ref.\ to}%
\index{Stern}%
by a galaxy of mathematicians, in which shone Dirichlet,
\index{Dirichlet!ref.\ to}%
Jacobi, Steiner, and Eisenstein. Returning to Göttingen in
\index{Eisenstein!ref.\ to}%
\index{Jacobi!ref.\ to}%
\index{Steiner!ref.\ to}%
1850, he studied physics under Weber, and obtained the
\index{Weber, W.~E.!ref.\ to}%
doctorate the following year. The thesis presented on that
occasion, \textit{Grundlagen für eine allgemeine Theorie der Funktionen
einer veränderlichen complexen Grösse}, excited the admiration of
Gauss to a very unusual degree, as did also Riemann's trial
lecture, \textit{Ueber die Hypothesen welche der Geometrie zu Grunde
liegen}. Riemann's Habilitationsschrift was on the Representation
of a Function by means of a Trigonometric Series, in
\index{Trigonometric series}%
which he advanced materially beyond the position of Dirichlet.
Our hearts are drawn to this extraordinarily gifted but
shy genius when we read of the timidity and nervousness
displayed when he began to lecture at Göttingen, and of his
jubilation over the unexpectedly large audience of eight students
at his first lecture on differential equations.

Later he lectured on Abelian functions to a class of three
\index{Abelian functions|)}%
only,---Schering, Bjerknes, and Dedekind. Gauss died in 1855,
\index{Bjerknes, C. A.}%
\index{Dedekind!ref.\ to}%
\index{Schering!ref.\ to}%
and was succeeded by Dirichlet. On the death of the latter,
in 1859, Riemann was made ordinary professor. In 1860 he
visited Paris, where he made the acquaintance of French
mathematicians. The delicate state of his health induced
him to go to Italy three times. He died on his last trip at
Selasca, and was buried at Biganzolo.

Like all of Riemann's researches, those on functions were
profound and far-reaching. He laid the foundation for a
%% -----File: 372.png---Folio 358-------
\index{Kelvin, Lord!ref.\ to}%
general theory of functions of a complex variable. The theory
of potential, which up to that time had been used only in
\index{Potential}%
mathematical physics, was applied by him in pure mathematics.
He accordingly based his theory of functions on the
partial differential equation, $\dfrac{\partial^2u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} = \Delta u = 0$, which must
hold for the analytical function $w = u + iv$ of $z = x + iy$. It
had been proved by Dirichlet that (for a plane) there is always
one, and only one, function of $x$~and~$y$, which satisfies $\Delta u = 0$,
and which, together with its differential quotients of the first
two orders, is for all values of $x$~and~$y$ within a given area
one-valued and continuous, and which has for points on the
boundary of the area arbitrarily given values.\Reference{86} Riemann
called this ``Dirichlet's principle,'' but the same theorem was
stated by Green and proved analytically by Sir William
\index{Green!ref.\ to}%
Thomson. It follows then that $w$ is uniquely determined for
\index{Thomson, Sir William!ref.\ to}%
all points within a closed surface, if $u$ is arbitrarily given
for all points on the curve, whilst $v$ is given for one point
within the curve. In order to treat the more complicated
case where $w$ has $n$~values for one value of~$z$, and to observe
the conditions about continuity, Riemann invented the celebrated
surfaces, known as ``Riemann's surfaces,'' consisting
\index{Riemann's surfaces}%
of $n$~coincident planes or sheets, such that the passage from
one sheet to another is made at the branch-points, and that the
$n$~sheets form together a multiply-connected surface, which
can be dissected by cross-cuts into a singly-connected surface.
The $n$-valued function~$w$ becomes thus a one-valued function.
Aided by researches of J.~Lüroth of Freiburg and of Clebsch,
\index{Clebsch!ref.\ to}%
\index{Luroth@{Lüroth}}%
W.~K. Clifford brought Riemann's surface for algebraic functions
\index{Clifford!ref.\ to}%
\Addendlabel{358}%
to a canonical form, in which only the two last of the $n$~leaves
are multiply-connected, and then transformed the surface into
the surface of a solid with $p$~holes. A.~Hurwitz of Zürich
\index{Hurwitz}%
discussed the question, how far a Riemann's surface is determinate
%% -----File: 373.png---Folio 359-------
\index{Thomson's theorem}%
by the assignment of its number of sheets, its branch-points
and branch-lines.\Reference{62}

Riemann's theory ascertains the criteria which will determine
an analytical function by aid of its discontinuities and
boundary conditions, and thus defines a function independently
of a mathematical expression. In order to show that
two different expressions are identical, it is not necessary to
transform one into the other, but it is sufficient to prove the
agreement to a far less extent, merely in certain critical points.

Riemann's theory, as based on Dirichlet's principle (Thomson's
\index{Dirichlet!ref.\ to}%
theorem), is not free from objections. It has become
evident that the existence of a derived function is not a consequence
of continuity, and that a function may be integrable
\index{Continuity}%
without being differentiable. It is not known how far the
methods of the infinitesimal calculus and the calculus of
variations (by which Dirichlet's principle is established) can
be applied to an unknown analytical function in its generality.
Hence the use of these methods will endow the functions with
properties which themselves require proof. Objections of this
kind to Riemann's theory have been raised by Kronecker,
\index{Kronecker!ref.\ to}%
Weierstrass, and others, and it has become doubtful whether
\index{Weierstrass}%
\index{Weierstrass!ref.\ to}%
his most important theorems are actually proved. In consequence
of this, attempts have been made to graft Riemann's
speculations on the more strongly rooted methods of Weierstrass.
The latter developed a theory of functions by starting,
not with the theory of potential, but with analytical
expressions and operations. Both applied their theories to
Abelian functions, but there Riemann's work is more general.\Reference{86}
\index{Abelian functions}%

The theory of functions of one complex variable has been
studied since Riemann's time mainly by \textbf{Karl Weierstrass} of
\index{Riemann|)}%
Berlin (born 1815), \textbf{Gustaf Mittag-Leffler} of Stockholm (born
\index{Mittag-Leffler}%
1846), and Poincaré of Paris. Of the three classes of such
\index{Poincaré!ref.\ to}%
%% -----File: 374.png---Folio 360-------
\index{Fuchsian functions}%
\index{Hyperelliptic functions}%
functions (viz.\ functions uniform throughout, functions uniform
only in lacunary spaces, and non-uniform functions)
Weierstrass showed that those functions of the first class
which can be developed according to ascending powers of~$x$
into converging series, can be decomposed into a product of an
infinite number of primary factors. A primary factor of the
\index{Primary factors, Weierstrass' theory of}%
species~$n$ is the product $\left(1- \dfrac{x}{a}\right)e^{\Change{P_{(x)}}{P(x)}}$, $\Change{P_{(x)}}{P(x)}$ being an entire polynomial
of the $n$th~degree. A function of the species~$n$ is one,
all the primary factors of which are of species~$n$. This classification
gave rise to many interesting problems studied also
by Poincaré.

The first of the three classes of functions of a complex
variable embraces, among others, functions having an infinite
number of singular points, but no singular lines, and at the
same time no isolated singular points. These are Fuchsian
functions, existing throughout the whole extent. Poincaré
first gave an example of such a function.

Uniform functions of two variables, unaltered by certain
linear substitutions, called hyperfuchsian functions, have been
studied by E.~Picard of Paris, and by Poincaré.\Reference{81}
\index{Picard, E.}%

Functions of the second class, uniform only in lacunary
spaces, were first pointed out by Weierstrass. The Fuchsian
and the Kleinian functions do not generally exist, except in
\index{Kleinian functions}%
the interior of a circle or of a domain otherwise bounded, and
are therefore examples of functions of the second class.
Poincaré has shown how to generate functions of this class,
and has studied them along the lines marked out by Weierstrass.
Important is his proof that there is no way of
generalising them so as to get rid of the lacunæ.

Non-uniform functions are much less developed than the
preceding classes, even though their properties in the vicinity
of a given point have been diligently studied, and though
%% -----File: 375.png---Folio 361-------
\index{Conform representation of surfaces}%
\index{Infinite series}%
much light has been thrown on them by the use of Riemann's
surfaces. With the view of reducing their study to that of
uniform transcendents, Poincaré proved that if $y$ is any
analytical non-uniform function of~$x$, one can always find a
variable~$z$, such that $x$~and~$y$ are uniform functions of~$z$.

Weierstrass and Darboux have each given examples of continuous
\index{Darboux}%
\index{Weierstrass!ref.\ to}%
functions having no derivatives. Formerly it had
been generally assumed that every function had a derivative.
\Addendlabel{361}%
Ampère was the first who attempted to prove analytically
\index{Ampère!ref.\ to}%
(1806) the existence of a derivative, but the demonstration
is not valid. In treating of discontinuous functions, Darboux
established rigorously the necessary and sufficient condition
that a continuous or discontinuous function be susceptible of
integration. He gave fresh evidence of the care that must
be exercised in the use of series by giving an example of a
series always convergent and continuous, such that the series
formed by the integrals of the terms is always convergent, and
yet does not represent the integral of the first series.\Reference{87}

The general theory of functions of two variables has been
investigated to some extent by Weierstrass and Poincaré.

\textbf{H.~A. Schwarz} of Berlin (born 1845), a pupil of Weierstrass,
\index{Schwarz}%
has given the conform\DPnote{** PP: [sic]} representation (\emph{Abbildung}) of various
surfaces on a circle. In transforming by aid of certain
substitutions a polygon bounded by circular arcs into another
also bounded by circular arcs, he was led to a remarkable
differential equation $\psi(u', t) = \psi(u, t)$, where $\psi(u, t)$ is the
expression which Cayley calls the ``Schwarzian derivative,''
\index{Schwarzian derivative}%
and which led Sylvester to the theory of reciprocants.
\index{Reciprocants}%
\index{Sylvester!ref.\ to}%
Schwarz's developments on minimum surfaces, his work on
hypergeometric series, his inquiries on the existence of solutions
\index{Hypergeometric series}%
to important partial differential equations under prescribed
conditions, have secured a prominent place in mathematical
literature.
%% -----File: 376.png---Folio 362-------
\index{Irrationals}%

The modern theory of functions of one real variable was
first worked out by H.~Hankel, Dedekind, G.~Cantor, Dini, and
\index{Cantor, G.}%
\index{Dedekind!ref.\ to}%
\index{Dini!ref.\ to}%
\index{Hankel!ref.\ to}%
Heine, and then carried further, principally, by Weierstrass,
\index{Heine!ref.\ to}%
\index{Weierstrass!ref.\ to}%
Schwarz, Du~Bois-Reymond, Thomae, and Darboux. Hankel
\index{Bois-Reymond, P.~du}%
\index{Darboux}%
\index{Schwarz!ref.\ to}%
\index{Thomae}%
established the principle of the condensation of singularities;
\index{Condensation of singularities}%
Dedekind and Cantor gave definitions for irrational numbers;
\index{Numbers!theory of numbers|(}%
definite integrals were studied by Thomae, Du~Bois-Reymond,
\index{Definite integrals}%
and Darboux along the lines indicated by the definitions of
such integrals given by Cauchy, Dirichlet, and Riemann. Dini
\index{Cauchy!ref.\ to}%
\index{Dirichlet!ref.\ to}%
\index{Riemann!ref.\ to}%
wrote a text-book on functions of a real variable (1878), which
was translated into German, with additions, by J.~Lüroth and
\index{Luroth@{Lüroth}!ref.\ to}%
A.~Schepp. Important works on the theory of functions are
\index{Schepp}%
the \textit{Cours de M.~Hermite}, \textit{Tannery's Théorie des Fonctions
\index{Hermite!ref.\ to}%
\index{Tannery!ref.\ to}%
d'une variable seule}, \textit{A Treatise on the Theory of Functions} by
\textit{James Harkness} and \textit{Frank Morley}, and \textit{Theory of Functions of
\index{Functions!theory of|)}%
\index{Harkness}%
\index{Morley}%
\index{Theory of functions|)}%
a Complex Variable} by \textit{A.~R. Forsyth}.
\index{Forsyth}%

\Section{Theory of Numbers}
\index{Theory of numbers|(}%

``Mathematics, the queen of the sciences, and arithmetic,
the queen of mathematics.'' Such was the dictum of Gauss,
\index{Gauss!ref.\ to}%
who was destined to revolutionise the theory of numbers.
When asked who was the greatest mathematician in Germany,
Laplace answered, Pfaff. When the questioner said
\index{Laplace!ref.\ to}%
\index{Pfaff!ref.\ to}%
he should have thought Gauss was, Laplace replied, ``Pfaff
is by far the greatest mathematician in Germany; but Gauss
is the greatest in all Europe.''\Reference{83} Gauss is one of the three
greatest masters of modern analysis,---Lagrange, Laplace,
\index{Lagrange!ref.\ to}%
Gauss. Of these three contemporaries he was the youngest.
While the first two belong to the period in mathematical history
preceding the one now under consideration, Gauss is the
one whose writings may truly be said to mark the beginning
%% -----File: 377.png---Folio 363-------
\index{Imaginary quantities}%
\index{Infinite series}%
\index{Kaestner}%
of our own epoch. In him that abundant fertility of invention,
displayed by mathematicians of the preceding period,
is combined with an absolute rigorousness in demonstration
which is too often wanting in their writings, and which the
ancient Greeks might have envied. Unlike Laplace, Gauss
\index{Gauss|(}%
\index{Laplace!ref.\ to}%
strove in his writings after perfection of form. He rivals
Lagrange in elegance, and surpasses this great Frenchman in
\index{Lagrange!ref.\ to}%
rigour. Wonderful was his richness of ideas; one thought followed
another so quickly that he had hardly time to write
down even the most meagre outline. At the age of twenty
Gauss had overturned old theories and old methods in all
branches of higher mathematics; but little pains did he take
to publish his results, and thereby to establish his priority.
He was the first to observe rigour in the treatment of infinite
series, the first to fully recognise and emphasise the importance,
and to make systematic use of determinants and of
\index{Determinants}%
imaginaries, the first to arrive at the method of least squares,
\index{Least squares}%
the first to observe the double periodicity of elliptic functions.
\index{Elliptic functions}%
He invented the heliotrope and, together with Weber, the
\index{Heliotrope}%
\index{Weber, W.~E.!ref.\ to}%
bifilar magnetometer and the declination instrument. He
reconstructed the whole of magnetic science.

\textbf{Carl Friedrich Gauss}\Reference[]{47} (1777--1855), the son of a bricklayer,
was born at Brunswick. He used to say, jokingly, that he
could reckon before he could talk. The marvellous aptitude
for calculation of the young boy attracted the attention of
Bartels, afterwards professor of mathematics at Dorpat, who
brought him under the notice of Charles William, Duke of
Brunswick. The duke undertook to educate the boy, and sent
him to the Collegium Carolinum. His progress in languages
there was quite equal to that in mathematics. In 1795 he
went to Göttingen, as yet undecided whether to pursue philology
or mathematics. Abraham Gotthelf Kästner, then professor
of mathematics there, and now chiefly remembered for
%% -----File: 378.png---Folio 364-------
his \textit{Geschichte der Mathematik} (1796), was not an inspiring
teacher. At the age of nineteen Gauss discovered a method of
inscribing in a circle a regular polygon of seventeen sides, and
this success encouraged him to pursue mathematics. He worked
quite independently of his teachers, and while a student at
Göttingen made several of his greatest discoveries. Higher
arithmetic was his favourite study. Among his small circle
of intimate friends was Wolfgang Bolyai. After completing
\index{Bolyai, Wolfgang!ref.\ to}%
his course he returned to Brunswick. In 1798 and 1799 he
repaired to the university at \Change{Helmstadt}{Helmstädt} to consult the library,
and there made the acquaintance of Pfaff, a mathematician of
much power. In 1807 the Emperor of Russia offered Gauss a
chair in the Academy at St.~Petersburg, but by the advice of
the astronomer Olbers, who desired to secure him as director
\index{Olbers}%
of a proposed new observatory at Göttingen, he declined the
offer, and accepted the place at Göttingen. Gauss had a
marked objection to a mathematical chair, and preferred the
post of astronomer, that he might give all his time to science.
He spent his life in Göttingen in the midst of continuous
work. In 1828 he went to Berlin to attend a meeting of
scientists, but after this he never again left Göttingen, except
in 1854, when a railroad was opened between Göttingen and
Hanover. He had a strong will, and his character showed
a curious mixture of self-conscious dignity and child-like
simplicity. He was little communicative, and at times
morose.

A new epoch in the theory of numbers dates from the publication
of his \textit{Disquisitiones Arithmeticæ}, Leipzig, 1801. The
beginning of this work dates back as far as 1795. Some of its
results had been previously given by Lagrange and Euler, but
\index{Euler!ref.\ to}%
\index{Lagrange!ref.\ to}%
were reached independently by Gauss, who had gone deeply
into the subject before he became acquainted with the writings
of his great predecessors. The \textit{Disquisitiones Arithmeticæ}
%% -----File: 379.png---Folio 365-------
\index{Congruencies, theory of}%
was already in print when Legendre's \textit{Théorie des Nombres}
\index{Legendre!ref.\ to}%
appeared. The great law of quadratic reciprocity, given in
\index{Quadratic reciprocity}%
the fourth section of Gauss' work, a law which involves the
whole theory of quadratic residues, was discovered by him by
induction before he was eighteen, and was proved by him one
year later. Afterwards he learned that Euler had imperfectly
\index{Euler!ref.\ to}%
enunciated that theorem, and that Legendre had attempted
to prove it, but met with apparently insuperable difficulties.
In the fifth section Gauss gave a second proof of this ``gem''
of higher arithmetic. In 1808 followed a third and fourth
demonstration; in 1817, a fifth and sixth. No wonder that
he felt a personal attachment to this theorem. Proofs were
given also by Jacobi, Eisenstein, Liouville, Lebesgue, A.~Genocchi,
\index{Eisenstein!ref.\ to}%
\index{Genocchi}%
\index{Jacobi!ref.\ to}%
\index{Lebesgue}%
\index{Liouville!ref.\ to}%
Kummer, M.~A. Stern, Chr.\ Zeller, Kronecker,
\index{Kronecker!ref.\ to}%
\index{Kummer!ref.\ to}%
\index{Stern}%
\index{Zeller}%
Bouniakowsky, E.~Schering, J.~Petersen, Voigt, E.~Busche,
\index{Bouniakowsky}%
\index{Busche}%
\index{Petersen}%
\index{Schering!ref.\ to}%
\index{Voigt}%
and Th.\ Pepin.\Reference{48} The solution of the problem of the representation
\index{Pepin}%
of numbers by binary quadratic forms is one of the
great achievements of Gauss. He created a new algorithm by
introducing the theory of congruences. The fourth section
of the \textit{Disquisitiones Arithmeticæ}, treating of congruences of
the second degree, and the fifth section, treating of quadratic
forms, were, until the time of Jacobi, passed over with universal
neglect, but they have since been the starting-point of a long
series of important researches. The seventh or last section,
developing the theory of the division of the circle, was received
\index{Circle!division of}%
\index{Division of the circle}%
from the start with deserved enthusiasm, and has since been
repeatedly elaborated for students. A standard work on
\textit{Kreistheilung} was published in 1872 by Paul Bachmann, then
\index{Bachmann!ref.\ to}%
of Breslau. Gauss had planned an eighth section, which was
omitted to lessen the expense of publication. His papers on
the theory of numbers were not all included in his great treatise.
Some of them were published for the first time after his death
in his collected works (1863--1871). He wrote two memoirs on
%% -----File: 380.png---Folio 366-------
\index{Action, least}%
\index{Capillarity}%
\index{Ellipsoid!(attraction of)}%
\index{Geodesy}%
the theory of biquadratic residues (1825 and 1831), the second
\index{Biquadratic residues}%
of which contains a theorem of biquadratic reciprocity.

Gauss was led to astronomy by the discovery of the planet
\index{Astronomy!more recent researches}%
Ceres at Palermo in 1801. His determination of the elements
of its orbit with sufficient accuracy to enable Olbers to \Change{rediscover}{re-discover}
it, made the name of Gauss generally known. In 1809
he published the \textit{Theoria motus corporum coelestium}, which
contains a discussion of the problems arising in the determination
of the movements of planets and comets from
observations made on them under any circumstances. In it
are found four formulæ in spherical trigonometry, now usually
called ``Gauss' Analogies,'' but which were published somewhat
\index{Gauss' Analogies}%
earlier by Karl Brandon Mollweide of Leipzig (1774--1825),
\index{Mollweide}%
and earlier still by Jean Baptiste Joseph Delambre
\index{Delambre}%
(1749--1822).\Reference{44} Many years of hard work were spent in the
astronomical and magnetic observatory. He founded the
German Magnetic Union, with the object of securing continuous
\index{German Magnetic Union}%
observations at fixed times. He took part in geodetic
observations, and in 1843 and 1846 wrote two memoirs, \textit{Ueber
Gegenstände der höheren Geodesie}. He wrote on the attraction
of homogeneous ellipsoids, 1813. In a memoir on capillary
attraction, 1833, he solves a problem in the calculus of
\index{Calculus!of variations}%
variations involving the variation of a certain double integral,
the limits of integration being also variable; it is the earliest
example of the solution of such a problem. He discussed the
problem of rays of light passing through a system of lenses.

Among Gauss' pupils were Christian Heinrich Schumacher,
\index{Schumacher}%
Christian Gerling, Friedrich Nicolai, August Ferdinand
\index{Gerling}%
\index{Nicolai}%
Möbius, Georg Wilhelm Struve,\DPnote{** PP: [sic] no `and'} Johann Frantz Encke.
\index{Encke}%
\index{Mobius@{Möbius}!ref.\ to}%
\index{Struve}%

Gauss' researches on the theory of numbers were the starting-point
for a school of writers, among the earliest of whom
was Jacobi. The latter contributed to \textit{Crelle's Journal} an article
on cubic residues, giving theorems without proofs. After the
\index{Cubic residues}%
%% -----File: 381.png---Folio 367-------
\index{Elliptic functions}%
publication of Gauss' paper on biquadratic residues, giving
the law of biquadratic reciprocity, and his treatment of complex
numbers, Jacobi found a similar law for cubic residues.
\index{Jacobi!ref.\ to}%
By the theory of elliptical functions, he was led to beautiful
theorems on the representation of numbers by $2$,~$4$,~$6$, and~$8$
squares. Next come the researches of Dirichlet, the expounder
\index{Dirichlet|(}%
of Gauss, and a contributor of rich results of his own.

\textbf{Peter Gustav Lejeune Dirichlet}\Reference[]{88} (1805--1859) was born in
Düren, attended the gymnasium in Bonn, and then the
Jesuit gymnasium in Cologne. In 1822 he was attracted to
Paris by the names of Laplace, Legendre, Fourier, Poisson,
\index{Fourier!ref.\ to}%
\index{Legendre!ref.\ to}%
Cauchy. The facilities for a mathematical education there
were far better than in Germany, where Gauss was the only
great figure. He read in Paris Gauss' \textit{Disquisitiones Arithmeticæ},
a work which he never ceased to admire and study.
Much in it was simplified by Dirichlet, and thereby placed
within easier reach of mathematicians. His first memoir on
the impossibility of certain indeterminate equations of the
fifth degree was presented to the French Academy in 1825.
He showed that Fermat's equation, $x^n + y^n = z^n$, cannot exist
\index{Fermat!ref.\ to}%
when $n = 5$. Some parts of the analysis are, however,
Legendre's. Euler and Lagrange had proved this when $n$~is~$3$
\index{Euler!ref.\ to}%
\index{Lagrange!ref.\ to}%
and~$4$, and Lamé proved it when $n = 7$. Dirichlet's acquaintance
\index{Lame@{Lamé}!ref.\ to}%
with Fourier led him to investigate Fourier's series. He
\index{Fourier's series}%
became docent in Breslau in 1827. In 1828 he accepted a
position in Berlin, and finally succeeded Gauss at Göttingen
in 1855. The general principles on which depends the average
number of classes of binary quadratic forms of positive
and negative determinant (a subject first investigated by
Gauss) were given by Dirichlet in a memoir, \textit{Ueber die Bestimmung
\index{Gauss|)}%
der mittleren Werthe in der Zahlentheorie}, 1849. More
recently F.~Mertens of Graz has determined the asymptotic
\index{Mertens}%
values of several numerical functions. Dirichlet gave some
%% -----File: 382.png---Folio 368-------
attention to prime numbers. Gauss and Legendre had given
\index{Prime numbers}%
expressions denoting approximately the asymptotic value of
the number of primes inferior to a given limit, but it remained
for Riemann in his memoir, \textit{Ueber die Anzahl der Primzahlen
\index{Riemann!ref.\ to}%
unter einer gegebenen Grösse}, 1859, to give an investigation of
the asymptotic frequency of primes which is rigorous. Approaching
the problem from a different direction, \textit{Patnutij
Tchebycheff}, formerly professor in the University of St.~Petersburg
\index{Tchebycheff}%
(born 1821), established, in a celebrated memoir, \textit{Sur les
Nombres Premiers}, 1850, the existence of limits within which
the sum of the logarithms of the primes~$P$, inferior to a given
number~$x$, must be comprised.\Reference{89} This paper depends on very
elementary considerations, and, in that respect, contrasts
strongly with Riemann's, which involves abstruse theorems
of the integral calculus. Poincaré's papers, Sylvester's contraction
\index{Integral calculus}%
\index{Poincaré!ref.\ to}%
\index{Sylvester!ref.\ to}%
of Tchebycheff's limits, with reference to the distribution
of primes, and researches of J.~Hadamard (awarded the
\index{Hadamard}%
\textit{Grand prix} of 1892), are among the latest researches in this
line. The enumeration of prime numbers has been undertaken
at different times by various mathematicians. In 1877 the
British Association began the preparation of factor-tables,
\index{Factor-tables}%
under the direction of J.~W.~L. Glaisher. The printing, by
\index{Glaisher!ref.\ to}%
the Association, of tables for the sixth million marked the
completion of tables, to the preparation of which Germany,
France, and England contributed, and which enable us to
resolve into prime factors every composite number less than
$9,000,000$.

Miscellaneous contributions to the theory of numbers were
made by \textit{Cauchy}. He showed, for instance, how to find all
\index{Cauchy!ref.\ to}%
the infinite solutions of a homogeneous indeterminate equation
of the second degree in three variables when one solution is
given. He established the theorem that if two congruences,
which have the same modulus, admit of a common solution,
%% -----File: 383.png---Folio 369-------
\index{Covariants}%
\index{Dirichlet|)}%
the modulus is a divisor of their resultant. \textbf{Joseph Liouville}
\index{Liouville}%
(1809--1882), professor at the \DPtypo{Collége}{Collège} de France, investigated
mainly questions on the theory of quadratic forms of two, and
of a greater number of variables. Profound researches were
instituted by \textbf{Ferdinand Gotthold Eisenstein} (1823--1852), of
\index{Eisenstein}%
Berlin. Ternary quadratic forms had been studied somewhat
by Gauss, but the extension from two to three indeterminates
was the work of Eisenstein who, in his memoir, \textit{Neue Theoreme
der höheren Arithmetik}, defined the ordinal and generic
characters of ternary quadratic forms of uneven determinant;
and, in case of definite forms, assigned the weight of any order
or genus. But he did not publish demonstrations of his results.
In inspecting the theory of binary cubic forms, he was
led to the discovery of the first covariant ever considered in
analysis. He showed that the series of theorems, relating to
the presentation of numbers by sums of squares, ceases when
the number of squares surpasses eight. Many of the proofs
omitted by Eisenstein were supplied by Henry Smith, who
\index{Smith, H.}%
was one of the few Englishmen who devoted themselves to the
study of higher arithmetic.

\textbf{Henry John Stephen Smith}\Reference[]{90} (1826--1883) was born in London,
and educated at Rugby and at Balliol College, Oxford.
Before 1847 he travelled much in Europe for his health,
and at one time attended lectures of Arago in Paris, but
after that year he was never absent from Oxford for a
single term. In 1861 he was elected Savilian professor of
geometry. His first paper on the theory of numbers appeared
in 1855. The results of ten years' study of everything published
on the theory of numbers are contained in his Reports
which appeared in the British Association volumes from 1859
to 1865. These reports are a model of clear and precise
exposition and perfection of form. They contain much original
matter, but the chief results of his own discoveries were
%% -----File: 384.png---Folio 370-------
printed in the \textit{Philosophical Transactions} for 1861 and 1867.
They treat of linear indeterminate equations and congruences,
and of the orders and genera of ternary quadratic forms. He
established the principles on which the extension to the general
case of $n$ indeterminates of quadratic forms depends.
He contributed also two memoirs to the \textit{Proceedings of the
Royal Society} of 1864 and 1868, in the second of which he
remarks that the theorems of Jacobi, Eisenstein, and Liouville,
\index{Eisenstein!ref.\ to}%
\index{Jacobi!ref.\ to}%
\index{Liouville!ref.\ to}%
relating to the representation of numbers by $4$,~$6$,~$8$
squares, and other simple quadratic forms are deducible by a
uniform method from the principles indicated in his paper.
Theorems relating to the case of $5$~squares were given by
Eisenstein, but Smith completed the enunciation of them, and
\index{Smith, H.}%
added the corresponding theorems for $7$~squares. The solution
of the cases of $2$,~$4$,~$6$ squares may be obtained by elliptic
functions, but when the number of squares is odd, it involves
processes peculiar to the theory of numbers. This class of
theorems is limited to $8$~squares, and Smith completed the
group. In ignorance of Smith's investigations, the French
Academy offered a prize for the demonstration and completion
of Eisenstein's theorems for $5$~squares. This Smith had
accomplished fifteen years earlier. He sent in a dissertation
in 1882, and next year, a month after his death, the prize was
awarded to him, another prize being also awarded to H.~Minkowsky
\index{Minkowsky}%
of Bonn. The theory of numbers led Smith to the
study of elliptic functions. He wrote also on modern geometry.
\index{Elliptic functions}%
His successor at Oxford was J.~J. Sylvester.
\index{Sylvester!ref.\ to}%

\textbf{Ernst Eduard Kummer} (1810--1893), professor in the University
\index{Kummer}%
of Berlin, is closely identified with the theory of numbers.
Dirichlet's work on complex numbers of the form $a + ib$,
introduced by Gauss, was extended by him, by Eisenstein,
and Dedekind. Instead of the equation $x^4 - 1 = 0$, the roots
of which yield Gauss' units, Eisenstein used the equation
%% -----File: 385.png---Folio 371-------
$x^3 - 1 = 0$ and complex numbers $a + b\rho$ ($\rho$~being a cube root
of unity), the theory of which resembles that of Gauss' numbers.
Kummer passed to the general case $x^n - 1 = 0$ and got
\index{Kummer}%
complex numbers of the form $\alpha = a_1A_1 + a_2A_2 + a_3A_3 + \dotsb$,
where $a_i$~are whole real numbers, and $A_i$~roots of the above
equation.\Reference{59} Euclid's theory of the greatest common divisor
is not applicable to such complex numbers, and their prime
factors cannot be defined in the same way as prime factors of
common integers are defined. In the effort to overcome this
difficulty, Kummer was led to introduce the conception of
``ideal numbers.'' These ideal numbers have been applied by
\index{Ideal numbers}%
G.~Zolotareff of St.~Petersburg to the solution of a problem
\index{Zolotareff}%
of the integral calculus, left unfinished by Abel (\textit{Liouville's
\index{Abel!ref.\ to}%
\index{Integral calculus}%
Journal}, Second Series, 1864, Vol.~IX.). \textbf{Julius Wilhelm Richard
Dedekind} of Braunschweig (born 1831) has given in the second
\index{Dedekind}%
edition of Dirichlet's \textit{Vorlesungen über Zahlentheorie} a new
\index{Dirichlet!ref.\ to}%
theory of complex numbers, in which he to some extent
deviates from the course of Kummer, and avoids the use of
ideal numbers. Dedekind has taken the roots of any irreducible
equation with integral coefficients as the units for his complex
numbers. Attracted by Kummer's investigations, his
pupil, \textbf{Leopold Kronecker} (1823--1891) made researches which
he applied to algebraic equations.

On the other hand, efforts have been made to utilise in the
theory of numbers the results of the modern higher algebra.
Following up researches of Hermite, \textbf{Paul Bachmann} of Münster
\index{Bachmann}%
investigated the arithmetical formula which gives the automorphics
of a ternary quadratic form.\Reference{89} The problem of the
equivalence of two positive or definite ternary quadratic forms
was solved by L.~Seeber; and that of the arithmetical automorphics
\index{Seeber}%
of such forms, by Eisenstein. The more difficult problem
\index{Eisenstein!ref.\ to}%
of the equivalence for indefinite ternary forms has been
investigated by Edward Selling of Würzburg. On quadratic
\index{Selling}%
%% -----File: 386.png---Folio 372-------
\index{Negative quantities}%
\index{Partition of numbers}%
\index{Theory of numbers|)}%
forms of four or more indeterminates little has yet been done.
Hermite showed that the number of non-equivalent classes of
\index{Hermite!ref.\ to}%
quadratic forms having integral coefficients and a given discriminant
is finite, while Zolotareff and A.~N. Korkine, both
\index{Korkine}%
\index{Zolotareff!ref.\ to}%
of St.~Petersburg, investigated the minima of positive quadratic
forms. In connection with binary quadratic forms, Smith
\index{Smith, H.!ref.\ to}%
established the theorem that if the joint invariant of two
properly primitive forms vanishes, the determinant of either
of them is represented primitively by the duplicate of the
other.

The interchange of theorems between arithmetic and algebra
is displayed in the recent researches of J.~W.~L. Glaisher
\index{Glaisher}%
of Trinity College (born 1848) and Sylvester. Sylvester gave
a Constructive Theory of Partitions, which received additions
from his pupils, F.~Franklin and G.~S. Ely.
\index{Ely}%
\index{Franklin}%

The conception of ``number'' has been much extended in
our time. With the Greeks it included only the ordinary
positive whole numbers; Diophantus added rational fractions
\index{Diophantus!ref.\ to}%
\index{Numbers!definitions of numbers}%
to the domain of numbers. Later negative numbers and
imaginaries came gradually to be recognised. Descartes fully
grasped the notion of the negative; Gauss, that of the imaginary.
\index{Imaginary quantities}%
With Euclid, a ratio, whether rational or irrational, was
not a number. The recognition of ratios and irrationals as
\index{Irrationals}%
\index{Ratios}%
numbers took place in the sixteenth century, and found expression
with Newton. By the ratio method, the continuity of the
\index{Continuity}%
\index{Newton!ref.\ to}%
real number system has been based on the continuity of space,
but in recent time three theories of irrationals have been
advanced by Weierstrass, J.~W.~R. Dedekind, G.~Cantor, and
\index{Cantor, G.}%
\index{Dedekind!ref.\ to}%
\index{Weierstrass!ref.\ to}%
Heine, which prove the continuity of numbers without borrowing
\index{Heine!ref.\ to}%
it from space. They are based on the definition of numbers
\index{Numbers!theory of numbers|)}%
by regular sequences, the use of series and limits, and some
new mathematical conceptions.
%% -----File: 387.png---Folio 373-------
\index{Astronomy!more recent researches|(}%


\Section{Applied Mathematics}
\index{Applied mathematics|(}%

Notwithstanding the beautiful developments of celestial
mechanics reached by Laplace at the close of the eighteenth
\index{Laplace!ref.\ to}%
century, there was made a discovery on the first day of the
present century which presented a problem seemingly beyond
the power of that analysis. We refer to the discovery of Ceres
by Piazzi in Italy, which became known in Germany just after
\index{Piazzi}%
the philosopher Hegel had published a dissertation proving \textit{a~priori}
\index{Hegel}%
that such a discovery could not be made. From the
positions of the planet observed by Piazzi its orbit could not
be satisfactorily calculated by the old methods, and it remained
for the genius of Gauss to devise a method of calculating
\index{Gauss!ref.\ to}%
elliptic orbits which was free from the assumption of a small
eccentricity and inclination. Gauss' method was developed
further in his \textit{Theoria Motus}. The new planet was re-discovered
with aid of Gauss' data by Olbers, an astronomer
\index{Olbers}%
who promoted science not only by his own astronomical
studies, but also by discerning and directing towards astronomical
pursuits the genius of Bessel.
\index{Bessel|(}%

\textbf{Friedrich Wilhelm Bessel}\Reference[]{91} (1784--1846) was a native of
Minden in Westphalia. Fondness for figures, and a distaste
for Latin grammar led him to the choice of a mercantile
career. In his fifteenth year he became an apprenticed clerk
in Bremen, and for nearly seven years he devoted his days to
mastering the details of his business, and part of his nights to
study. Hoping some day to become a supercargo on trading
expeditions, he became interested in observations at sea. With
a sextant constructed by him and an ordinary clock he determined
the latitude of Bremen. His success in this inspired
him for astronomical study. One work after another was
mastered by him, unaided, during the hours snatched from
%% -----File: 388.png---Folio 374-------
sleep. From old observations he calculated the orbit of
Halley's comet. Bessel introduced himself to Olbers, and
\index{Halley's Comet}%
submitted to him the calculation, which Olbers immediately
sent for publication. Encouraged by Olbers, Bessel turned
his back to the prospect of affluence, chose poverty and the
stars, and became assistant in J.~H. Schröter's observatory at
\index{SchroterJH@{Schröter, J.~H.}}%
Lilienthal. Four years later he was chosen to superintend
the construction of the new observatory at Königsberg.\Reference{92} In
the absence of an adequate mathematical teaching force, Bessel
was obliged to lecture on mathematics to prepare students for
astronomy. He was relieved of this work in 1825 by the
arrival of Jacobi. We shall not recount the labours by which
\index{Jacobi!ref.\ to}%
Bessel earned the title of founder of modern practical astronomy
and geodesy. As an observer he towered far above
Gauss, but as a mathematician he reverently bowed before the
genius of his great contemporary. Of Bessel's papers, the one
of greatest mathematical interest is an ``\textit{Untersuchung des
Theils der planetarischen Störungen, welcher aus der Bewegung
der Sonne ensteht}'' (1824), in which he introduces a class of
transcendental functions, $J_n(x)$, much used in applied mathematics,
and known as ``Bessel's functions.'' He gave their
\index{Bessel's functions}%
principal properties, and constructed tables for their evaluation.
Recently it has been observed that Bessel's functions
appear much earlier in mathematical literature.\Reference{98}
Such functions of the zero order occur in papers of Daniel
Bernoulli (1732) and Euler on vibration of heavy strings suspended
\index{Euler!ref.\ to}%
from one end. All of Bessel's functions of the first
kind and of integral orders occur in a paper by Euler (1764) on
the vibration of a stretched elastic membrane. In 1878 Lord
Rayleigh proved that Bessel's functions are merely particular
\index{Rayleigh, Lord!ref.\ to}%
cases of Laplace's functions. J.~W.~L. Glaisher illustrates
\index{Glaisher!ref.\ to}%
by Bessel's functions his assertion that mathematical branches
growing out of physical inquiries as a rule ``lack the easy flow
%% -----File: 389.png---Folio 375-------
\index{Bessel|)}%
or homogeneity of form which is characteristic of a mathematical
theory properly so called.'' These functions have been
studied by C.~Th.\ Anger of Danzig, O.~Schlömilch of Dresden,
\index{Anger}%
\index{Schlomilch@{Schlömilch}}%
R.~Lipschitz of Bonn (born 1832), Carl Neumann of Leipzig
\index{Lipschitz!ref.\ to}%
\index{Neumann, C.}%
(born 1832), Eugen Lommel of Leipzig, I.~Todhunter of St.~John's
\index{Lommel}%
\index{Todhunter!ref.\ to}%
College, Cambridge.

Prominent among the successors of Laplace are the following:
\index{Laplace!ref.\ to}%
\textit{Siméon Denis Poisson} (1781--1840), who wrote in 1808
\index{Poisson!ref.\ to}%
a classic \textit{Mémoire sur les inégalités séculaires des moyens mouvements
des planètes}. \textit{Giovanni Antonio Amadeo Plana} (1781--1864)
\index{Plana}%
of Turin, a nephew of Lagrange, who published in 1811
a \Ita{\textit{Memoria sulla teoria dell'\DPnote{** PP: [sic] space} attrazione degli sferoidi ellitici}}, and
contributed to the theory of the moon. \textbf{Peter Andreas Hansen}
\index{Hansen}%
(1795--1874) of Gotha, at one time a clockmaker in Tondern,
then Schumacher's assistant at Altona, and finally director of
the observatory at Gotha, wrote on various astronomical subjects,
but mainly on the lunar theory, which he elaborated in
his work \textit{Fundamenta nova investigationes orbitæ veræ quam
Luna perlustrat} (1838), and in subsequent investigations
embracing extensive lunar tables. \textbf{George Biddel Airy} (1801--1892),
\index{Airy}%
royal astronomer at Greenwich, published in 1826 his
\textit{Mathematical Tracts on the Lunar and Planetary Theories}.
These researches have since been greatly extended by him.
\textbf{August Ferdinand Möbius} (1790--1868) of Leipzig wrote, in 1842,
\index{Mobius@{Möbius}!ref.\ to}%
\textit{Elemente der Mechanik des Himmels}. \textbf{Urbain Jean Joseph Le
Verrier} (1811--1877) of Paris wrote\DPtypo{,}{} the \textit{Recherches Astronomiques},
\index{Leverrier@{Le Verrier}}%
constituting in part a new elaboration of celestial
mechanics, and is famous for his theoretical discovery of
Neptune. \textbf{John Couch Adams} (1819--1892) of Cambridge
\index{Adams}%
\index{Neptune, discovery of}%
divided with Le~Verrier the honour of the mathematical discovery
of Neptune, and pointed out in 1853 that Laplace's
explanation of the secular acceleration of the moon's mean
motion accounted for only half the observed acceleration.
%% -----File: 390.png---Folio 376-------
\textbf{Charles Eugène Delaunay} (born 1816, and drowned off Cherbourg
\index{Delaunay}%
in 1872), professor of mechanics at the Sorbonne in
Paris, explained most of the remaining acceleration of the
moon, unaccounted for by Laplace's theory as corrected by
\index{Laplace!ref.\ to}%
Adams, by tracing the effect of tidal friction, a theory
previously suggested independently by Kant, Robert Mayer,
\index{Kant}%
\index{Mayer!ref.\ to}%
and William Ferrel of Kentucky. \textbf{George Howard Darwin} of
\index{Darwin}%
\index{Ferrel!ref.\ to}%
Cambridge (born 1845) made some very remarkable investigations
in 1879 on tidal friction, which trace with great
certainty the history of the moon from its origin. He has
since studied also the effects of tidal friction upon other
bodies in the solar system. Criticisms on some parts of his
researches have been made by James Nolan of Victoria. \textbf{Simon
\index{Nolan}%
Newcomb} (born 1835), superintendent of the \textit{Nautical Almanac}
\index{Nautical almanac, United States}%
\index{Newcomb}%
at Washington, and professor of mathematics at the Johns
Hopkins University, investigated the errors in Hansen's tables
of the moon. For the last twelve years the main work of the
\textit{U.~S. Nautical Almanac} office has been to collect and discuss
data for new tables of the planets which will supplant the
tables of Le~Verrier. \textit{G.~W. Hill} of that office has contributed
\index{Hill}%
\index{Leverrier@{Le Verrier}!ref.\ to}%
an elegant paper on certain possible abbreviations in the computation
of the long-period of the moon's motion due to the
direct action of the planets, and has made the most elaborate
determination yet undertaken of the inequalities of the moon's
motion due to the figure of the earth. He has also computed
certain lunar inequalities due to the action of Jupiter.

The mathematical discussion of Saturn's rings was taken up
\index{Saturn's rings}%
first by Laplace, who demonstrated that a homogeneous solid
ring could not be in equilibrium, and in 1851 by B.~Peirce,
\index{Peirce, B.!ref.\ to}%
who proved their non-solidity by showing that even an irregular
solid ring could not be in equilibrium about Saturn. The
mechanism of these rings was investigated by James Clerk
Maxwell in an essay to which the Adams prize was awarded.
\index{Maxwell!ref.\ to}%
%% -----File: 391.png---Folio 377-------
\index{Algebraic functions!integrals}%
\index{Mechanics!more recent work|(}%
He concluded that they consisted of an aggregate of unconnected
particles.

The problem of three bodies has been treated in various
\index{Problem of three bodies}%
\index{Three bodies, problem of}%
ways since the time of Lagrange, but no decided advance
towards a more complete algebraic solution has been made,
and the problem stands substantially where it was left by him.
He had made a reduction in the differential equations to the
seventh order. This was elegantly accomplished in a different
way by Jacobi in 1843. \textit{R.~Radau} (\textit{Comptes Rendus}, LXVII.,
\index{Jacobi!ref.\ to}%
\index{Radau}%
1868, p.~841) and \textit{Allégret} (\textit{Journal de Mathématiques}, 1875,
\index{Allégret}%
p.~277) showed that the reduction can be performed on the
equations in their original form. Noteworthy transformations
and discussions of the problem have been given by J.~L.~F.
Bertrand, by Émile Bour (1831--1866) of the Polytechnic School
\index{Bertrand}%
\index{Bour}%
in Paris, by Mathieu, Hesse, J.~A. Serret. H.~Bruns of Leipzig
\index{Bruns}%
\index{Hesse!ref.\ to}%
\index{Mathieu!ref.\ to}%
\index{Serret!ref.\ to}%
has shown that no advance in the problem of three or of $n$
bodies may be expected by algebraic integrals, and that we
must look to the modern theory of functions for a complete
solution (\textit{Acta Math.}, XI., p.~43).\Reference{93}

Among valuable text-books on mathematical astronomy rank
the following works: \textit{Manual of Spherical and Practical Astronomy}
by \textit{Chauvenet} (1863), \textit{Practical and Spherical Astronomy}
\index{Chauvenet}%
by \textit{Robert Main} of Cambridge, \textit{Theoretical Astronomy} by \textit{James~C.
\index{Astronomy!more recent researches|)}%
\index{Main}%
Watson} of Ann Arbor (1868), \textit{Traité élémentaire de Mécanique
\index{Watson, J.~C.}%
Céleste} of \textit{H.~Resal} of the Polytechnic School in Paris,
\index{Resal}%
\textit{Cours d'Astronomie de l'École Polytechnique} by \textit{Faye}, \textit{Traité
\index{Faye}%
de Mécanique Céleste} by \textit{Tisserand}, \textit{Lehrbuch der Bahnbestimmung}
\index{Tisserand}%
by \textit{T.~Oppolzer}, \textit{Mathematische Theorien der Planetenbewegung}
\index{Oppolzer}%
by \textit{O.~Dziobek}, translated into English by M.~W.
\index{Dziobek}%
Harrington and W.~J. Hussey.
\index{Harrington}%
\index{Hussey}%

During the present century we have come to recognise the
advantages frequently arising from a geometrical treatment of
mechanical problems. To Poinsot, Chasles, and Möbius we
\index{Chasles!ref.\ to}%
\index{Mobius@{Möbius}!ref.\ to}%
\index{Poinsot!ref.\ to}%
%% -----File: 392.png---Folio 378-------
\index{Dynamics|(}%
\index{Kowalevsky!ref.\ to}%
\index{Variation of arbitrary consonants}%
owe the most important developments made in geometrical
mechanics. \textbf{Louis Poinsot} (1777--1859), a graduate of the
\index{Poinsot}%
Polytechnic School in Paris, and for many years member of
the superior council of public instruction, published in 1804
his \textit{Éléments de Statique}. This work is remarkable not only
as being the earliest introduction to synthetic mechanics, but
also as containing for the first time the idea of couples, which
was applied by Poinsot in a publication of 1834 to the theory
of rotation. A clear conception of the nature of rotary
motion was conveyed by Poinsot's elegant geometrical representation
by means of an ellipsoid rolling on a certain fixed
\index{Ellipsoid!(attraction of)}%
plane. This construction was extended by Sylvester so as
\index{Sylvester!ref.\ to}%
to measure the rate of rotation of the ellipsoid on the plane.

A particular class of dynamical problems has recently been
treated geometrically by \textbf{Sir Robert Stawell Ball}, formerly
\index{Ball, Sir~R.~S.}%
astronomer royal of Ireland, now Lowndean Professor of
Astronomy and Geometry at Cambridge. His method is given
in a work entitled \textit{Theory of Screws}, Dublin, 1876, and in
\index{Screws, theory of}%
subsequent articles. Modern geometry is here drawn upon,
as was done also by Clifford in the related subject of Biquaternions.
\index{Biquaternions}%
\index{Clifford!ref.\ to}%
Arthur Buchheim of Manchester (1859--1888),
\index{Buchheim}%
showed that Grassmann's Ausdehnungslehre supplies all the
\index{Ausdehnungslehre}%
\index{Grassmann!ref.\ to}%
necessary materials for a simple calculus of screws in elliptic
space. Horace Lamb applied the theory of screws to the question
\index{Lamb}%
of the steady motion of any solid in a fluid.

Advances in theoretical mechanics, bearing on the integration
and the alteration in form of dynamical equations,
were made since Lagrange by Poisson, William Rowan Hamilton,
\index{Hamilton, W.~R.!ref.\ to}%
\index{Lagrange!ref.\ to}%
\index{Poisson!ref.\ to}%
Jacobi, Madame Kowalevski, and others. Lagrange had
\index{Jacobi!ref.\ to}%
established the ``Lagrangian form'' of the equations of
motion. He had given a theory of the variation of the
arbitrary constants which, however, turned out to be less
fruitful in results than a theory advanced by Poisson.\Reference{99} Poisson's
%% -----File: 393.png---Folio 379-------
\index{Geodesics}%
\index{Ultimate multiplier, theory of}%
theory of the variation of the arbitrary constants and
the method of integration thereby afforded marked the first
onward step since Lagrange. Then came the researches of
Sir William Rowan Hamilton. His discovery that the integration
\index{Hamilton, W.~R.!ref.\ to}%
of the dynamic differential equations is connected with
the integration of a certain partial differential equation of the
\index{Partial differential equations}%
first order and second degree, grew out of an attempt to deduce,
by the undulatory theory, results in geometrical optics previously
\index{Undulatory theory of light}%
based on the conceptions of the emission theory. The
\textit{Philosophical Transactions} of 1833 and 1834 contain Hamilton's
papers, in which appear the first applications to mechanics
of the principle of varying action and the characteristic
\index{Action!varying}%
\index{Varying action, principle of}%
function, established by him some years previously. The
object which Hamilton proposed to himself is indicated by
the title of his first paper, viz.\ the discovery of a function
by means of which all integral equations can be actually
represented. The new form obtained by him for the equation
of motion is a result of no less importance than that which
was the professed object of the memoir. Hamilton's method
of integration was freed by Jacobi of an unnecessary complication,
\index{Jacobi!ref.\ to}%
and was then applied by him to the determination of a
geodetic line on the general ellipsoid. With aid of elliptic \DPtypo{coordinates}{co-ordinates}
\index{Co-ordinates}%
\index{Elliptic co-ordinates}%
\index{Ellipsoid!(attraction of)}%
Jacobi integrated the partial differential equation
and expressed the equation of the geodetic in form of a
relation between two Abelian integrals. Jacobi applied to
\index{Abelian integrals}%
differential equations of dynamics the theory of the ultimate
multiplier. The differential equations of dynamics are only
one of the classes of differential equations considered by
Jacobi. Dynamic investigations along the lines of Lagrange,
Hamilton, and Jacobi were made by Liouville, A.~Desboves,
\index{Desboves}%
\index{Liouville!ref.\ to}%
Serret, J.~C.~F. Sturm, Ostrogradsky, J.~Bertrand, Donkin,
\index{Bertrand}%
\index{Donkin}%
\index{Ostrogradsky}%
\index{Serret!ref.\ to}%
\index{Sturm, J.~C.~F.!ref.\ to}%
Brioschi, leading up to the development of the theory of a
\index{Brioschi!ref.\ to}%
system of canonical integrals.
%% -----File: 394.png---Folio 380-------
\index{Hydrodynamics}%
\index{Ignoration of co-ordinates}%
\index{Kowalevsky}%

An important addition to the theory of the motion of a solid
body about a fixed point was made by Madame \textbf{Sophie de
Kowalevski}\Reference[]{96} (1853--1891), who discovered a new case in which
the differential equations of motion can be integrated. By
the use of theta-functions of two independent variables she
\index{Theta-functions}%
furnished a remarkable example of how the modern theory of
functions may become useful in mechanical problems. She
was a native of Moscow, studied under Weierstrass, obtained
the doctor's degree at Göttingen, and from 1884 until her
death was professor of higher mathematics at the University
of Stockholm. The research above mentioned received the
Bordin prize of the French Academy in 1888, which was
doubled on account of the exceptional merit of the paper.

There are in vogue three forms for the expression of the
kinetic energy of a dynamical system: the Lagrangian, the
Hamiltonian, and a modified form of Lagrange's equations in
which certain velocities are omitted. The kinetic energy
is expressed in the first form as a homogeneous quadratic
function of the velocities, which are the time-variations of the
co-ordinates of the system; in the second form, as a homogeneous
quadratic function of the momenta of the system;
the third form, elaborated recently by Edward John Routh
\index{Routh}%
of Cambridge, in connection with his theory of ``ignoration of
co-ordinates,'' and by A.~B. Basset, is of importance in hydrodynamical
\index{Basset}%
problems relating to the motion of perforated solids
in a liquid, and in other branches of physics.

In recent time great practical importance has come to be
attached to the principle of mechanical similitude. By it one
\index{Similitude (mechanical)}%
can determine from the performance of a model the action of
the machine constructed on a larger scale. The principle was
first enunciated by Newton (\textit{Principia}, Bk.~II., Sec.~VIII.,
\index{Newton!ref.\ to}%
Prop.~32), and was derived by Bertrand from the principle
\index{Bertrand}%
of virtual velocities. A corollary to it, applied in ship-building,
%% -----File: 395.png---Folio 381-------
\index{Hydrodynamics|(}%
goes by the name of William Froude's law, but was enunciated
\index{Froude}%
also by Reech.
\index{Reech}%

The present problems of dynamics differ materially from
those of the last century. The explanation of the orbital and
axial motions of the heavenly bodies by the law of universal
gravitation was the great problem solved by Clairaut, Euler,
D'Alembert, Lagrange, and Laplace. It did not involve the
consideration of frictional resistances. In the present time
the aid of dynamics has been invoked by the physical
sciences. The problems there arising are often complicated
by the presence of friction. Unlike astronomical problems of
a century ago, they refer to phenomena of matter and motion
that are usually concealed from direct observation. The great
pioneer in such problems is Lord Kelvin. While yet an
\index{Kelvin, Lord!ref.\ to}%
undergraduate at Cambridge, during holidays spent at the
seaside, he entered upon researches of this kind by working
out the theory of spinning tops, which previously had been
only partially explained by Jellet in his \textit{Treatise on the Theory
\index{Jellet!ref.\ to}%
of Friction} (1872), and by Archibald Smith.
\index{Smith, A.}%

Among standard works on mechanics are \textbf{Jacobi's} \textit{Vorlesungen
\index{Jacobi!ref.\ to}%
über Dynamik}, edited by Clebsch, 1866; \textbf{Kirchhoff's} \textit{Vorlesungen
\index{Clebsch!ref.\ to}%
\index{Kirchhoff!ref.\ to}%
über mathematische Physik}, 1876; \textbf{Benjamin Peirce's}
\index{Peirce, B.!ref.\ to}%
\textit{Analytic Mechanics}, 1855; \textbf{Somoff's} \textit{Theoretische Mechanik},
\index{Mechanics!more recent work|)}%
\index{Somoff}%
1879; \textbf{Tait and Steele's} \textit{Dynamics of a Particle}, 1856; \textbf{Minchin's}
\index{Minchin}%
\index{Steele}%
\index{Tait}%
\textit{Treatise on Statics}; \textbf{Routh's} \textit{Dynamics of a System of Rigid
\index{Dynamics|)}%
\index{Routh!ref.\ to}%
Bodies}; \textbf{Sturm's} \textit{Cours de Mécanique de l'École Polytechnique}.
\index{Sturm, J.~C.~F.!ref.\ to}%

The equations which constitute the foundation of the theory
of fluid motion were fully laid down at the time of Lagrange,
but the solutions actually worked out were few and mainly
of the irrotational type. A powerful method of attacking
problems in fluid motion is that of images, introduced in 1843
\index{Images, theory of}%
by George Gabriel Stokes of Pembroke College, Cambridge.
\index{Stokes}%
It received little attention until Sir William Thomson's discovery
\index{Thomson, Sir William!ref.\ to}%
%% -----File: 396.png---Folio 382-------
\index{Kelvin, Lord!ref.\ to}%
\index{Vortex rings}%
of electrical images, whereupon the theory was extended
by Stokes, Hicks, and Lewis. In 1849, Thomson gave the
\index{Hicks}%
\index{Lewis}%
\index{Stokes!ref.\ to}%
\index{Thomson, J.~J.}%
\index{Thomson, Sir William!ref.\ to}%
maximum and minimum theorem peculiar to hydrodynamics,
which was afterwards extended to dynamical problems in
general.

A new epoch in the progress of hydrodynamics was created,
in 1856, by Helmholtz, who worked out remarkable properties
\index{Helmholtz!ref.\ to}%
of rotational motion in a homogeneous, incompressible fluid,
devoid of viscosity. He showed that the vortex filaments in
such a medium may possess any number of knottings and twistings,
but are either endless or the ends are in the free surface
of the medium; they are indivisible. These results suggested
to Sir William Thomson the possibility of founding on them a
new form of the atomic theory, according to which every atom
\index{Atomic theory}%
is a vortex ring in a non-frictional ether, and as such must be
absolutely permanent in substance and duration. The vortex-atom
theory is discussed by J.~J. Thomson of Cambridge
(born 1856) in his classical treatise on the \textit{Motion of Vortex
Rings}, to which the Adams Prize was awarded in 1882.
Papers on vortex motion have been published also by Horace
\index{Vortex motion}%
Lamb, Thomas Craig, Henry~A. Rowland, and Charles Chree.
\index{Chree}%
\index{Craig, T.}%
\index{Lamb}%
\index{Rowland}%

The subject of jets was investigated by Helmholtz, Kirchhoff,
\index{Jets}%
\index{Kirchhoff!ref.\ to}%
Plateau, and Rayleigh; the motion of fluids in a fluid by
\index{Plateau}%
Stokes, Sir W.~Thomson, Köpcke, Greenhill, and Lamb; the
\index{Greenhill}%
\index{Kopcke@{Köpcke}}%
theory of viscous fluids by Navier, Poisson, Saint-Venant,
\index{Navier!ref.\ to}%
\index{Poisson!ref.\ to}%
\index{Saint-Venant!ref.\ to}%
Stokes, O.~E. Meyer, Stefano, Maxwell, Lipschitz, Craig,
\index{Lipschitz!ref.\ to}%
\index{Maxwell!ref.\ to}%
\index{Meyer, O.~E.}%
\index{Stefano}%
Helmholtz, and A.~B. Basset. Viscous fluids present great
\index{Basset}%
difficulties, because the equations of motion have not the same
degree of certainty as in perfect fluids, on account of a deficient
theory of friction, and of the difficulty of connecting
\index{Friction, theory of}%
oblique pressures on a small area with the differentials of the
velocities.

Waves in liquids have been a favourite subject with English
\index{Waves|(}%
%% -----File: 397.png---Folio 383-------
\index{Kelvin, Lord!ref.\ to}%
\index{Solitary wave}%
mathematicians. The early inquiries of Poisson and
\index{Poisson!ref.\ to}%
Cauchy were directed to the investigation of waves produced
\index{Cauchy!ref.\ to}%
by disturbing causes acting arbitrarily on a small portion
of the fluid. The velocity of the long wave was given
\index{Long wave}%
approximately by Lagrange in 1786 in case of a channel of
\index{Lagrange!ref.\ to}%
rectangular cross-section, by Green in 1839 for a channel of
\index{Green!ref.\ to}%
triangular section, and by P.~Kelland for a channel of any
\index{Kelland}%
uniform section. Sir George B. Airy, in his treatise on \textit{Tides
\index{Airy!ref.\ to}%
\index{Tides}%
and Waves}, discarded mere approximations, and gave the exact
equation on which the theory of the long wave in a channel of
uniform rectangular section depends. But he gave no general
solutions. J.~McCowan of University College at Dundee
\index{McCowan}%
discusses this topic more fully, and arrives at exact and
complete solutions for certain cases. The most important
application of the theory of the long wave is to the explanation
of tidal phenomena in rivers and estuaries.

The mathematical treatment of solitary waves was first
taken up by S.~Earnshaw in 1845, then by Stokes; but the first
\index{Earnshaw}%
\index{Stokes!ref.\ to}%
sound approximate theory was given by J.~Boussinesq in 1871,
\index{Boussinesq}%
who obtained an equation for their form, and a value for the
velocity in agreement with experiment. Other methods of
approximation were given by Rayleigh and J.~McCowan. In
\index{Rayleigh, Lord!ref.\ to}%
connection with deep-water waves, Osborne Reynolds gave in
\index{Reynolds}%
1877 the dynamical explanation for the fact that a group
of such waves advances with only half the rapidity of the
individual waves.

The solution of the problem of the general motion of an
ellipsoid in a fluid is due to the successive labours of Green
\index{Ellipsoid!motion of}%
(1833), Clebsch (1856), and Bjerknes (1873). The free
\index{Bjerknes, C. A.}%
\index{Clebsch!ref.\ to}%
motion of a solid in a liquid has been investigated by W.~Thomson,
\index{Thomson, Sir William!ref.\ to}%
Kirchhoff, and Horace Lamb. By these labours, the
\index{Kirchhoff!ref.\ to}%
\index{Lamb}%
motion of a single solid in a fluid has come to be pretty well
understood, but the case of two solids in a fluid is not developed
%% -----File: 398.png---Folio 384-------
\index{Hydrodynamics|)}%
\index{Spheroid (liquid)}%
so fully. The problem has been attacked by W.~M.
Hicks.
\index{Hicks}%

The determination of the period of oscillation of a rotating
liquid spheroid has important bearings on the question of the
origin of the moon. G.~H. Darwin's investigations thereon,
\index{Darwin!ref.\ to}%
viewed in the light of Riemann's and Poincaré's researches,
\index{Poincaré!ref.\ to}%
\index{Riemann!ref.\ to}%
seem to disprove Laplace's hypothesis that the moon separated
\index{Laplace!ref.\ to}%
from the earth as a ring, because the angular velocity was too
great for stability; Darwin finds no instability.

The explanation of the contracted vein has been a point of
\index{Contracted vein}%
much controversy, but has been put in a much better light by
the application of the principle of momentum, originated by
Froude and Rayleigh. Rayleigh considered also the reflection
\index{Froude}%
\index{Rayleigh, Lord!ref.\ to}%
of waves, not at the surface of separation of two uniform
media, where the transition is abrupt, but at the confines of
two media between which the transition is gradual.

The first serious study of the circulation of winds on the
\index{Winds|(}%
earth's surface was instituted at the beginning of the second
quarter of this century by \textit{H.~W. Dové}, \textit{William~C. Redfield}, and
\index{Dové}%
\index{Redfield}%
\textit{James~P. Espy}, followed by researches of \textit{W.~Reid}, \textit{Piddington},
\index{Espy}%
\index{Piddington}%
\index{Reid}%
and \textit{Elias Loomis}. But the deepest insight into the wonderful
\index{Loomis}%
correlations that exist among the varied motions of the
atmosphere was obtained by \textbf{William Ferrel} (1817--1891). He
\index{Ferrel}%
was born in Fulton County, Pa., and brought up on a farm.
Though in unfavourable surroundings, a burning thirst for
knowledge spurred the boy to the mastery of one branch after
another. He attended Marshall College, Pa., and graduated
in 1844 from Bethany College. While teaching school he
became interested in meteorology and in the subject of tides.
\index{Meteorology|(}%
In 1856 he wrote an article on ``the winds and currents of the
ocean.'' The following year he became connected with the
\textit{Nautical Almanac}. A mathematical paper followed in 1858
on ``the motion of fluids and solids relative to the earth's
%% -----File: 399.png---Folio 385-------
surface.'' The subject was extended afterwards so as to
embrace the mathematical theory of cyclones, tornadoes,
water-spouts, etc. In 1885 appeared his \textit{Recent Advances in
Meteorology}. In the opinion of a leading European meteorologist
(\textit{Julius Hann} of Vienna), Ferrel has ``contributed more
\index{Hann}%
to the advance of the physics of the atmosphere than any
other living physicist or meteorologist.''

Ferrel teaches that the air flows in great spirals toward the
poles, both in the upper strata of the atmosphere and on the
earth's surface beyond the 30th~degree of latitude; while
the return current blows at nearly right angles to the above
spirals, in the middle strata as well as on the earth's surface,
in a zone comprised between the parallels $30°$~N. and $30°$~S. The
idea of three superposed currents blowing spirals was first
advanced by James Thomson, but was published in very
\index{Thomson, J.}%
meagre abstract.

Ferrel's views have given a strong impulse to theoretical
research in America, Austria, and Germany. Several objections
raised against his argument have been abandoned, or
have been answered by W.~M. Davis of Harvard. The mathematical
\index{Davis, W.~M.}%
analysis of F.~Waldo of Washington, and of others,
\index{Waldo}%
has further confirmed the accuracy of the theory. The transport
of Krakatoa dust and observations made on clouds point
toward the existence of an upper east current on the equator,
and Pernter has mathematically deduced from Ferrel's theory
\index{Pernter, J.~M.}%
the existence of such a current.

Another theory of the general circulation of the atmosphere
was propounded by Werner Siemens of Berlin, in which an
\index{Siemens}%
attempt is made to apply thermodynamics to aërial currents.
\index{Thermodynamics}%
Important new points of view have been introduced recently
by Helmholtz, who concludes that when two air currents blow
\index{Helmholtz!ref.\ to}%
one above the other in different directions, a system of air
waves must arise in the same way as waves are formed on the
\index{Waves|)}%
%% -----File: 400.png---Folio 386-------
\index{Meteorology|)}%
\index{Winds|)}%
sea. He and A.~Oberbeck showed that when the waves on the
\index{Oberbeck}%
sea attain lengths of from $16$ to $33$~feet, the air waves must
attain lengths of from $10$ to $20$~miles, and proportional depths.
Superposed strata would thus mix more thoroughly, and their
energy would be partly dissipated. From hydrodynamical
equations of rotation Helmholtz established the reason why
\index{Helmholtz!ref.\ to}%
the observed velocity from equatorial regions is much less in
a latitude of, say, $20°$ or~$30°$, than it would be were the movements
unchecked.

About 1860 acoustics began to be studied with renewed
\index{Acoustics}%
zeal. The mathematical theory of pipes and vibrating strings
had been elaborated in the eighteenth century by Daniel Bernoulli,
\index{Bernoulli, Daniel!ref.\ to}%
D'Alembert, Euler, and Lagrange. In the first part of
\index{D'Alembert!ref.\ to}%
\index{Euler!ref.\ to}%
\index{Lagrange!ref.\ to}%
the present century Laplace corrected Newton's theory on the
\index{Laplace!ref.\ to}%
\index{Newton!ref.\ to}%
velocity of sound in gases, Poisson gave a mathematical discussion
\index{Poisson!ref.\ to}%
of torsional vibrations; Poisson, Sophie Germain, and
\index{Germain, Sophie!ref.\ to}%
Wheatstone studied Chladni's figures; Thomas Young and the
\index{Chladni's figures}%
\index{Wheatstone}%
\index{Young!ref.\ to}%
brothers Weber developed the wave-theory of sound. \textbf{Sir J.~F.~W.
Herschel} wrote on the mathematical theory of sound for
\index{Herschel, J.~F.~W.}%
the \textit{Encyclopædia Metropolitana}, 1845. Epoch-making were
Helmholtz's experimental and mathematical researches. In
his hands and Rayleigh's, Fourier's series received due
\index{Fourier's series}%
\index{Rayleigh, Lord}%
attention. Helmholtz gave the mathematical theory of beats,
difference tones, and summation tones. \textbf{Lord Rayleigh} (John
William Strutt) of Cambridge (born 1842) made extensive
\index{Strutt, J.~W.}%
mathematical researches in acoustics as a part of the theory of
vibration in general. Particular mention may be made of his
discussion of the disturbance produced by a spherical obstacle
on the waves of sound, and of phenomena, such as sensitive
flames, connected with the instability of jets of fluid. In 1877
\index{Jets}%
and 1878 he published in two volumes a treatise on \textit{The Theory
of Sound}. Other mathematical researches on this subject have
been made in England by Donkin and Stokes.
\index{Stokes!ref.\ to}%
%% -----File: 401.png---Folio 387-------

The theory of elasticity\Reference[]{42} belongs to this century. Before
\index{Elasticity|(}%
1800 no attempt had been made to form general equations for
the motion or equilibrium of an elastic solid. Particular problems
had been solved by special hypotheses. Thus, James
Bernoulli considered elastic laminæ; Daniel Bernoulli and
\index{Bernoulli, James (born 1758)}%
Euler investigated vibrating rods; Lagrange and Euler, the
\index{Euler!ref.\ to}%
\index{Lagrange!ref.\ to}%
\index{Vibrating rods}%
equilibrium of springs and columns. The earliest investigations
of this century, by Thomas Young (``Young's modulus of
\index{Young!ref.\ to}%
elasticity'') in England, J.~Binet in France, and G.~A.~A. Plana
\index{Binet}%
\index{Plana}%
in Italy, were chiefly occupied in extending and correcting the
earlier labours. Between 1830 and 1840 the broad outline of the
modern theory of elasticity was established. This was accomplished
almost exclusively by French writers,---Louis-Marie-Henri
Navier (1785--1836), Poisson, Cauchy, Mademoiselle
\index{Cauchy!ref.\ to}%
\index{Navier}%
\index{Poisson}%
\index{Poisson!ref.\ to}%
Sophie Germain (1776--1831), Félix Savart (1791--1841).
\index{Germain, Sophie}%
\index{Savart}%

\textbf{Siméon Denis Poisson}\Reference[]{94} (1781--1840) was born at Pithiviers.
The boy was put out to a nurse, and he used to tell that when
his father (a common soldier) came to see him one day, the
nurse had gone out and left him suspended by a thin cord to a
nail in the wall in order to protect him from perishing under
the teeth of the carnivorous and unclean animals that roamed
on the floor. Poisson used to add that his gymnastic efforts
when thus suspended caused him to swing back and forth, and
thus to gain an early familiarity with the pendulum, the study
of which occupied him much in his maturer life. His father
destined him for the medical profession, but so repugnant was
this to him that he was permitted to enter the Polytechnic
School at the age of seventeen. His talents excited the interest
of Lagrange and Laplace. At eighteen he wrote a memoir
on finite differences which was printed on the recommendation
of Legendre. He soon became a lecturer at the school, and
continued through life to hold various government scientific
posts and professorships. He prepared some 400 publications,
%% -----File: 402.png---Folio 388-------
\index{Capillarity}%
\index{Kelvin, Lord!ref.\ to}%
mainly on applied mathematics. His \textit{Traité de Mécanique},
2~vols., 1811 and 1833, was long a standard work. He wrote
on the mathematical theory of heat, capillary action, probability
of judgment, the mathematical theory of electricity and
magnetism, physical astronomy, the attraction of ellipsoids,
definite integrals, series, and the theory of elasticity. He was
considered one of the leading analysts of his time.

His work on elasticity is hardly excelled by that of Cauchy,
\index{Cauchy!ref.\ to}%
and second only to that of Saint-Venant. There is hardly a
problem in elasticity to which he has not contributed, while
many of his inquiries were new. The equilibrium and motion
of a circular plate was first successfully treated by him.
Instead of the definite integrals of earlier writers, he used
preferably finite summations. Poisson's contour conditions
for elastic plates were objected to by Gustav Kirchhoff of
\index{Kirchhoff!ref.\ to}%
Berlin, who established new conditions. But Thomson and
\index{Thomson, Sir William!ref.\ to}%
Tait in their \textit{Treatise on Natural Philosophy} have explained
\index{Tait}%
the discrepancy between Poisson's and Kirchhoff's boundary
conditions, and established a reconciliation between them.

Important contributions to the theory of elasticity were
made by Cauchy. To him we owe the origin of the theory
of stress, and the transition from the consideration of the
force upon a molecule exerted by its neighbours to the consideration
of the stress upon a small plane at a point. He
anticipated Green and Stokes in giving the equations of isotropic
\index{Green!ref.\ to}%
\index{Stokes!ref.\ to}%
elasticity with two constants. The theory of elasticity
was presented by Gabrio Piola of Italy according to the principles
\index{Piola}%
of Lagrange's \textit{Mécanique Analytique}, but the superiority
of this method over that of Poisson and Cauchy is far from
evident. The influence of temperature on stress was first
investigated experimentally by Wilhelm Weber of Göttingen,
\index{Weber, W.~E.!ref.\ to}%
and afterwards mathematically by Duhamel, who, assuming
\index{Duhamel}%
Poisson's theory of elasticity, examined the alterations of
%% -----File: 403.png---Folio 389-------
form which the formulæ undergo when we allow for changes
of temperature. Weber was also the first to experiment
on elastic after-strain. Other important experiments were
made by different scientists, which disclosed a wider range
of phenomena, and demanded a more comprehensive theory.
Set was investigated by Gerstner (1756--1832) and Eaton
\index{Gerstner}%
Hodgkinson, while the latter physicist in England and Vicat
\index{Hodgkinson}%
\index{Vicat}%
(1786--1861) in France experimented extensively on absolute
strength. Vicat boldly attacked the mathematical theories of
flexure because they failed to consider shear and the time-element.
\index{Flexure, theory of}%
As a result, a truer theory of flexure was soon propounded
by Saint-Venant. Poncelet advanced the theories of
\index{Poncelet!ref.\ to}%
\index{Saint-Venant!ref.\ to}%
resilience and cohesion.

\textbf{Gabriel Lamé}\Reference[]{94} (1795--1870) was born at Tours, and graduated
\index{Lame@{Lamé}}%
\index{Lame@{Lamé}!ref.\ to}%
at the Polytechnic School. He was called to Russia
with Clapeyron and others to superintend the construction of
bridges and roads. On his return, in 1832, he was elected
professor of physics at the Polytechnic School. Subsequently
he held various engineering posts and professorships in Paris.
As engineer he took an active part in the construction of the
first railroads in France. Lamé devoted his fine mathematical
talents mainly to mathematical physics. In four works:
\textit{Leçons sur les fonctions inverses des transcendantes et les surfaces
isothermes}; \textit{Sur les coordonnées curvilignes et leurs diverses
applications}; \textit{Sur la théorie analytique de la chaleur}; \textit{Sur la
théorie mathématique de l'élasticité des corps solides} (1852), and
in various memoirs he displays fine analytical powers; but a
certain want of physical touch sometimes reduces the value of
his contributions to elasticity and other physical subjects. In
considering the temperature in the interior of an ellipsoid
under certain conditions, he employed functions analogous to
Laplace's functions, and known by the name of ``Lamé's functions.''
\index{Lame's@{Lamé's functions}}%
A problem in elasticity called by Lamé's name, viz.\
%% -----File: 404.png---Folio 390-------
\index{Jellet!ref.\ to}%
to investigate the conditions for equilibrium of a spherical
elastic envelope subject to a given distribution of load on
the bounding spherical surfaces, and the determination of the
resulting shifts is the only completely general problem on
elasticity which can be said to be completely solved. He
deserves much credit for his derivation and transformation
of the general elastic equations, and for his application of
them to double refraction. Rectangular and triangular membranes
were shown by him to be connected with questions in
the theory of numbers. The field of photo-elasticity was
entered upon by Lamé, F.~E. Neumann, Clerk Maxwell.
\index{Maxwell!ref.\ to}%
\index{Neumann, F.~E.!ref.\ to}%
Stokes, Wertheim, R.~Clausius, Jellett, threw new light upon
\index{Clausius!ref.\ to}%
\index{Light, theory of}%
\index{Stokes!ref.\ to}%
\index{Wertheim}%
the subject of ``rari-constancy'' and ``multi-constancy,'' which
\index{Multi-constancy}%
\index{Rari-constancy}%
has long divided elasticians into two opposing factions. The
uni-constant isotropy of Navier and Poisson had been questioned
\index{Navier!ref.\ to}%
\index{Poisson!ref.\ to}%
by Cauchy, and was now severely criticised by Green
\index{Cauchy!ref.\ to}%
\index{Green!ref.\ to}%
and Stokes.

\textbf{Barré de Saint-Venant} (1797--1886), ingénieur des ponts et
\index{Saint-Venant}%
chaussées, made it his life-work to render the theory of
elasticity of practical value. The charge brought by practical
engineers, like Vicat, against the theorists led Saint-Venant to
\index{Vicat!ref.\ to}%
place the theory in its true place as a guide to the practical
man. Numerous errors committed by his predecessors were
removed. He corrected the theory of flexure by the consideration
of slide, the theory of elastic rods of double curvature by
the introduction of the third moment, and the theory of torsion
by the discovery of the distortion of the primitively
plane section. His results on torsion abound in beautiful
graphic illustrations. In case of a rod, upon the side surfaces
of which no forces act, he showed that the problems of flexure
and torsion can be solved, if the end-forces are distributed
over the end-surfaces by a definite law. Clebsch, in his
\index{Clebsch!ref.\ to|(}%
\textit{Lehrbuch der Elasticität}, 1862, showed that this problem is
%% -----File: 405.png---Folio 391-------
reversible to the case of side-forces without end-forces.
Clebsch\Reference[]{68} extended the research to very thin rods and to very
thin plates. Saint-Venant considered problems arising in the
scientific design of built-up artillery, and his solution of them
differs considerably from Lamé's solution, which was popularised
by Rankine, and much used by gun-designers. In Saint-Venant's
translation into French of Clebsch's \textit{Elasticität}, he
develops extensively a double-suffix notation for strain and
stresses. Though often advantageous, this notation is cumbrous,
and has not been generally adopted. \textit{Karl Pearson},
\index{Pearson}%
professor in University College, London, has recently examined
mathematically the permissible limits of the application
of the ordinary theory of flexure of a beam.

The mathematical theory of elasticity is still in an unsettled
condition. Not only are scientists still divided into two
schools of ``rari-constancy'' and ``multi-constancy,'' but difference
\index{Multi-constancy}%
of opinion exists on other vital questions. Among the
numerous modern writers on elasticity may be mentioned
Émile Mathieu (1835--1891), professor at Besançon, Maurice
\index{Mathieu}%
Levy of Paris, Charles Chree, superintendent of the Kew Observatory,
\index{Chree}%
\index{Levy}%
A.~B. Basset, Sir William Thomson (Lord Kelvin)
\index{Kelvin, Lord!ref.\ to}%
\index{Thomson, Sir William!ref.\ to}%
of Glasgow, J.~Boussinesq of Paris, and others. Sir William
\index{Boussinesq}%
Thomson applied the laws of elasticity of solids to the investigation
of the earth's elasticity, which is an important element
\index{Earth!rigidity of}%
in the theory of ocean-tides. If the earth is a solid, then its
elasticity co-operates with gravity in opposing deformation
due to the attraction of the sun and moon. Laplace had
shown how the earth would behave if it resisted deformation
only by gravity. Lamé had investigated how a solid sphere
would change if its elasticity only came into play. Sir
William Thomson combined the two results, and compared
them with the actual deformation. Thomson, and afterwards
G.~H. Darwin, computed that the resistance of the earth to
\index{Darwin!ref.\ to}%
%% -----File: 406.png---Folio 392-------
tidal deformation is nearly as great as though it were of steel.
This conclusion has been confirmed recently by Simon Newcomb,
\index{Newcomb!ref.\ to}%
from the study of the observed periodic changes in
latitude. For an ideally rigid earth the period would be $360$~days,
\index{Latitude, periodic changes in}%
but if as rigid as steel, it would be~$441$, the observed
period being $430$~days.

Among text-books on elasticity may be mentioned the works
\index{Elasticity|)}%
of Lamé, Clebsch, Winkler, Beer, Mathieu, W.~J. Ibbetson, and
\index{Beer}%
\index{Clebsch!ref.\ to|)}%
\index{Ibbetson}%
\index{Lame@{Lamé}!ref.\ to}%
\index{Mathieu!ref.\ to}%
\index{Winkler}%
F.~Neumann, edited by O.~E. Meyer.
\index{Meyer, O.~E.}%
\index{Neumann, F.~E.!ref.\ to}%

Riemann's opinion that a \emph{science} of physics only exists since
\index{Riemann!ref.\ to}%
the invention of differential equations finds corroboration even
in this brief and fragmentary outline of the progress of mathematical
physics. The undulatory theory of light, first advanced
\index{Undulatory theory of light|(}%
by Huygens, owes much to the power of mathematics:
\index{Huygens!ref.\ to}%
by mathematical analysis its assumptions were worked out
to their last consequences. \textbf{Thomas Young}\Reference[]{95} (1773--1829) was
\index{Young}%
the first to explain the principle of interference, both of
light and sound, and the first to bring forward the idea
of transverse vibrations in light waves. Young's explanations,
not being verified by him by extensive numerical calculations,
attracted little notice, and it was not until \textbf{Augustin
Fresnel} (1788--1827) applied mathematical analysis to a much
\index{Fresnel}%
greater extent than Young had done, that the undulatory
theory began to carry conviction. Some of Fresnel's mathematical
assumptions were not satisfactory; hence Laplace,
\index{Laplace!ref.\ to}%
Poisson, and others belonging to the strictly mathematical
\index{Poisson!ref.\ to}%
school, at first disdained to consider the theory. By their
opposition Fresnel was spurred to greater exertion. Arago
\index{Arago}%
was the first great convert made by Fresnel. When polarisation
and double refraction were explained by Young and
Fresnel, then Laplace was at last won over. Poisson drew
from Fresnel's formulæ the seemingly paradoxical deduction
that a small circular disc, illuminated by a luminous point,
%% -----File: 407.png---Folio 393-------
\index{Kelvin, Lord!ref.\ to}%
\index{MacCullagh!ref.\ to}%
must cast a shadow with a bright spot in the centre. But
this was found to be in accordance with fact. The theory
was taken up by another great mathematician, Hamilton, who
\index{Hamilton, W.~R.!ref.\ to}%
from his formulæ predicted conical refraction, verified experimentally
by Lloyd. These predictions do not prove, however,
\index{Lloyd}%
that Fresnel's formulæ are correct, for these prophecies might
have been made by other forms of the wave-theory. The
theory was placed on a sounder dynamical basis by the writings
of Cauchy, Biot, Green, C.~Neumann, Kirchhoff, McCullagh,
\index{Biot}%
\index{Cauchy!ref.\ to}%
\index{Green!ref.\ to}%
\index{Kirchhoff!ref.\ to}%
\index{McCullagh}%
\index{Neumann, C.!ref.\ to}%
Stokes, Saint-Venant, Sarrau, Lorenz, and Sir William Thomson.
\index{Lorenz}%
\index{Saint-Venant!ref.\ to}%
\index{Sarrau}%
\index{Stokes!ref.\ to}%
\index{Thomson, Sir William!ref.\ to}%
In the wave-theory, as taught by Green and others, the
luminiferous ether was an incompressible elastic solid, for
\index{Ether, luminiferous}%
the reason that fluids could not propagate transverse vibrations.
But, according to Green, such an elastic solid would
transmit a longitudinal disturbance with infinite velocity.
Stokes remarked, however, that the ether might act like
a fluid in case of finite disturbances, and like an elastic solid
in case of the infinitesimal disturbances in light propagation.

Fresnel postulated the density of ether to be different in
different media, but the elasticity the same, while C.~Neumann
and McCullagh assume the density uniform and the elasticity
different in all substances. On the latter assumption the
direction of vibration lies in the plane of polarisation, and not
perpendicular to it, as in the theory of Fresnel.

While the above writers endeavoured to explain all optical
properties of a medium on the supposition that they arise
entirely from difference in rigidity or density of the ether in
the medium, there is another school advancing theories in
which the mutual action between the molecules of the body
and the ether is considered the main cause of refraction and
dispersion.\Reference{100} The chief workers in this field are J.~Boussinesq,
\index{Boussinesq}%
W.~Sellmeyer, Helmholtz, E.~Lommel, E.~Ketteler, W.~Voigt,
\index{Helmholtz!ref.\ to}%
\index{Ketteler}%
\index{Lommel}%
\index{Sellmeyer}%
\index{Voigt}%
and Sir William Thomson in his lectures delivered at the
%% -----File: 408.png---Folio 394-------
\index{Kelvin, Lord!ref.\ to}%
\index{Undulatory theory of light|)}%
Johns Hopkins University in 1884. Neither this nor the
first-named school succeeded in explaining all the phenomena.
A third school was founded by Maxwell. He proposed the
\index{Maxwell!ref.\ to}%
electro-magnetic theory, which has received extensive development
\index{Electro-magnetic theory of light}%
recently. It will be mentioned again later. According
to Maxwell's theory, the direction of vibration does not lie
exclusively in the plane of polarisation, nor in a plane perpendicular
to it, but something occurs in both planes---a magnetic
vibration in one, and an electric in the other. Fitzgerald and
\index{Fitzgerald}%
Trouton in Dublin verified this conclusion of Maxwell by
\index{Trouton}%
experiments on electro-magnetic waves.

Of recent mathematical and experimental contributions to
optics, mention must be made of H.~A. Rowland's theory of
\index{Rowland}%
concave gratings, and of A.~A. Michelson's work on interference,
\index{Michelson}%
and his application of interference methods to astronomical
measurements.

In electricity the mathematical theory and the measurements
\index{Electricity|(}%
of \textbf{Henry Cavendish} (1731--1810), and in magnetism
\index{Cavendish}%
the measurements of \textbf{Charles Augustin Coulomb} (1736--1806),
\index{Coulomb}%
became the foundations for a system of measurement. For
electro-magnetism the same thing was done by \textbf{Andrè Marie
Ampère} (1775--1836). The first complete method of measurement
\index{Ampère}%
was the system of absolute measurements of terrestrial
magnetism introduced by \textit{Gauss} and \textbf{Wilhelm Weber} (1804--1891)
\index{Weber, W.~E.}%
and afterwards extended by Wilhelm Weber and F.~Kohlrausch
\index{Kohlrausch}%
to electro-magnetism and electro-statics. In 1861
the British Association and the Royal Society appointed a
special commission with Sir William Thomson at the head, to
\index{Thomson, Sir William!ref.\ to}%
consider the unit of electrical resistance. The commission
recommended a unit in principle like W. Weber's, but greater
than Weber's by a factor of~$10^7$.\Reference{101} The discussions and labours
on this subject continued for twenty years, until in 1881 a
general agreement was reached at an electrical congress in Paris.
%% -----File: 409.png---Folio 395-------
\index{Kelvin, Lord|(}%
\index{Kelvin, Lord!ref.\ to}%

A function of fundamental importance in the mathematical
theories of electricity and magnetism is the ``potential.'' It
\index{Potential}%
was first used by Lagrange in the determination of gravitational
attractions in 1773. Soon after, Laplace gave the
\index{Laplace!ref.\ to}%
celebrated differential equation,
\[
\frac{\partial^2V}{\DPtypo{d}{\partial}x^2} +
\frac{\partial^2V}{\DPtypo{d}{\partial}y^2} +
\frac{\partial^2V}{\DPtypo{d}{\partial}z^2} = 0,
\]
which was extended by Poisson by writing $-4\pi k$ in place of
\index{Poisson!ref.\ to}%
zero in the right-hand member of the equation, so that it
applies not only to a point external to the attracting mass,
but to any point whatever. The first to apply the potential
function to other than gravitation problems was \textbf{George Green}
\index{Green}%
\index{Green!ref.\ to}%
(1793--1841). He introduced it into the mathematical theory
of electricity and magnetism. Green was a self-educated man
who started out as a baker, and at his death was fellow of
Caius College, Cambridge. In 1828 he published by subscription
at Nottingham a paper entitled \textit{Essay on the application
of mathematical analysis to the theory of electricity and magnetism}.
It escaped the notice even of English mathematicians
until 1846, when Sir William Thomson had it reprinted in
\index{Thomson, Sir William}%
\index{Thomson, Sir William!ref.\ to}%
\textit{Crelle's Journal}, vols.~xliv.\ and xlv. It contained what is now
known as ``Green's theorem'' for the treatment of potential.
Meanwhile all of Green's general theorems had been re-discovered
by Sir William Thomson, Chasles, Sturm, and Gauss.
The term \emph{potential function} is due to Green. Hamilton used
the word \emph{force-function}, while Gauss, who about 1840 secured
\index{Force-function}%
the general adoption of the function, called it simply \emph{potential}.

Large contributions to electricity and magnetism have been
made by \textbf{William Thomson}. He was born in 1824 at Belfast,
Ireland, but is of Scotch descent. He and his brother James
studied in Glasgow. From there he entered Cambridge, and
was graduated as Second Wrangler in 1845. William Thomson,
%% -----File: 410.png---Folio 396-------
\index{Addition theorem of elliptic integrals}%
Sylvester, Maxwell, Clifford, and J.~J. Thomson are a group
\index{Clifford!ref.\ to}%
\index{Maxwell!ref.\ to}%
\index{Sylvester!ref.\ to}%
\index{Thomson, J.~J.!ref.\ to}%
\index{Thomson, Sir William}%
of great men who were Second Wranglers at Cambridge. At
the age of twenty-two W.~Thomson was elected professor of
natural philosophy in the University of Glasgow, a position
which he has held ever since. For his brilliant mathematical
and physical achievements he was knighted, and in 1892 was
made Lord Kelvin. His researches on the theory of potential
\index{Kelvin, Lord|)}%
are epoch-making. What is called ``Dirichlet's principle''
\index{Dirichlet!ref.\ to}%
was discovered by him in 1848, somewhat earlier than by
Dirichlet. We owe to Sir William Thomson new synthetical
methods of great elegance, viz.\ the theory of electric images
and the method of electric inversion founded thereon. By
them he determined the distribution of electricity on a bowl,
a problem previously considered insolvable. The distribution
of static electricity on conductors had been studied before this
mainly by Poisson and Plana. In 1845 F.~E. Neumann of
\index{Neumann, F.~E.!ref.\ to}%
\index{Plana}%
\index{Poisson!ref.\ to}%
Königsberg developed from the experimental laws of Lenz the
mathematical theory of magneto-electric induction. In 1855
W.~Thomson predicted by mathematical analysis that the discharge
of a Leyden jar through a linear conductor would in
\index{Leyden jar}%
certain cases consist of a series of decaying oscillations. This
was first established experimentally by Joseph Henry of
\index{Henry}%
Washington. William Thomson worked out the electro-static
induction in submarine cables. The subject of the screening
effect against induction, due to sheets of different metals, was
worked out mathematically by Horace Lamb and also by
\index{Lamb}%
Charles Niven. W.~Weber's chief researches were on electro-dynamics.
\index{Niven}%
\index{Weber, W.~E.!ref.\ to}%
Helmholtz in 1851 gave the mathematical theory
\index{Helmholtz!ref.\ to}%
of the course of induced currents in various cases. \textbf{Gustav
Robert Kirchhoff}\Reference[]{97} (1824--1887) investigated the distribution of
\index{Kirchhoff}%
\index{Kirchhoff!ref.\ to}%
a current over a flat conductor, and also the strength of current
in each branch of a network of linear conductors.

The entire subject of electro-magnetism was revolutionised
%% -----File: 411.png---Folio 397-------
\index{Conservation!of energy}%
\index{Energy, conservation of}%
by \textbf{James Clerk Maxwell} (1831--1879). He was born near
\index{Maxwell}%
Edinburgh, entered the University of Edinburgh, and became
a pupil of Kelland and Forbes. In 1850 he went to Trinity
\index{Forbes}%
\index{Kelland}%
College, Cambridge, and came out Second Wrangler, E.~Routh
\index{Routh!ref.\ to}%
being Senior Wrangler. Maxwell then became lecturer at
Cambridge, in 1856 professor at Aberdeen, and in 1860
professor at King's College, London. In 1865 he retired to
private life until 1871, when he became professor of physics
at Cambridge. Maxwell not only translated into mathematical
language the experimental results of Faraday, but established
the electro-magnetic theory of light, since verified experimentally
by Hertz. His first researches thereon were published
in 1864. In 1871 appeared his great \textit{Treatise on Electricity
and Magnetism}. He constructed the electro-magnetic theory
from general equations, which are established upon purely
dynamical principles, and which determine the state of the
electric field. It is a mathematical discussion of the stresses
and strains in a dielectric medium subjected to electro-magnetic
forces. The electro-magnetic theory has received developments
from Lord Rayleigh, J.~J. Thomson, H.~A. Rowland, R.~T.
\index{Rayleigh, Lord!ref.\ to}%
\index{Rowland}%
\index{Thomson, J.~J.!ref.\ to}%
Glazebrook, H.~Helmholtz, L.~Boltzmann, O.~Heaviside, J.~H.
\index{Boltzmann}%
\index{Glazebrook}%
\index{Heaviside}%
\index{Helmholtz}%
Poynting, and others. \textbf{Hermann von~Helmholtz} turned his
\index{Poynting}%
attention to this part of the subject in 1871. He was born
in 1821 at Potsdam, studied at the University of Berlin, and
published in 1847 his pamphlet \textit{Ueber die Erhaltung der Kraft}.
He became teacher of anatomy in the Academy of Art in
Berlin. He was elected professor of physiology at Königsberg
in 1849, at Bonn in 1855, at Heidelberg in 1858. It was
at Heidelberg that he produced his work on \textit{Tonempfindung}.
In 1871 he accepted the chair of physics at the University of
Berlin. From this time on he has been engaged chiefly on
inquiries in electricity and hydrodynamics. Helmholtz aimed
to determine in what direction experiments should be made to
%% -----File: 412.png---Folio 398-------
\index{Electricity|)}%
\index{Helmholtz}%
\index{Thermodynamics|(}%
decide between the theories of W.~Weber, F.~E. Neumann,
\index{Neumann, F.~E.}%
\index{Weber, W.~E.!ref.\ to}%
Riemann, and Clausius, who had attempted to explain electro-dynamic
\index{Clausius!ref.\ to}%
\index{Riemann!ref.\ to}%
phenomena by the assumption of forces acting at a distance
between two portions of the hypothetical electrical fluid,---the
intensity being dependent not only on the distance, but also
on the velocity and acceleration,---and the theory of Faraday
\index{Faraday}%
and Maxwell, which discarded action at a distance and assumed
\index{Maxwell!ref.\ to}%
stresses and strains in the dielectric. His experiments favoured
the British theory. He wrote on abnormal dispersion, and
created analogies between electro-dynamics and hydrodynamics.
Lord Rayleigh compared electro-magnetic problems
\index{Rayleigh, Lord!ref.\ to}%
with their mechanical analogues, gave a dynamical theory of
diffraction, and applied Laplace's coefficients to the theory of
\index{Laplace!ref.\ to}%
radiation. Rowland made some emendations on Stokes' paper
\index{Rowland}%
\index{Stokes!ref.\ to}%
on diffraction and considered the propagation of an arbitrary
electro-magnetic disturbance and spherical waves of light.
Electro-magnetic induction has been investigated mathematically
by Oliver Heaviside, and he showed that in a cable it is
\index{Heaviside}%
an actual benefit. Heaviside and Poynting have reached
\index{Poynting}%
remarkable mathematical results in their interpretation and
development of Maxwell's theory. Most of Heaviside's papers
have been published since 1882; they cover a wide field.

One part of the theory of capillary attraction, left defective
by Laplace, namely, the action of a solid upon a liquid, and
the mutual action between two liquids, was made dynamically
perfect by Gauss. He stated the rule for angles of contact
\index{Gauss!ref.\ to}%
between liquids and solids. A similar rule for liquids was
established by Ernst Franz Neumann. Chief among recent
workers on the mathematical theory of capillarity are Lord
\index{Capillarity}%
Rayleigh and E.~Mathieu.
\index{Mathieu!ref.\ to}%

The great principle of the conservation of energy was
\index{Conservation!of energy}%
\index{Energy, conservation of}%
established by \textbf{Robert Mayer} (1814--1878), a physician in
\index{Mayer}%
Heilbronn, and again independently by Colding of Copenhagen,
\index{Colding}%
%% -----File: 413.png---Folio 399-------
\index{Kelvin, Lord!ref.\ to}%
Joule, and Helmholtz. \textbf{James Prescott Joule} (1818--1889)
\index{Joule}%
determined experimentally the mechanical equivalent
of heat. Helmholtz in 1847 applied the conceptions of the
\index{Heat, theory of|(}%
transformation and conservation of energy to the various
branches of physics, and thereby linked together many well-known
phenomena. These labours led to the abandonment
of the corpuscular theory of heat. The mathematical treatment
of thermic problems was demanded by practical considerations.
Thermodynamics grew out of the attempt to
determine mathematically how much work can be gotten out
of a steam engine. \textbf{Sadi-Carnot}, an adherent of the corpuscular
\index{Carnot, Sadi}%
theory, gave the first impulse to this. The principle known
by his name was published in 1824. Though the importance
of his work was emphasised by \textit{B.~P.~E. Clapeyron}, it did not
\index{Clapeyron}%
meet with general recognition until it was brought forward
by William Thomson. The latter pointed out the necessity
\index{Thomson, Sir William!ref.\ to}%
of modifying Carnot's reasoning so as to bring it into accord
with the new theory of heat. William Thomson showed in
1848 that Carnot's principle led to the conception of an
absolute scale of temperature. In 1849 he published ``an
account of Carnot's theory of the motive power of heat, with
numerical results deduced from Regnault's experiments.'' In
February, 1850, \textbf{Rudolph Clausius} (1822--1888), then in Zürich
\index{Clausius}%
(afterwards professor in Bonn), communicated to the Berlin
Academy a paper on the same subject which contains the
Protean second law of thermodynamics. In the same month
\textbf{William John~M. Rankine} (1820--1872), professor of engineering
\index{Rankine}%
and mechanics at Glasgow, read before the Royal Society
of Edinburgh a paper in which he declares the nature of
heat to consist in the rotational motion of molecules, and
arrives at some of the results reached previously by Clausius.
He does not mention the second law of thermodynamics, but
in a subsequent paper he declares that it could be derived
%% -----File: 414.png---Folio 400-------
\index{Kelvin, Lord!ref.\ to}%
from equations contained in his first paper. His proof of
the second law is not free from objections. In March, 1851,
appeared a paper of William Thomson which contained a
\index{Thomson, Sir William!ref.\ to}%
perfectly rigorous proof of the second law. He obtained it
before he had seen the researches of Clausius. The statement
\index{Clausius!ref.\ to|(}%
of this law, as given by Clausius, has been much
criticised, particularly by Rankine, Theodor Wand, P.~G.
\index{Rankine!ref.\ to}%
\index{Wand}%
Tait, and Tolver Preston. Repeated efforts to deduce it from
\index{Preston}%
\index{Tait}%
general mechanical principles have remained fruitless. The
science of thermodynamics was developed with great success
by Thomson, Clausius, and Rankine. As early as 1852
Thomson discovered the law of the dissipation of energy,
\index{Dissipation of energy}%
deduced at a later period also by Clausius. The latter designated
the non-transformable energy by the name \emph{entropy},
\index{Entropy}%
and then stated that the entropy of the universe tends
toward a maximum. For entropy Rankine used the term
\emph{thermodynamic function}. Thermodynamic investigations have
been carried on also by G.~Ad.\ Hirn of Colmar, and Helmholtz
\index{Helmholtz!ref.\ to}%
\index{Hirn}%
(monocyclic and polycyclic systems). Valuable graphic
methods for the study of thermodynamic relations were devised
in 1873--1878 by \textbf{J.~Willard Gibbs} of Yale College.
\index{Gibbs}%
Gibbs first gives an account of the advantages of using
various pairs of the five fundamental thermodynamic quantities
for graphical representation, then discusses the entropy-temperature
and entropy-volume diagrams, and the volume-energy-entropy
surface (described in Maxwell's \textit{Theory of
\index{Maxwell!ref.\ to}%
Heat}). Gibbs formulated the energy-entropy criterion of
equilibrium and stability, and expressed it in a form applicable
to complicated problems of dissociation. Important
works on thermodynamics have been prepared by Clausius
in 1875, by R.~Rühlmann in 1875, and by Poincaré in 1892.
\index{Poincaré!ref.\ to}%
\index{Ruhlmann@{Rühlmann}}%

In the study of the law of dissipation of energy and the
principle of least action, mathematics and metaphysics met on
%% -----File: 415.png---Folio 401-------
\index{Action, least}%
\index{Thermodynamics|)}%
common ground. The doctrine of least action was first propounded
\index{Least action}%
by Maupertius in 1744. Two years later he proclaimed
\index{Maupertius}%
it to be a universal law of nature, and the first
scientific proof of the existence of God. It was weakly supported
by him, violently attacked by König of Leipzig, and
\index{Konig@{König}}%
keenly defended by Euler. Lagrange's conception of the principle
\index{Euler!ref.\ to}%
\index{Lagrange!ref.\ to}%
of least action became the mother of analytic mechanics,
\index{Mechanics!more recent work}%
but his statement of it was inaccurate, as has been remarked
by Josef Bertrand in the third edition of the \textit{Mécanique Analytique}.
\index{Bertrand}%
The form of the principle of least action, as it now
exists, was given by Hamilton, and was extended to electro-dynamics
\index{Hamilton, W.~R.!ref.\ to}%
by F.~E. Neumann, Clausius, Maxwell, and Helmholtz.
\index{Helmholtz!ref.\ to}%
\index{Maxwell!ref.\ to}%
\index{Neumann, F.~E.!ref.\ to}%
To subordinate the principle to all reversible processes,
Helmholtz introduced into it the conception of the ``kinetic
potential.'' In this form the principle has universal validity.

An offshoot of the mechanical theory of heat is the modern
kinetic theory of gases, developed mathematically by \textit{Clausius},
\index{Gases!Kinetic theory of|(}%
\index{Kinetic theory of gases|(}%
\textit{Maxwell}, \textit{Ludwig Boltzmann} of Munich, and others. The first
suggestions of a kinetic theory of matter go back as far as the
time of the Greeks. The earliest work to be mentioned here is
that of Daniel Bernoulli, 1738. He attributed to gas-molecules
\index{Bernoulli, Daniel!ref.\ to}%
great velocity, explained the pressure of a gas by molecular
bombardment, and deduced Boyle's law as a consequence of
\index{Boyle's law}%
his assumptions. Over a century later his ideas were taken
up by Joule (in~1846), A.~K. Krönig (in~1856), and Clausius
\index{Joule!ref.\ to}%
\index{Krönig}%
(in~1857). Joule dropped his speculations on this subject
when he began his experimental work on heat. Krönig
\index{Heat, theory of|)}%
explained by the kinetic theory the fact determined experimentally
by Joule that the internal energy of a gas is not
altered by expansion when no external work is done. Clausius
took an important step in supposing that molecules may have
rotary motion, and that atoms in a molecule may move relatively
to each other. He assumed that the force acting
%% -----File: 416.png---Folio 402-------
between molecules is a function of their distances, that temperature
depends solely upon the kinetic energy of molecular
motions, and that the number of molecules which at any
moment are so near to each other that they perceptibly influence
each other is comparatively so small that it may be
neglected. He calculated the average velocities of molecules,
and explained evaporation. Objections to his theory, raised
by Buy's-Ballot and by Jochmann, were satisfactorily answered
\index{Buy's-Ballot}%
\index{Jochmann}%
by Clausius and Maxwell, except in one case where an additional
\index{Maxwell!ref.\ to}%
hypothesis had to be made. Maxwell proposed to himself
the problem to determine the average number of molecules,
the velocities of which lie between given limits. His expression
therefor constitutes the important law of distribution of
velocities named after him. By this law the distribution of
molecules according to their velocities is determined by the
same formula (given in the theory of probability) as the distribution
of empirical observations according to the magnitude
of their errors. The average molecular velocity as deduced
by Maxwell differs from that of Clausius by a constant factor.
\index{Clausius!ref.\ to|)}%
Maxwell's first deduction of this average from his law of distribution
was not rigorous. A sound derivation was given by
O.~E. Meyer in 1866. Maxwell predicted that so long as
\index{Meyer, O.~E.}%
Boyle's law is true, the coefficient of viscosity and the coefficient
of thermal conductivity remain independent of the pressure.
His deduction that the coefficient of viscosity should
be proportional to the square root of the absolute temperature
appeared to be at variance with results obtained from pendulum
experiments. This induced him to alter the very foundation
of his kinetic theory of gases by assuming between
the molecules a repelling force varying inversely as the fifth
power of their distances. The founders of the kinetic theory
had assumed the molecules of a gas to be hard elastic spheres;
but Maxwell, in his second presentation of the theory in 1866,
%% -----File: 417.png---Folio 403-------
\index{Applied mathematics|)}%
\index{Thomson, Sir William!ref.\ to}%
went on the assumption that the molecules behave like centres
of forces. He demonstrated anew the law of distribution
of velocities; but the proof had a flaw in argument, pointed
out by Boltzmann, and recognised by Maxwell, who adopted
\index{Boltzmann}%
\index{Maxwell!ref.\ to}%
a somewhat different form of the distributive function in a
paper of 1879, intended to explain mathematically the effects
observed in Crookes' radiometer. Boltzmann gave a rigorous
\index{Radiometer}%
general proof of Maxwell's law of the distribution of velocities.

None of the fundamental assumptions in the kinetic theory
of gases leads by the laws of probability to results in very
close agreement with observation. Boltzmann tried to establish
kinetic theories of gases by assuming the forces between
molecules to act according to different laws from those previously
assumed. Clausius, Maxwell, and their predecessors
took the mutual action of molecules in collision as repulsive,
but Boltzmann assumed that they may be attractive. \DPtypo{Experiment}{Experiments}
of Joule and Lord Kelvin seem to support the latter
\index{Joule!ref.\ to}%
\index{Kelvin, Lord!ref.\ to}%
assumption.

Among the latest researches on the kinetic theory is Lord
\index{Gases!Kinetic theory of|)}%
\index{Kinetic theory of gases|)}%
Kelvin's disproof of a general theorem of Maxwell and Boltzmann,
asserting that the average kinetic energy of two given
portions of a system must be in the ratio of the number of
degrees of freedom of those portions.
%% -----File: 418.png---Folio 404-------

\clearpage
\phantomsection
\pdfbookmark[-1]{Back Matter}{Back Matter}

\NChapter[1]{Addenda}

\footnotesize
\Addendref{14}. The new \textit{Akhmim papyrus}, written in Greek, is probably the
copy of an older papyrus, antedating Heron's works, and is the oldest
extant text-book on practical Greek arithmetic. It contains, besides
arithmetical examples, a table for finding ``unit-fractions,'' identical in
scope with that of Ahmes, and, like Ahmes's, without a clue as to its mode
of construction. See \textsc{Biblioth.\ Math.}, 1893, p.~79--89. The papyrus is
edited by J.~Baillet (\textit{Mémoires publiés par les membres de la mission
archéologique française au Caire}, T.~IX., 1\textsuperscript{r}~\textit{fascicule}, Paris, 1892, p.~1--88).

\Addendref{39}. Chasles's or Simson's definition of a Porism is preferable to
Proclus's, given in the text. See Gow, p.~217--221.

\Addendref{114}. Nasir Eddin for the first time elaborated trigonometry independently
of astronomy and to such great perfection that, had his work
been known, Europeans of the 15th~century might have spared their
labours. See \textsc{Biblioth.\ Math.}, 1893, p.~6.

\Addendref{116}. This law of sines was probably known before Gabir ben
Aflah to Tabit ben Korra and others. See \textsc{Biblioth.\ Math.}, 1893, p.~7.

\Addendref{125}. Athelard was probably not the first to translate Euclid's
\textit{Elements} from the Arabic. See M.~Cantor's \textsc{Vorlesungen}, Vol.~II.,
p.~91,~92.

\Addendref{240}. G.~Eneström argues that Taylor and not Nicole is the real
inventor of finite differences. See \textsc{Biblioth.\ Math.}, 1893, p.~91.

\Addendref{250}. An earlier publication in which $3.14159\ldots$ is designated by~$\pi$,
is W.~Jones's \textit{Synopsis palmariorum matheseos}, London, 1706, p.~243,
263~\textit{et~seq.} See \textsc{Biblioth.\ Math.}, 1894, p.~106.

\Addendref{335}. Before Gauss a theorem on convergence, usually attributed
to Cauchy, was given by Maclaurin (\textit{Fluxions}, §~350). A rule of convergence
was deduced also by Stirling. See \textit{Bull.\ N.~Y. Math.\ Soc.}, Vol.~III.,
p.~186.

\Addendref{358}. The surface of a solid with $p$~holes was considered before
Clifford by Tonelli, and was probably used by Riemann himself. See
\textsc{Math.\ Annalen}, Vol.~45, p.~142.

\Addendref{361}. As early as 1835, Lobachevsky showed in a memoir the
necessity of distinguishing between continuity and differentiability. See
G.~B. Halsted's transl.\ of A.~Vasiliev's \textit{Address} on Lobachevsky, p.~23.

\textit{Recent deaths.} Johann Rudolf Wolf, Dec.~6, 1893; Heinrich Hertz,
Jan.~1, 1894; Eugène Catalan, Feb.~14, 1894; Hermann von~Helmholtz,
Sept.~8, 1894; Arthur Cayley, Jan.~26, 1895.
%% -----File: 419.png---Folio 405-------

\normalsize
\cleardoublepage

\IndexBookmark
\fancyhead[C]{\textsc{INDEX}}
\printindex


% **** Need to replace index text

%% -----File: 437.png---Folio 423-------

\Catalog

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%% -----File: 438.png---Folio 424-------
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%% -----File: 439.png---Folio 425-------

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%% -----File: 440.png---Folio 426-------

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%% -----File: 441.png---Folio 427-------

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%[** PG etext 29913]
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%% -----File: 442.png---Folio 428-------

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%% -----File: 445.png---Folio 431-------

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%[** PG etext 26839]
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%% -----File: 446.png---Folio 432-------

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%% -----File: 447.png---Folio 433-------

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%% -----File: 448.png---Folio 434-------

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\$\Pr{2.00}.

% [** PG etext 25155]
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% [** PG etext #26262]
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% End of Project Gutenberg's A History of Mathematics, by Florian Cajori  %
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% *** END OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS ***    %
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[]\OT1/cmtt/m/n/8 The Project Gutenberg EBook of A History of Mathematics, by F
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PDF statistics:
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