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\begin{verbatim}
Project Gutenberg's Number-System of Algebra, by Henry Fine

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: The Number-System of Algebra (2nd edition)
       Treated Theoretically and Historically

Author: Henry Fine

Release Date: March 4, 2006 [EBook #17920]

Language: English

Character set encoding: TeX

*** START OF THIS PROJECT GUTENBERG EBOOK NUMBER-SYSTEM OF ALGEBRA ***




Produced by Jonathan Ingram, Susan Skinner and the
Online Distributed Proofreading Team at https://www.pgdp.net



\end{verbatim}
\normalsize
\newpage


\begin{titlepage}
\begin{center}
THE\\[2.5cm]

{\LARGE\bfseries NUMBER-SYSTEM OF ALGEBRA}\\[2.5cm]

TREATED THEORETICALLY AND HISTORICALLY\\[2.5cm]

BY\\[2.5cm]

{\large HENRY B. FINE, PH.~D.}\\
{\small PROFESSOR OF MATHEMATICS IN PRINCETON UNIVERSITY}\\[2.5cm]

{\small \textit{SECOND EDITION, WITH CORRECTIONS}}\\[2.5cm]

BOSTON, U.~S.~A.\\
D.~C.~HEATH \& CO., PUBLISHERS\\

1907
\end{center}
\end{titlepage}
\frontmatter
\begin{center}
COPYRIGHT, 1890,\\

BY HENRY B.~FINE.
\end{center}
\newpage
\section*{PREFACE.}

\small{The theoretical part of this little book is an elementary
exposition of the nature of the number concept, of the positive
integer, and of the four artificial forms of number which, with
the positive integer, constitute the ``number-system'' of algebra,
viz.\ the negative, the fraction, the irrational, and the
imaginary. The discussion of the artificial numbers follows, in
general, the same lines as my pamphlet: \textit{On the Forms of
Number arising in Common Algebra}, but it is much more exhaustive
and thorough-going. The point of view is the one first suggested
by Peacock and Gregory, and accepted by mathematicians generally
since the discovery of quaternions and the Ausdehnungslehre of
Grassmann, that algebra is completely defined formally by the laws
of combination to which its fundamental operations are subject;
that, speaking generally, these laws alone define the operations,
and the operations the various artificial numbers, as their formal
or symbolic results. This doctrine was fully developed for the
negative, the fraction, and the imaginary by Hankel, in his
\textit{Complexe Zahlensystemen}, in 1867, and made complete by
Cantor's beautiful theory of the irrational in 1871, but it has
not as yet received adequate treatment in English.

Any large degree of originality in work of this kind is naturally
out of the question. I have borrowed from a great many sources,
especially from Peacock, Grassmann, Hankel, Weierstrass, Cantor,
and Thomae (\textit{Theorie der analytischen Functionen einer
complexen Veränderlichen}). I may mention, however, as more or
less distinctive features of my discussion, the treatment of
number, counting (§§~1--5), and the equation (§§~4,~12), and the
prominence given the laws of the determinateness of subtraction
and division.

Much care and labor have been expended on the historical chapters
of the book. These were meant at the outset to contain only a
brief account of the origin and history of the artificial numbers.
But I could not bring myself to ignore primitive counting and the
development of numeral notation, and I soon found that a clear and
connected account of the origin of the negative and imaginary is
possible only when embodied in a sketch of the early history of
the equation. I have thus been led to write a \textit{résumé} of
the history of the most important parts of elementary arithmetic
and algebra.

Moritz Cantor's \textit{Vorlesungen über die Geschichte der
Mathematik}, Vol. I, has been my principal authority for the
entire period which it covers, \textit{i.~e.} to 1200 \textsc{a.~d.~} For the
little I have to say on the period 1200 to 1600, I have depended
chiefly, though by no means absolutely, on Hankel: \textit{Zur
Geschichte der Mathematik in Altertum und Mittelalter}. The
remainder of my sketch is for the most part based on the original
sources.

\begin{flushright}
HENRY B. FINE.
\end{flushright}
\textsc{Princeton}, April, 1891.

\begin{center}
\rule{5.0cm}{0.1mm}
\end{center}

In this second edition a number of important corrections have been
made. But there has been no attempt at a complete revision of the
book.

\begin{flushright}
HENRY B. FINE.
\end{flushright}
\textsc{Princeton}, September, 1902.}

\tableofcontents \bigskip

\begin{center}
\begin{tabular}{lr}
\multicolumn{2}{l}
{\textbf{PRINCIPAL FOOTNOTES}} \\
Instances of quinary and vigesimal systems of notation \dotfill&
\pageref{Instances of quinary and vigesimal systems of notation}\\
Instances of digit numerals \dotfill& \pageref{Instances of digit
numerals}\\
Summary of the history of Greek mathematics \dotfill
&\pageref{Summary of the history of Greek mathematics}\\
Old Greek demonstration that the side and diagonal of a square are
incommensurable \dotfill & \pageref{Old Greek demonstration that
the side and diagonal of a square are incommensurable}\\
Greek methods of approximation \dotfill &\pageref{Greek methods of
approximation}\\
Diophantine equations\dotfill & \pageref{Diophantine equations}\\
Alchayyâmî's method of solving cubics by the intersections of
conics\dotfill&\pageref{Alchayyami method of solving cubics by the
intersections of conics}\\
Jordanus Nemorarius \dotfill & \pageref{Jordanus Nemorarius}\\
The \textit{summa} of Luca Pacioli \dotfill & \pageref{The summa
of Luca
Pacioli}\\
Regiomontanus \dotfill & \pageref{Regiomontanus}\\
Algebraic symbolism \dotfill & \pageref{Jordanus Nemorarius},
\pageref{Algebraic symbolism}\\
The irrationality of $e$ and $\pi$. Lindemann \dotfill &
\pageref{irrationality}
\end{tabular}
\end{center}

\mainmatter

\part{THEORETICAL}


\chapter{THE POSITIVE INTEGER,\\
AND THE LAWS WHICH REGULATE THE ADDITION AND MULTIPLICATION
OF POSITIVE INTEGERS.}

\addcontentsline{toc}{section}{\numberline{}The number concept}

\textbf{1. Number}. We say of certain distinct things that they
form a group\footnote{By group we mean \textit{finite} group, that
is, one which cannot be brought into one-to-one correspondence (§~2)
with any part of itself.} when we make them collectively a
single object of our attention.

The \textit{number of things} in a group is that property of the
group which remains unchanged during every change in the group
which does not destroy the separateness of the things from one
another or their common separateness from all other things.

Such changes may be changes in the characteristics of the things
or in their arrangement within the group. Again, changes of
arrangement may be changes either in the order of the things or in
the manner in which they are associated with one another in
smaller groups.

We may therefore say:

\textit{The number of things in any group of distinct things is
independent of the characters of these things, of the order in
which they may be arranged in the group, and of the manner in
which they may be associated with one another in smaller groups.}

\addcontentsline{toc}{section}{\numberline{}Numerical equality }

\textbf{2. Numerical Equality}. The number of things in any two
groups of distinct things is the same, when for each thing in the
first group there is one in the second, and reciprocally, for each
thing in the second group, one in the first.

Thus, the number of letters in the two groups, $A$, $B$, $C$; $D$,
$E$, $F$, is the same. In the second group there is a letter which
may be assigned to each of the letters in the first: as $D$ to
$A$, $E$ to $B$, $F$ to $C$; and reciprocally, a letter in the
first which may be assigned to each in the second: as $A$ to $D$,
$B$ to $E$, $C$ to $F$.

Two groups thus related are said to be in \textit{one-to-one}
(1--1) \textit{correspondence}.

Underlying the statement just made is the assumption that if the
two groups correspond in the manner described for one order of the
things in each, they will correspond if the things be taken in any
other order also; thus, in the example given, that if $E$ instead
of $D$ be assigned to $A$, there will again be a letter in the
group $D$, $E$, $F$, viz. $D$ or $F$, for each of the remaining
letters $B$ and $C$, and reciprocally. This is an immediate
consequence of §~1, foot-note.

The number of things in the first group is \textit{greater than}
that in the second, or the number of things in the second
\textit{less than} that in the first, when there is one thing in
the first group for each thing in the second, but \textit{not}
reciprocally one in the second for each in the first.

\addcontentsline{toc}{section}{\numberline{}Numeral symbols }

\textbf{3. Numeral Symbols}. As regards the number of things which
it contains, therefore, a group may be represented by any other
group, \textit{e.~g.} of the fingers or of simple marks, $|$'s,
which stands to it in the relation of correspondence described in
§~2. This is the primitive method of representing the number of
things in a group and, like the modern method, makes it possible
to compare numerically groups which are separated in time or
space.

The modern method of representing the number of things in a group
differs from the primitive only in the substitution of symbols, as
1, 2, 3, etc., or numeral words, as \textit{one, two, three},
etc., for the various groups of marks $|$, $||$, $|||$, etc. These
symbols are the positive integers of arithmetic.

\textit{A positive integer is a symbol for the number of things in
a group of distinct things}.

For convenience we shall call the positive integer which
represents the number of things in any group its numeral symbol,
or when not likely to cause confusion, its number simply,---this
being, in fact, the primary use of the word ``number'' in
arithmetic.

In the following discussion, for the sake of giving our statements
a general form, we shall represent these numeral symbols by
letters, $a$, $b$, $c$, etc.

\addcontentsline{toc}{section}{\numberline{}The numerical
equation}

\textbf{4. The Equation.} The numeral symbols of two groups being
$a$ and $b$; when the number of things in the groups is the same,
this relation is expressed by the \textit{equation}
\[
a = b;
\]
when the first group is greater than the second, by the
\textit{inequality}
\[
a > b;
\]
when the first group is less than the second, by the
\textit{inequality}
\[
a < b.
\]
\textit{A numerical equation is thus a declaration in terms of the
numeral symbols of two groups and the symbol} = \textit{that these
groups are in one-to-one correspondence} (§2).

\addcontentsline{toc}{section}{\numberline{}Counting}

\textbf{5. Counting.} The fundamental operation of arithmetic is
counting.

To count a group is to set up a one-to-one correspondence between
the individuals of this group and the individuals of some
representative group.

Counting leads to an expression for the number of things in any
group in terms of the representative group: if the representative
group be the fingers, to a group of fingers; if marks, to a group
of marks; if the numeral words or symbols in common use, to one of
these words or symbols.

There is a difference between counting with numeral words and the
earlier methods of counting, due to the fact that the numeral
words have a certain recognized order. As in finger-counting one
finger is attached to each thing counted, so here one word; but
that word represents numerically not the thing to which it is
attached, but the entire group of which this is the last. The same
sort of counting may be done on the fingers when there is an
agreement as to the order in which the fingers are to be used;
thus if it were understood that the fingers were always to be
taken in normal order from thumb to little finger, the little
finger would be as good a symbol for 5 as the entire hand.

\addcontentsline{toc}{section}{\numberline{}Addition and its laws}

\textbf{6. Addition.} If two or more groups of things be brought
together so as to form a single group, the numeral symbol of this
group is called the \textit{sum} of the numbers of the separate
groups.

If the sum be $s$, and the numbers of the separate groups $a$,
$b$, $c$, etc., respectively, the relation between them is
symbolically expressed by the equation
\[
s = a + b + c + \textrm{etc.,}
\]
where the sum-group is supposed to be formed by joining the second
group---to which $b$ belongs---to the first, the third group---to
which $c$ belongs---to the resulting group, and so on.

The operation of finding $s$ when $a$, $b$, $c$, etc., are known,
is \textit{addition}.

Addition is abbreviated counting.

Addition is subject to the two following laws, called the
\textit{commutative} and \textit{associative} laws respectively,
viz.:

\[
\begin{array}{rl}
\textrm{I.} & a + b = b + a.\\

\textrm{II.} &a + (b + c) = a + b + c.
\end{array}
\]

Or,

\begin{tabular}{rl}
I. & To add $b$ to $a$ is the same as to add $a$ to $b$.\\

II. & To add the sum of $b$ and $c$ to $a$ is the same as to add
$c$ to the sum of $a$ and $b$.
\end{tabular}

Both these laws are immediate consequences of the fact that the
sum-group will consist of the same individual things, and the
number of things in it therefore be the same, whatever the order
or the combinations in which the separate groups are brought
together (§1).

\addcontentsline{toc}{section}{\numberline{}Multiplication and its
laws}

\textbf{7. Multiplication.} The sum of $b$ numbers each of which
is $a$ is called the \textit{product} of $a$ by $b$, and is
written $a \times b$, or $a \cdot b$, or simply $ab$.

The operation by which the product of $a$ by $b$ is found, when
$a$ and $b$ are known, is called \textit{multiplication}.

Multiplication is an abbreviated addition.

Multiplication is subject to the three following laws, called
respectively the \textit{commutative, associative}, and
\textit{distributive} laws for multiplication, viz.:

\begin{tabular}{rl}
III. & $ab = ba$.\\

IV. & $a(bc) = abc$.\\

V. & $a(b + c) = ab +ac$.
\end{tabular}

Or,

\begin{tabular}{rl}
III. &The product of $a$ by $b$ is the same as the product of $b$
by $a$.\\

IV. & The product of $a$ by $bc$ is the same as the product of
$ab$ by $c$.\\

V. & The product of $a$ by the sum of $b$ and $c$ is the same as
the sum of the product of $a$ by $b$ and of $a$ by $c$.
\end{tabular}

These laws are consequences of the commutative and associative
laws for addition. Thus,

III. \textit{The Commutative Law}. The units of the group which
corresponds to the sum of $b$ numbers each equal to $a$ may be
arranged in $b$ rows containing $a$ units each. But in such an
arrangement there are $a$ columns containing $b$ units each; so
that if this same set of units be grouped by columns instead of
rows, the sum becomes that of $a$ numbers each equal to $b$, or
$ba$. Therefore $ab = ba$, by the commutative and associative laws
for addition.

IV. \textit{The Associative Law}.
\begin{align*}
abc & = c \ \textrm{sums such as} \ (a + a + \cdots \ \textrm{to} \ b \ \textrm{terms}) \\
    & = a + a + a + \cdots \ \textrm{to} \ bc \ \textrm{terms (by the associative law for addition)} \\
    & = a(bc).
\end{align*}

V. \textit{The Distributive Law}.
\begin{align*}
a(b + c) & = a + a + a + \cdots \textrm{to} \ (b + c) \ \textrm{terms} \\
         & =  a + a + \cdots \ \textrm{to} \ b \ \textrm{terms}) + (a + a + \cdots \ \textrm{to} \  c \ \textrm{terms}) \\
         &    \qquad \textrm{(by the associative law for addition),} \\
         & =  ab + ac.
\end{align*}

The commutative, associative, and distributive laws for sums of
any number of terms and products of any number of factors follow
immediately from I--V. Thus the product of the factors $a$, $b$,
$c$, $d$, taken in any two orders, is the same, since the one
order can be transformed into the other by successive interchanges
of consecutive letters.


\chapter{SUBTRACTION AND THE NEGATIVE INTEGER.}

\addcontentsline{toc}{section}{\numberline{}Numerical subtraction}

\textbf{8. Numerical Subtraction.} Corresponding to every
mathematical operation there is another, commonly called its
\textit{inverse}, which exactly undoes what the operation itself
does. Subtraction stands in this relation to addition, and
division to multiplication.

To \textit{subtract b} from $a$ is to find a number to which if
$b$ be added, the sum will be $a$. The result is written $a - b$;
by definition, it identically satisfies the equation

VI. $(a - b) + b = a$;

that is to say, $a - b$ is the number belonging to the group which
with the $b$-group makes up the $a$-group.

Obviously subtraction is always possible when $b$ is less than
$a$, but then only. Unlike addition, in each application of this
operation regard must be had to the relative size of the two
numbers concerned.

\addcontentsline{toc}{section}{\numberline{}Determinateness of
numerical subtraction}

\textbf{9. Determinateness of Numerical Subtraction}. Subtraction,
when possible, is a \textit{determinate} operation. There is but
\textit{one} number which will satisfy the equation $x + b = a$,
but one number the sum of which and $b$ is $a$. In other words, $a
- b$ is one-valued.

For if $c$ and $d$ both satisfy the equation $x + b = a$, since
then $c + b = a$ and $d + b = a$, $c + b = d + b$; that is, a
one-to-one correspondence may be set up between the individuals of
the $(c + b)$ and $(d + b)$ groups (§4). The same sort of
correspondence, however, exists between any $b$ individuals of the
first group and any $b$ individuals of the second; it must,
therefore, exist between the remaining $c$ of the first and the
remaining $d$ of the second, or $c = d$.

This characteristic of subtraction is of the same order of
importance as the commutative and associative laws, and we shall
add to the group of laws I--V and definition VI---as being, like
them, a fundamental principle in the following discussion---the
theorem

VII. $ \qquad \left\{ \begin{array}{rcl} \textrm{If} \ a + c & = & b + c \\
                          a & = & b,
                      \end{array} \right.$

which may also be stated in the form: If one term of a sum changes
while the other remains constant, the sum changes. The same
reasoning proves, also, that

VIII. $ \qquad \left\{ \begin{array}{rcl} \textrm{As} \ a + c  > & \textrm{or} & < b + c \\
                          a & \textrm{or} & b,
                       \end{array} \right.$

\addcontentsline{toc}{section}{\numberline{}Formal rules of
subtraction}

\textbf{10. Formal Rules of Subtraction.} All the rules of
subtraction are purely \textit{formal} consequences of the
fundamental laws I--V, VII, and definition VI\@. They must follow,
whatever the meaning of the symbols $a$, $b$, $c$, $+$, $-$, $=$;
a fact which has an important bearing on the following discussion.

It will be sufficient to consider the equations which follow. For,
properly combined, they determine the result of any series of
subtractions or of any complex operation made up of additions,
subtractions, and multiplications.

\begin{enumerate}
\item $a - (b + c) = a - b - c = a - c - b$.

\item $a - (b - c) = a - b + c$.

\item $a + b - b = a$.

\item $a + (b - c) = a + b - c = a - c + b$.

\item $a(b - c) = ab - ac$.
\end{enumerate}

For

\begin{enumerate}

\item $a - b - c$ is the form to which if first $c$ and then $b$
be added; or, what is the same thing (by I), first $b$ and then
$c$; or, what is again the same thing (by II), $b + c$ at
once,---the sum produced is $a$ (by VI). $a - b - c$ is therefore
the same as $a - c - b$, which is as it stands the form to which
if $b$, then $c$, be added the sum is $a$; also the same as $a -
(b + c)$, which is the form to which if $b + c$ be added the sum
is $a$.

\item
\[
\begin{array}{rlr}
a - (b - c) &= a - (b - c) - c + c, &\textrm{Def. VI.}\\

&= a - (b - c + c) + c, & \textrm{ Eq. 1.}\\

&= a - b - c. & \textrm{Def. VI.}\\
\end{array}
\]

\item
\[
\begin{array}{rlr}
a + b - b + b &= a + b.& \textrm{Def. VI.}\\
\textrm{ But }\quad a + b &= a + b.\\
\therefore a + b - b &= a. & \textrm{Law VII.}
\end{array}
\]

\item
\[ \begin{array}{rlr}
a + b - c &= a + (b - c + c) - c,& \textrm{Def. VI.}\\
&= a + (b - c). & \textrm{Law II, Eq. 3.}
\end{array}
\]

\item
\[
\begin{array}{rlr}
ab - ac &= a(b - c + c)- ac, &\textrm{Def. VI.}\\
&= a(b - c) + ac - ac,& \textrm{Law V.}\\
&= a(b - c).& \textrm{Eq. 3.}
\end{array}
\]
\end{enumerate}

Equation 3 is particularly interesting in that it defines addition
as the inverse of subtraction. Equation 1 declares that two
consecutive subtractions may change places, are commutative.
Equations 1, 2, 4 together supplement law II, constituting with it
a complete associative law of addition and subtraction; and
equation 5 in like manner supplements law V.


\addcontentsline{toc}{section}{\numberline{}Limitations of
numerical subtraction}

 \textbf{11. Limitations of Numerical Subtraction}.
Judged by the equations 1--5, subtraction is the exact counterpart
of addition. It conforms to the same general laws as that
operation, and the two could with fairness be made to interchange
their rôles of direct and inverse operation.

But this equality proves to be only apparent when we attempt to
interpret these equations. The requirement that subtrahend be less
than minuend then becomes a serious restriction. It makes the
range of subtraction much narrower than that of addition. It
renders the equations 1--5 available for special classes of values
of $a$, $b$, $c$ only. If it must be insisted on, even so simple
an inference as that $a - (a + b) + 2b$ is equal to $b$ cannot be
drawn, and the use of subtraction in any reckoning with symbols
whose relative values are not at all times known must be
pronounced unwarranted.

One is thus naturally led to ask whether to be valid an algebraic
reckoning must be interpretable numerically and, if not, to seek
to free subtraction and the rules of reckoning with the results of
subtraction from a restriction which we have found to be so
serious.

\addcontentsline{toc}{section}{\numberline{}Symbolic equations}
\addcontentsline{toc}{section}{\numberline{}Principle of
permanence. Symbolic subtraction}

\textbf{12. Symbolic Equations. Principle of Permanence. Symbolic
Subtraction.} In pursuance of this inquiry one turns first to the
equation $(a - b) + b = a$, which serves as a definition of
subtraction when $b$ is less than $a$.

This is an equation in the primary sense (§~4) only when $a - b$
is a number. But in the broader sense, that

\textit{An equation is any declaration of the equivalence of
definite combinations of symbols---equivalence in the sense that
one may be substituted for the other,---} $(a - b) + b = a$ may be
an equation, whatever the values of $a$ and $b$.

And if no different meaning has been attached to $a - b$, and it
is declared that $a - b$ is the symbol which associated with $b$
in the combination $(a - b) + b$ is equivalent to $a$, this
declaration, or the \textit{equation}

\[
(a - b) + b = a,
\]

is a \textit{definition}\footnote{A definition in terms of
symbolic, not numerical addition. The sign + can, of course,
indicate numerical addition only when both the symbols which it
connects are numbers.} of this symbol.

By the assumption of the \textit{permanence of form} of the
numerical equation in which the definition of subtraction
resulted, one is thus put immediately in possession of a
\textit{symbolic} definition of subtraction which is general.

The numerical definition is subordinate to the symbolic
definition, being the interpretation of which it admits when $b$
is less than $a$.

But from the standpoint of the symbolic definition,
interpretability---the question whether $a - b$ is a number or
not---is irrelevant; only such properties may be attached to $a -
b$, by itself considered, as flow immediately from the generalized
equation

\[
(a - b) + b = a.
\]

In like manner each of the fundamental laws I--V, VII, on the
assumption of the \textit{permanence of its form} after it has
ceased to be interpretable numerically, becomes a declaration of
the equivalence of certain definite combinations of symbols, and
the formal consequences of these laws---the equations 1--5 of §~10---become 
definitions of addition, subtraction, multiplication,
and their mutual relations---definitions which are purely
symbolic, it may be, but which are unrestricted in their
application.

These definitions are legitimate from a logical point of view. For
they are merely the laws I--VII, and we may assume that these laws
are \textit{mutually consistent} since we have proved that they
hold good for positive integers. Hence, if \textit{used
correctly}, there is no more possibility of their leading to false
results than there is of the more tangible numerical definitions
leading to false results. The laws of correct thinking are as
applicable to mere symbols as to numbers.

What the value of these symbolic definitions is, to what extent
they add to the power to draw inferences concerning numbers, the
elementary algebra abundantly illustrates.

One of their immediate consequences is the introduction into
algebra of two new symbols, \textit{zero} and the
\textit{negative}, which contribute greatly to increase the
simplicity, comprehensiveness, and power of its operations.

\addcontentsline{toc}{section}{\numberline{}Zero}

\textbf{13. Zero.} When $b$ is set equal to $a$ in the general
equation

\[
(a - b) + b = a,
\]

it takes one of the forms

\[
(a - a) + a = a,
\]
\[
(b - b) + b = b.
\]

It may be proved that

\[
\begin{array}{rlr}
a - a &= b - b.\\

\textrm{ For} \quad   (a - a) + (a + b) &= (a - a) + a + b, &
\textrm{Law II.}\\
&= a + b,\\

\textrm{since} \quad      (a - a) + a &= a.\\

\textrm{And}\quad   (b - b) + (a + b) &= (b - b) + b + a,
&\textrm{ Laws I, II.}\\
&   = b + a,\\
\textrm{since}\quad      (b - b) + b &= b.\\

\textrm{Therefore}\quad      a - a &= b - b.  &  \textrm{Law VII.}
\end{array}
\]


$a - a$ is therefore altogether independent of $a$ and may
properly be represented by a symbol unrelated to a. The symbol
which has been chosen for it is 0, called \textit{zero}.

\textit{Addition} is defined for this symbol by the equations

\begin{enumerate}
\item

\[
\begin{array}{rlr}
      0 + a &= a,  &     \textrm{definition of 0.}\\
       a + 0 &= a.  &                   \textrm{Law I.}
\end{array}
\]

\textit{Subtraction} (partially), by the equation

\item
\[
\begin{array}{rlr}
a - 0 &= a.\\

\textrm{For}\quad  (a - 0) + 0 &= a. &        \textrm{Def. VI.}
\end{array}
\]

\textit{Multiplication} (partially), by the equations

\item
\[
\begin{array}{rlr}
    a \times 0 &= 0 \times a = 0.\\

\textrm{For}\quad    a \times 0 &= a (b - b), & \textrm{definition
of 0.}\\
                    &= ab - ab,   &     \textrm{§~10, 5.}\\
                    &= 0.      &        \textrm{definition of 0.}
\end{array}
\]
\end{enumerate}

\addcontentsline{toc}{section}{\numberline{}The negative}

\textbf{14. The Negative.} When $b$ is greater than $a$, equal say
to $a + d$, so that $b - a = d$, then


\[
\begin{array}{rlr}

       a - b &= a - (a + d),\\
                  &= a - a - d, &       \textrm{§~10, 1.}\\
                  &= 0 - d.     &        \textrm{definition of 0.}
\end{array}
\]

For $0 - d$ the briefer symbol $-d$ has been substituted; with
propriety, certainly, in view of the lack of significance of 0 in
relation to addition and subtraction. The equation $0 - d = -d$,
moreover, supplies the missing rule of subtraction for 0. (Compare
§~13, 2.)

The symbol $-d$ is called the \textit{negative}, and in opposition
to it, the number $d$ is called \textit{positive}.

Though in its origin a sign of operation (subtraction from 0), the
sign $-$ is here to be regarded merely as part of the symbol $-d$.

$-d$ is as serviceable a substitute for $a - b$ when $a < b$, as
is a single numeral symbol when $a > b$.

The rules for reckoning with the new symbol---definitions of its
addition, subtraction, multiplication---are readily deduced from
the laws I--V, VII, definition VI, and the equations 1--5 of §~10,
as follows:

\begin{enumerate}
\item

\[
\begin{array}{rlr}

b + (-b) &= -b + b = 0.\\

\textrm{For} \quad  -b + b  &= (0 - b) + b,& \textrm{definition of
negative.}\\

&= 0.   &       \textrm{Def. VI.}
\end{array}
\]

$-b$ may therefore be defined as the symbol the sum of which and
$b$ is 0.

\item

\[
\begin{array}{rlr}
  a + (-b) &= -b + a = a -b.\\
  \textrm {For} \quad a + (-b) &= a + (0-b),& \textrm{definition of
  negative.}\\
&= a + 0 - b,   &      \textrm{ §~10, 4.}\\
&= a - b.   & \textrm  {§~13, 1.}
\end{array}
\]

\item

\[
\begin{array}{rlr}

-a + (-b) &= - (a + b).\\
\textrm{For} \quad -a+ (-b) &= 0 -a-b, & \textrm{by the reasoning
in §~14, 2.}\\
&= 0 - (a + b), &   \textrm{ §10,1.}\\
&= -(a + b).& \textrm{ definition of negative.}
\end{array}
\]


\item

\[
\begin{array}{rlr}

a-(-b) &= a + b.\\
\textrm{ For} \quad a-(-b) &= a - (0-b), & \textrm{definition of
negative.}\\
& = a -0 + b, &\textrm{  §~10, 2.}\\
& = a + b.  & \textrm{§13, 2.}
\end{array}
\]

\item

\[
\begin{array}{rlr}

(-a) - (-b) &= b - a.\\
\textrm{ For} \quad -a - (-b) &= -a + b,& \textrm{ by the
reasoning in §~14, 4.}\\
&   = b - a.     & \textrm{  §14, 2.}\\

\textrm{COR.} \quad (-a) - (-a) &= 0.
\end{array}
\]

\item

\[
\begin{array}{rlr}
a(-b) &= (-b)a = -ab.\\
\textrm{For} \quad 0 &= a(b - b), &\textrm{ §13, 3.}\\
&= ab + a(-b).&           \textrm{ Law V.}\\
\therefore a(-b) &= -ab.  & \textrm { §~14, 1; Law VII.}
\end{array}
\]

\item
\[
\begin{array}{lrlr}
&(-a)\times0 & = 0\times(-a)=0. \\
\text{For}  &(-a)\times0 & =(-a)(b-b),  & \text{definition of 0}. \\
&& =(-a)b-(-a)b,  & \S~10,5. \\
&& =0. & \S~14, 6, \text{and} \; 5, \text{Cor}.\\
\end{array}
\]

\item
\[
\begin{array}{lrlr}
& (-a)(-b) & =ab. \\
\text{For} & 0 & =(-a)(b-b),  & \S~14, 7.\\
&& = (-a)b+(-a)(-b),  & \text{Law V}.\\
&& = -ab+(-a)(-b).  & \S~14, 6.\\
&\therefore (-a)(-b) & = ab.  & \S~14, 1; \text{Law VII.}
\end{array}
\]
By this method one is led, also, to definitions of
\textit{equality} and greater or lesser \textit{inequality} of
negatives. Thus

\item
\[
\begin{array}{lrlr}
 & -a >, & = \; \text{or} \; < -b,\\
\text{according as } & b >, & = \; \text{or} \; < a.\footnotemark[1]\\
\text{For as} & b>, & =,<a,\\
&-a+a+b>, & =,<-b+b+a, & \S~14, 1; \S~13, 1.\\
\text{or} & -a>, & =,<-b, & \text{Law VII or VII$^\prime$}.\\
\text{In like manner} &-a&<0<b.
\end{array}
\]
\end{enumerate}

\footnotetext[1]{On the other hand, $-a$ is said to be
\textit{numerically} greater than, equal to, or less than $-b$,
according as $a$ is itself greater than, equal to, or less than
$b$.}

\addcontentsline{toc}{section}{\numberline{}Recapitulation of the
argument of the chapter }

\textbf{15. Recapitulation.} The nature of the argument which has
been developed in the present chapter should be carefully
observed.

From the definitions of the positive integer, addition, and
subtraction, the associative and commutative laws and the
determinateness of subtraction followed. The assumption of the
permanence of the result $a-b$, as defined by $(a-b)+b=a$, for all
values of $a$ and $b$, led to definitions of the two symbols $0$,
$-d$, zero and the negative; and from the assumption of the
permanence of the laws I--V, VII were derived definitions of the
addition, subtraction, and multiplication of these symbols,---the
assumptions being just sufficient to determine the meanings of
these operations unambiguously.

In the case of numbers, the laws I--V, VII, and definition VI were
deduced from the characteristics of numbers and the definitions of
their operations; in the case of the symbols $0$, $-d$, on the
other hand, the characteristics of these symbols and the
definitions of their operations were deduced from the laws.

With the acceptance of the negative the character of arithmetic
undergoes a radical change.\footnote{In this connection see §~25.}
It was already in a sense symbolic, expressed itself in equations
and inequalities, and investigated the results of certain
operations. But its symbols, equations, and operations were all
interpretable in terms of the reality which gave rise to it, the
number of things in actually existing groups of things. Its
connection with this reality was as immediate as that of the
elementary geometry with actually existing space relations.

But the negative severs this connection. The negative is a symbol
for the result of an operation which cannot be effected with
actually existing groups of things, which is, therefore, purely
symbolic. And not only do the fundamental operations and the
symbols on which they are performed lose reality; the equation,
the fundamental judgment in all mathematical reasoning, suffers
the same loss. From being a declaration that two groups of things
are in one-to-one correspondence, it becomes a mere declaration
regarding two combinations of symbols, that in any reckoning one
may be substituted for the other.

\chapter{DIVISION AND THE FRACTION.}

\addcontentsline{toc}{section}{\numberline{}Numerical division }

\textbf{16. Numerical Division.} The inverse operation to
multiplication is division.

To divide $a$ by $b$ is to find a number which multiplied by $b$
produces $a$. The result is called the quotient of $a$ by $b$, and
is written $\frac{a}{b}$. By definition
\[
\left(\frac{a}{b}\right)b=a
\]
Like subtraction, division cannot be always effected. Only in
exceptional cases can the $a$-group be subdivided into $b$ equal
groups.

\addcontentsline{toc}{section}{\numberline{}Determinateness of
numerical division }

\textbf{17. Determinateness of Numerical Division.} When division
can be effected at all, it can lead to but a single result; it is
\textit{determinate}.

For there can be but one number the product of which by $b$ is
$a$; in other words,
\[
\left\{
\begin{array}{rl}
\textrm{If} \quad cb &= db,\\
c &= d.\footnotemark
\end{array}
\right.
\]

\footnotetext{The case $b = 0$ is excluded, 0 not being a number
in the sense in which that word is here used.}

For $b$ groups each containing $c$ individuals cannot be equal to
$b$ groups each containing $d$ individuals unless $c$ = $d$ (§4).

This is a theorem of fundamental importance. It may be called the
law of determinateness of division. It declares that if a product
and one of its factors be determined, the remaining factor is
definitely determined also; or that if one of the factors of a
product changes while the other remains unchanged, the product
changes. It alone makes division in the arithmetical sense
possible. The fact that it does not hold for the symbol 0, but
that rather a product remains unchanged (being always 0) when one
of its factors is 0, however the other factor be changed, makes
division by 0 impossible, rendering unjustifiable the conclusions
which can be drawn in the case of other divisors.

The reasoning which proved law IX proves also that

\[
\textrm{IX'.} \qquad \left\{
\begin{array}{rl}
\textrm{As} \quad cb > &\textrm{ or } < db,\\
c  > &\textrm{ or } < d.
\end{array}
\right.
\]

\addcontentsline{toc}{section}{\numberline{}Formal rules of
division }

\textbf{18. Formal Rules of Division.} The fundamental laws of the
multiplication of numbers are

\[
\begin{array}{lrl}

\textrm{III.} & ab&=ba,\\

\textrm{IV.} & a(bc)&=abc,\\

\textrm{V.} & a(b+c)&=ab+ac.
\end{array}
\]

Of these, the definition

\[
\textrm{VIII.} \qquad \left(\frac{a}{b}\right)b=a,
\]

the theorem

\[
\textrm{IX.} \qquad \left\{
\begin{array}{rlr}
 \textrm{If } ac&=bc,\\
a&=b, &\textrm{unless } c=0,
\end{array}
\right.
\]


and the corresponding laws of addition and subtraction, the rules
of division are purely \textit{formal} consequences, deducible
precisely as the rules of subtraction 1--5 of §10 in the preceding
chapter. They follow without regard to the meaning of the symbols
$a$, $b$, $c$, $=$, $+$, $-$, $ab$, $\frac{a}{b}$. Thus:

\begin{enumerate}
\item

\[
\begin{array}{rlr}

\dfrac{a}{b} \cdot \dfrac{c}{d}& = \dfrac{ac}{bd}.\\

\textrm{ For} \quad \dfrac{a}{b} \cdot \dfrac{c}{d} \cdot bd &=
\dfrac{a}{b}b \cdot \dfrac{c}{d}d, & \textrm{ Laws IV, III.}\\

&=ac, &\textrm{Def. VIII.}\\

\textrm{and} \quad \dfrac{ac}{bd} \cdot bd &=ac. & \textrm{Def
VIII.}
\end{array}
\]

The theorem follows by law IX.

\item

\[
\begin{array}{rlr}

\dfrac{\frac{a}{b}}{\frac{c}{d}}&=d\frac{ad}{bc}.\\

\textrm{ For} \quad \dfrac{\frac{a}{b}}{\frac{c}{d}} \cdot
\dfrac{c}{d}&=\dfrac{a}{b}, & \textrm{Def. VIII.}\\

\textrm{and} \quad \dfrac{ad}{bc} \cdot \dfrac{c}{d} &=
\dfrac{a}{b} \cdot \dfrac{dc}{cd}, & \textrm{§18, 1; Law IV.}\\
&=\dfrac{a}{b},\\

\textrm {since} \quad \dfrac{dc}{cd}&= dc=1 \times cd.
&\textrm{Def. VIII, Law IX.}
\end{array}
\]

The theorem follows by law IX.

\item

\[
\begin{array}{rlr}
\dfrac{a}{b}± \dfrac{c}{d}&=\dfrac{ad±bc}{bd}.\\

\textrm{For} \quad \left(\dfrac{a}{b}±\dfrac{c}{d}\right)bd
&=\dfrac{a}{b}b
\cdot d± \dfrac{c}{d}d \cdot b, & \textrm{ Laws III--V: §10, 5.}\\

&=ad±bc, & \textrm{Def. VIII.}\\

\textrm{and} \quad \left(\dfrac{ad±bc}{bd}\right)bd& =ad±bc.
&\textrm{Def. VIII.}
\end{array}
\]

The theorem follows by law IX.

By the same method it may be inferred that

\item

\[
\begin{array}{rlr}
 \dfrac{a}{b} > , &= , < \dfrac{c}{d},\\

\textrm{as} \quad ad > , &= , < bc.& \textrm{Def. VIII, Laws III,
IV, IX, IX'.}
\end{array}
\]
\end{enumerate}

\addcontentsline{toc}{section}{\numberline{}Limitations of
numerical division }

\addcontentsline{toc}{section}{\numberline{}Symbolic division. The
fraction}

\textbf{19. Limitations of Numerical Division. Symbolic Division.
The Fraction.} General as is the form of the preceding equations,
they are capable of numerical interpretation only when
$\frac{a}{b}$, $\frac{c}{d}$ are numbers, a case of comparatively
rare occurrence. The narrow limits set the quotient in the
numerical definition render division an unimportant operation as
compared with addition, multiplication, or the generalized
subtraction discussed in the preceding chapter.

But the way which led to an unrestricted subtraction lies open
also to the removal of this restriction; and the reasons for
following it there are even more cogent here.

We accept as the quotient of  $a$  divided by any number  $b$,
which is not 0, the symbol  $\frac{a}{b}$  defined by the equation
\[
\left(\frac{a}{b}\right) b = a,
\]
regarding this equation merely as a declaration of the equivalence
of the symbols $(\frac{a}{b}) b$  and  $a$, of the right to
substitute one for the other in any reckoning.

Whether $\frac{a}{b}$ be a number or not is to this definition
irrelevant. When a mere symbol, $\frac{a}{b}$ is called a
\textit{fraction}, and in opposition to this a number is called an
\textit{integer}.

We then put ourselves in immediate possession of definitions of
the addition, subtraction, multiplication, and division of this
symbol, as well as of the relations of equality and greater and
lesser inequality---definitions which are consistent with the
corresponding numerical definitions and with one another---by
assuming the permanence of form of the equations 1, 2, 3 and of
the test 4 of §~18 as symbolic statements, when they cease to be
interpretable as numerical statements.

The purely symbolic character of  $\frac{a}{b}$  and its
operations detracts nothing from their legitimacy, and they
establish division on a footing of at least formal equality with
the other three fundamental operations of arithmetic.\footnote{The
doctrine of symbolic division admits of being presented in the
very same form as that of symbolic subtraction.

The equations of Chapter II immediately pass over into theorems
respecting division when the signs of multiplication and division
are substituted for those of addition and subtraction; so, for
instance,
\[
a - (b + c) = a - b - c = a - c - b \text{ gives } \frac{a}{bc} =
\frac{(\frac{a}{b})}{c}=\frac{(\frac{a}{c})}{b}
\]

In particular, to $(a - a) + a = a$ corresponds $\frac{a}{a}a =
a$. Thus a purely symbolic definition may be given 1. It plays the
same rôle in multiplication as 0 in addition. Again, it has the
same exceptional character in involution---an operation related to
multiplication quite as multiplication to addition---as 0 in
multiplication; for $1^m = 1^n$, whatever the values of $m$ and
$n$.

Similarly, to the equation $(- a) + a = 0$, or $(0 - a) + a = 0$,
corresponds $(\frac{1}{a})a = 1$, which answers as a definition of
the unit fraction $\frac{1}{a}$; and in terms of these unit
fractions and integers all other fractions may be expressed.}

\addcontentsline{toc}{section}{\numberline{}Negative fractions}

\textbf{20. Negative Fractions.} Inasmuch as negatives conform to
the laws and definitions I--IX, the equations 1, 2, 3 and the test
4 of §18 are valid when any of the numbers $a$, $b$, $c$, $d$ are
replaced by negatives. In particular, it follows from the
definition of quotient and its determinateness, that
\[
\dfrac{a}{-b} = -\dfrac{a}{b}; \dfrac{-a}{b} = -\dfrac{a}{b};
\dfrac{-a}{-b} = \dfrac{a}{b}.
\]

It ought, perhaps, to be said that the determinateness of division
of negatives has not been formally demonstrated. The theorem,
however, that if $(\pm a)(\pm c) = (\pm b)(\pm c),
 \pm a = \pm b$, follows for every selection of the signs $\pm$ from
the one selection $+$, $+$, $+$, $+$ by §14, 6, 8.

\addcontentsline{toc}{section}{\numberline{}General test of the
equality or inequality of fractions}

\textbf{21. General Test of the Equality or Inequality of
Fractions.}

Given any two fractions $\pm \frac{a}{b},\pm \frac{c}{d}$.
\[
\begin{array}{rlr}
\pm \frac{a}{b} > , &= \text{ or } < \pm \frac{c}{d},\\
\text{ according as }  \pm ad > , &= \text{ or } < \pm bc. \\
&\text{Laws IX, IX'.} &  \text{ Compare §4, §14, 9.}
\end{array}
\]

\addcontentsline{toc}{section}{\numberline{}Indeterminateness of
division by zero}

\textbf{22. Indeterminateness of Division by Zero.} Division by 0
does not conform to the law of determinateness; the equations 1,
2, 3 and the test 4 of \S \, 18 are, therefore, not valid when 0
is one of the divisors.

The symbols $\displaystyle\frac{0}{0},\, \frac{a}{0},$ of which
some use is made in mathematics, are indeterminate.\footnote{In
this connection see \S \, 32.}

1. $\displaystyle\frac{0}{0}$ is indeterminate.   For
$\displaystyle\frac{0}{0} $ is completely defined by the equation
$\displaystyle\left(\frac{0}{0}\right)0 = 0$;  but since $ x \;
\text{x} \; 0 = 0$, whatever the value of $x$, any number
whatsoever will satisfy this equation.

2. $\displaystyle\frac{a}{0}$ is indeterminate. For, by
definition, $\displaystyle\left(\frac{a}{0}\right)0 = a$. Were
$\displaystyle\frac{a}{0}$  determinate, therefore,---since then
$\displaystyle \left(\frac{a}{0}\right)0 $ would, by \S \,18, 1,
be equal to $\displaystyle\frac{a \, \text{x} \, 0 }{0},$ or to
$\displaystyle\frac{0}{0}$,---the number $a$ would be equal to
$\displaystyle\frac{0}{0}$, or indeterminate.

\emph{Division by 0 is not an admissible operation.}

\addcontentsline{toc}{section}{\numberline{}Determinateness of
symbolic division}

\textbf{23.  Determinateness of Symbolic Division.} This exception
to the determinateness of division may seem to raise an objection
to the legitimacy of assuming---as is done when the demonstrations
1--4 of \S \, 18 are made to apply to symbolic quotients---that
symbolic division is determinate.

It must be observed, however, that $\displaystyle\frac{0}{0}$,
$\displaystyle\frac{a}{0}$ are indeterminate in the
\textit{numerical} sense, whereas by the determinateness of
symbolic division is, of course, not meant actual numerical
determinateness, but ``symbolic determinateness,'' conformity to
law IX, taken merely as a symbolic statement. For, as has been
already frequently said, from the present standpoint the
\emph{fraction} $\displaystyle\frac{a}{b}$ is a mere symbol,
altogether without numerical meaning apart from the equation
$\displaystyle\left(\frac{a}{b}\right)b=a$, with which, therefore,
the property of numerical determinateness has no possible
connection. The same is true of the product, sum or difference of
two fractions, and of the quotient of one fraction by another.

As for symbolic determinateness, it needs no justification when
assumed, as in the case of the fraction and the demonstrations
1--4, of symbols whose definitions do not preclude it. The
inference, for instance, that because

\begin{align*}
\left( \frac{a}{b}\frac{c}{d} \right)bd & = \left(\frac{ac}{bd}\right)bd, \\
\frac{a}{b} \frac{c}{d} & = \frac{ac}{bd},
\end{align*}

\noindent which depends on this principle of symbolic
determinateness, is of precisely the same character as the
inference that

\[
\left(\frac{a}{b}\frac{c}{d}\right)=\frac{a}{b}b \cdot
\frac{c}{d}d,
\]

\noindent which depends on the associative and commutative laws.

Both are pure assumptions made of the \textit{undefined} symbol
$\displaystyle \frac{a}{b} \frac{c}{d}$ for the sake of securing
it a definition identical in form with that of the product of two
numerical quotients.\footnote{These remarks, \textit{mutatis
mutandis}, apply with equal force to subtraction.}

\addcontentsline{toc}{section}{\numberline{}The vanishing of a
product}

 \textbf{24. The Vanishing of a Product.} It has already
been shown (\S~13, 3, \S~14, 7, \S~18, 1) that the sufficient
condition for the vanishing of a product is the vanishing of one
of its factors. From the determinateness of division it follows
that this is also the necessary condition, that is to say:

\textit{If a product vanish, one of its factors must vanish.}

Let $xy = 0$, where $x$, $y$ may represent numbers or any of the
symbols we have been considering.

\begin{flalign*}
&\text{\indent Since }&             xy &= 0,  &&
\\
&&                             xy + xz &= xz,  &\text{ \S 13, 1.}&
\\
&\text{or }&                  x(y + z) &= xz,  &\text{ Law V.}&
\\
&\text{whence, if $x$ be not $0$, }&  y + z &= z,   &\text{ Law
IX.}&
\\
&\text{or }&                         y &= 0.   &\text{ Law VII.}&
\end{flalign*}

\addcontentsline{toc}{section}{\numberline{}The system of rational
numbers }

\textbf{25. The System of Rational Numbers.} Three symbols, $0$,
$-d$, $\frac{a}{b}$, have thus been found which can be reckoned
with by the same rules as numbers, and in terms of which it is
possible to express the result of every addition, subtraction,
multiplication or division, whether performed on numbers or on
these symbols themselves; therefore, also, the result of any
complex operation which can be resolved into a finite combination
of these four operations.

Inasmuch as these symbols play the same r\^ole as numbers in
relation to the fundamental operations of arithmetic, it is
natural to class them with numbers. The word ``number,''
originally applicable to the positive integer only, has come to
apply to zero, the negative integer, the positive and negative
fraction also, this entire group of symbols being called the
system of \emph{rational numbers}.\footnote{It hardly need be said
that the fraction, zero, and the negative actually made their way
into the number-system for quite a different reason from
this;---because they admitted of certain ``real'' interpretations,
the fraction in measurements of lines, the negative in debit where
the corresponding positive meant credit or in a length measured to
the left where the corresponding positive meant a length measured
to the right. Such interpretations, or correspondences to existing
things which lie entirely outside of pure arithmetic, are ignored
in the present discussion as being irrelevant to a pure
arithmetical doctrine of the artificial forms of number.} This
involves, of course, a radical change of the number concept, in
consequence of which numbers become merely part of the symbolic
equipment of certain operations, admitting, for the most part, of
only such definitions as these operations lend them.

In accepting these symbols as its numbers, arithmetic ceases to be
occupied exclusively or even principally with the properties of
numbers in the strict sense. It becomes an \emph{algebra}, whose
immediate concern is with certain operations defined, as addition
by the equations $a + b = b + a$, $a + (b + c) = a + b + c$,
formally only, without reference to the meaning of the symbols
operated on.\footnote{The word ``algebra'' is here used in the
general sense, the sense in which \emph{quaternions} and the
\textit{Ausdehungslehre} (see \S\S~127, 128) are algebras.
Inasmuch as elementary arithmetic, as actually constituted,
accepts the fraction, there is no essential difference between it
and elementary algebra with respect to the kinds of number with
which it deals; algebra merely goes further in the use of
artificial numbers. The elementary algebra differs from arithmetic
in employing literal symbols for numbers, but chiefly in making
the equation an object of investigation.}



\chapter{THE IRRATIONAL.}

\addcontentsline{toc}{section}{\numberline{}Inadequateness of the
system of rational numbers  }

\textbf{26. The System of Rational Numbers Inadequate.} The system
of rational numbers, while it suffices for the four fundamental
operations of arithmetic and finite combinations of these
operations, does not fully meet the needs of algebra.

The great central problem of algebra is the equation, and that
only is an adequate number-system for algebra which supplies the
means of expressing the roots of all possible equations. The
system of rational numbers, however, is equal to the requirements
of equations of the first degree only; it contains symbols not
even for the roots of such elementary equations of higher degrees
as $x^2 = 2$, $x^2 = -1$.

But how is the system of rational numbers to be enlarged into an
algebraic system which shall be adequate and at the same time
sufficiently simple?

The roots of the equation
\[
x^{n} + p_{1}x^{n-1} + p_{2}x^{n-2} + \dotsb + p_{n-1}x + p_{n} =
0
\]
are not the results of single elementary operations, as are the
negative of subtraction and the fraction of division; for though
the roots of the quadratic are results of ``evolution,'' and the
same operation often enough repeated yields the roots of the cubic
and biquadratic also, it fails to yield the roots of higher
equations. A system built up as the rational system was built, by
accepting indiscriminately every new symbol which could show cause
for recognition, would, therefore, fall in pieces of its own
weight.

The most general characteristics of the roots must be discovered
and defined and embodied in symbols---by a method which does not
depend on processes for solving equations. These symbols, of
course, however characterized otherwise, must stand in consistent
relations with the system of rational numbers and their
operations.

An investigation shows that the forms of number necessary to
complete the algebraic system may be reduced to two: the symbol
$\displaystyle\sqrt{-1}$, called the \textit{imaginary} (an
indicated root of the equation $x^2 + 1 = 0$), and the class of
symbols called \textit{irrational}, to which the roots of the
equation $x^2-2=0$ belong.

\addcontentsline{toc}{section}{\numberline{}Numbers defined by
``regular sequences.'' The irrational}

\textbf{27. Numbers Defined by Regular Sequences. The Irrational.}
On applying to 2 the ordinary method for extracting the square
root of a number, there is obtained the following sequence of
numbers, the results of carrying the reckoning out to 0, 1, 2, 3,
4, \ldots places of decimals, viz.:

\[
 1, 1.4, 1.41, 1.414, 1.4142,\; \ldots
\]

These numbers are rational; the first of them differs from each
that follows it by less than 1, the second by less than
$\displaystyle\frac{1}{10}$, the third by less than
$\displaystyle\frac{1}{100}$, \ldots the $n$th by less than
$\displaystyle\frac{1}{10^{n-1}}$. And
$\displaystyle\frac{1}{10^{n-1}}$ is a fraction which may be made
less than any assignable number whatsoever by taking $n$ great
enough.

This sequence may be regarded as a definition of the square root
of 2. It is such in the sense that a term may be found in it the
square of which, as well as of each following term, differs from 2
by less than any assignable number.

\textit{Any sequence of rational numbers}
\[\alpha_1,\alpha_2,\alpha_3,\cdots,\alpha_{\mu},\alpha_{\mu+1},\cdots\alpha_{\mu+\nu},\cdots\]
\textit{in which, as in the above sequence, the term
$\alpha_{\mu}$ may, by taking $\mu$ great enough, be made to
differ numerically from each term that follows it by less than any
assignable number, so that, for all values of $\nu$, the
difference, $\alpha_{\mu+\nu}-\alpha_{\mu}$, is numerically less
than $\delta$, however small $\delta$ be taken, is called a
regular sequence.}

The entire class of operations which lead to regular sequences may
be called \textit{regular sequence-building}. Evolution is only
one of many operations belonging to this class.

\textit{Any regular sequence is said to ``define a
number,''}---this ``number'' being merely the symbolic, ideal,
result of the operation which led to the sequence. It will
sometimes be convenient to represent numbers thus defined by the
single letters $a$, $b$, $c$, etc., which have heretofore
represented positive integers only.

After some particular term all terms of the sequence $\alpha_1$,
$\alpha_2,\cdots$ may be the same, say $\alpha$. The number
defined by the sequence is then $\alpha$ itself. A place is thus
provided for rational numbers in the general scheme of numbers
which the definition contemplates.

When not a rational, the number defined by a regular sequence is
called \textit{irrational}.

The regular sequence .3, .33, \ldots, has a \textit{limiting
value}, viz., $\displaystyle\frac{1}{3}$; which is to say that a
term can be found in this sequence which itself, as well as each
term which follows it, differs from $\displaystyle\frac{1}{3}$ by
less than any assignable number. In other words, the difference
between $\displaystyle\frac{1}{3}$ and the $\mu$th term of the
sequence may be made less than any assignable number whatsoever by
taking $\mu$ great enough. It will be shown presently that the
number defined by any regular sequence, $\alpha_{1}$,
$\alpha_{2},\cdots$ stands in this same relation to its term
$\alpha_{\mu}$.

\addcontentsline{toc}{section}{\numberline{}Generalized
definitions of zero, positive, negative }

 \textbf{28. Zero, Positive, Negative.} In any regular
sequence $\alpha_{1}, \alpha_{2}, \cdots$ a term $\alpha_{\mu}$
may always be found which itself, as well as each term which
follows it, is either

(1) numerically less than any assignable number,\\
or (2) greater than some definite positive rational number,\\
or (3) less than some definite negative rational number.\\

In the first case the number $a$, which the sequence defines, is
said to be \emph{zero}, in the second \emph{positive}, in the
third \emph{negative}.

\addcontentsline{toc}{section}{\numberline{}Of the four
fundamental operations}

\textbf{29. The Four Fundamental Operations.} \textit{Of the
numbers defined by the two sequences:}
\begin{align*}
&\alpha_{1},\alpha_{2},\alpha_{3},\cdots,\alpha_{\mu},
\alpha_{\mu+1},\cdots,\alpha_{\mu+\nu},\cdots, \\
&\beta_{1},\beta_{2},\beta_{3},\cdots,\beta_{\mu},
\beta_{\mu+1},\cdots,\beta_{\mu+\nu},\cdots
\end{align*}

(1) \textit{The sum is the number defined by the sequence:}
\[\alpha_{1}+\beta_{1},\alpha_{2}+\beta_{2},\cdots
\alpha_{\mu}+\beta_{\mu},\alpha_{\mu+1}+\beta_{\mu+1},\cdots
\alpha_{\mu+\nu}+\beta_{\mu+\nu},\cdots\]

(2) \textit{The difference is the number defined by the sequence:}
\[\alpha_{1}-\beta_{1},\alpha_{2}-\beta_{2},\cdots
\alpha_{\mu}-\beta_{\mu},\alpha_{\mu+1}-\beta_{\mu+1},\cdots
\alpha_{\mu+\nu}-\beta_{\mu+\nu},\cdots\]

(3) \textit{The product is the number defined by the sequence:}
\[\alpha_{1}\beta_{1},\alpha_{2}\beta_{2},\cdots
\alpha_{\mu}\beta_{\mu},\alpha_{\mu+1}\beta_{\mu+1},\cdots
\alpha_{\mu+\nu}\beta_{\mu+\nu},\cdots\]

(4) \textit{The quotient is the number defined by the sequence:}
\[\frac{\alpha_{1}}{\beta_{1}},
\frac{\alpha_{2}}{\beta_{2}},\cdots
\frac{\alpha_{\mu}}{\beta_{\mu}},
\frac{\alpha_{\mu+1}}{\beta_{\mu+1}},\cdots
\frac{\alpha_{\mu+\nu}}{\beta_{\mu+\nu}},\cdots\]

For these definitions are consistent with the corresponding
definitions for rational numbers; they reduce to these elementary
definitions, in fact, whenever the sequences $\alpha_1$,
$\alpha_2, \ldots$; $\beta_1$, $\beta_2, \ldots$ either reduce to
the forms $\alpha$, $\alpha,\ldots$; $\beta$, $\beta, \ldots$ or
have rational limiting values.

They conform to the fundamental laws I--IX\@. This is immediately
obvious with respect to the commutative, associative, and
distributive laws, the corresponding terms of the two sequences
$\alpha_1\beta_1$, $\alpha_2\beta_2,\ldots$; $\beta_1\alpha_1$,
$\beta_2\alpha_2, \ldots$, for instance, being identically equal,
by the commutative law for rationals.

But again division as just defined is determinate. For division
can be indeterminate only when a product may vanish without either
factor vanishing (cf. \S~24); whereas $\alpha_1\beta_1$,
$\alpha_2\beta_2,\ldots$ can define 0, or its terms after the
$n$th fall below any assignable number whatsoever, only when the
same is true of one of the sequences $\alpha_1$, $\alpha_2,
\ldots$; $\beta_1$, $\beta_2, \ldots$\footnote{It is worth
noticing that the determinateness of division is here not an
independent assumption, but a consequence of the definition of
multiplication and the determinateness of the division of
rationals. The same thing is true of the other fundamental laws
I--V, VII. }

It only remains to prove, therefore, that the sequences (1), (2),
(3), (4) are qualified to define numbers (\S~27).

(1) and (2) Since the sequences $\alpha_1$, $\alpha_2,\ldots$;
$\beta_1$, $\beta_2,\ldots$ are, by hypothesis, such as define
numbers, corresponding terms in the two, $\alpha_\mu$, $\beta_\mu$
may be found, such that

\begin{tabular}{ll}
& $\alpha_{\mu+\nu}-\alpha_\mu$ \; is numerically \; $< \delta$, \\
and & $ \; \beta_{\mu+\nu}-\beta_\mu$ \; is numerically \; $ < \delta$, \\
and, therefore, & $ \; (\alpha_{\mu+\nu} \pm \beta_{\mu+\nu})-(\alpha_\mu \pm \beta_\mu) < 2\delta$,\\
\end{tabular}

\noindent for all values of $\nu$, and that however small $\delta$
may be.

Therefore each of the sequences $\alpha_1+\beta_1$,
$\alpha_2+\beta_2,\ldots$; $\alpha_1-\beta_1$,
$\alpha_2-\beta_2,\ldots$ is regular.

(3) Let $\alpha_\mu$ and $\beta_\mu$ be chosen as before.

Then  $\alpha_{\mu+\nu}\beta_{\mu+\nu} - \alpha_\mu \beta_\mu$,

since it is identically equal to
\[
\alpha_{\mu+\nu}(\beta_{\mu+\nu}-\beta_\mu) +
\beta_\mu(\alpha_{\mu+\nu}-\alpha_\mu),
\]
is numerically less than $\alpha_{\mu+\nu}\delta+\beta_\mu\delta$,
and may, therefore, be made less than any assignable number by
taking $\delta$ small enough; and that for all values of $\nu$.

Therefore the sequence $\alpha_1\beta_1, \alpha_2\beta_2,\ldots$
is regular.
\[
(4) \qquad
\frac{\alpha_{\mu+\nu}}{\beta_{\mu+\nu}}-\frac{\alpha_\mu}{\beta_\mu}
=
\frac{\alpha_{\mu+\nu}\beta_\mu-\beta_{\mu+\nu}\alpha_\mu}{\beta_{\mu+\nu}\beta_\mu},
\]
which is identically equal to
\[
\frac{\beta_{\mu+\nu}(\alpha_{\mu+\nu}-\alpha_\mu)-\alpha_{\mu+\nu}(\beta_{\mu+\nu}-\beta_\mu)}{\beta_{\mu+\nu}\beta_\mu}.
\]

By choosing $\alpha_\mu$ and $\beta_\mu$ as before the numerator
of this fraction, and therefore the fraction itself, may be made
less than any assignable number; and that for all values of $\nu$.

Therefore the sequence $\displaystyle\frac{\alpha_1}{\beta_1},
\frac{\alpha_2}{\beta_2}, \ldots$ is regular.

\addcontentsline{toc}{section}{\numberline{}Of equality and
greater and lesser inequality  }

\textbf{30.   Equality.   Greater and Lesser Inequality.}
\textit{Of two numbers, $a$ and $b$, defined by regular sequences
$\alpha_1, \alpha_2,\ldots,$; $\beta_1,\beta_2, \ldots$, the first
is greater than, equal to or less than the second, according as
the number defined by $\alpha_1-\beta_1, \alpha_1-\beta_2,\ldots$
is greater than, equal to or less than $0$.}

This definition is to be justified exactly as the definitions of
the fundamental operations on numbers defined by regular sequences
were justified in \S~29.

From this definition, and the definition of $0$ in \S~28, it
immediately follows that

COR. \textit{Two numbers which differ by less than any assignable
number are equal.}

\addcontentsline{toc}{section}{\numberline{}The number defined by
a regular sequence its limiting value }

\textbf{31.  The Number Defined by a Regular Sequence is its
Limiting Value.}  The difference between a number $a$ and the term
$\alpha_{\mu}$ of the sequence by which it is defined may be made
less than any assignable number by taking $\mu$ great enough.


For it is only a restatement of the definition of a regular
sequence $\alpha_1,\alpha_2,\ldots$ to say that the sequence
\[
\alpha_1-\alpha_{\mu},\alpha_2-\alpha_{\mu},\ldots,\alpha_{\mu+\nu}-\alpha_\mu,\ldots,
\]
which defines the difference $a-\alpha_{\mu}$ (\S~29, 2), is one
whose terms after the $\mu$th can be made less than any assignable
number by choosing $\mu$ great enough, and which, therefore,
becomes, as $\mu$ is indefinitely increased, a sequence which
defines 0 (\S~28).

In other words, the \textit{limit} of $a-\alpha_{\mu}$ as $\mu$ is
indefinitely increased is 0, or $a=\text{limit}\,(\alpha_{\mu})$.
Hence

\textit{The number defined by a regular sequence is the limit to
which the $\mu$th term of this sequence approaches as $\mu$ is
indefinitely increased.}\footnote{What the above demonstration
proves is that $a$ stands in the same relation to $\alpha_{\mu}$
when irrational as when rational. The principle of permanence (cf.
\S~12), therefore, justifies one in regarding $a$ as the ideal
limit in the former case since it is the actual limit in the
latter (\S~27). $a$, when irrational, is limit $(\alpha_{\mu})$ in
precisely the same sense that $\displaystyle\frac{c}{d}$ is the
quotient of $c$ by $d$, when $c$ is a positive integer not
containing $d$. It follows from the demonstration that if there be
a reality corresponding to $a$, as in geometry we assume there is
(\S~40), that reality will be the actual limit of the reality of
the same kind corresponding to $\alpha_{\mu}$.

The notion of irrational limiting values was not immediately
available because, prior to \S\S~28, 29, 30, the meaning of
difference and greater and lesser inequality had not been
determined for numbers defined by sequences.}

The definitions (1), (2), (3), (4) of \S~29 may, therefore, be
stated in the form:

\begin{equation*}
\begin{aligned}
& \text{limit}\,(\alpha_{\mu}) \pm \text{limit}\,(\beta_{\mu})  &= &\text{limit}\,(\alpha_{\mu}\pm\beta_{\mu}),\\
& \text{limit}\,(\alpha_{\mu})\cdot\text{limit}\,(\beta_{\mu}) &= & \text{limit}\,(\alpha_{\mu}\beta_{\mu}),\\
& \frac{\text{limit}\,(\alpha_{\mu})}{\text{limit}\,(\beta_{\mu})} &=&  \text{limit}\,\left(\frac{\alpha_{\mu}}{\beta_{\mu}}\right).\\
\end{aligned}
\end{equation*}

For limit ($\alpha_{\mu}$) the more complete symbol
$\displaystyle\lim_{\mu\doteq\infty}(\alpha_{\mu})$ is also used,
read ``the limit which $\alpha_{\mu}$ approaches as $\mu$
approaches infinity''; the phrase ``approaches infinity'' meaning
only, ``becomes greater than any assignable number.''

\addcontentsline{toc}{section}{\numberline{}Division by zero }

\textbf{32. Division by Zero.} (1) The sequence
$\displaystyle\frac{\alpha_1}{\beta_1},\frac{\alpha_2}{\beta_2},\ldots$
cannot define a number when the number defined by
$\beta_1,\beta_2,\ldots$ is 0, unless the number defined by
$\alpha_1,\alpha_2,\ldots$ be also 0. In this case it may;
$\displaystyle\frac{\alpha_{\mu}}{\beta_{\mu}}$ may approach a
definite limit as $\mu$ increases, however small $\alpha_{\mu}$
and $\beta_{\mu}$ become. But this number is not to be regarded as
the mere quotient $\displaystyle\frac{0}{0}$. Its value is not at
all determined by the fact that the numbers defined by
$\alpha_1,\alpha_2,\ldots$; $\beta_1,\beta_2,\ldots$ are 0; for
there is an indefinite number of different sequences which define
0, and by properly choosing $\alpha_1,\alpha_2,\ldots$;
$\beta_1,\beta_2,\ldots$ from among them, the terms of the
sequence
$\displaystyle\frac{\alpha_1}{\beta_1},\frac{\alpha_2}{\beta_2},\ldots$
may be made to take any value whatsoever.

(2) The sequence
$\displaystyle\frac{\alpha_1}{\beta_1},\frac{\alpha_2}{\beta_2},\ldots$
is not regular when $\beta_1,\beta_2,\ldots$ defines 0 and
$\alpha_1,\alpha_2,\ldots$ defines a number different from 0.

No term $\displaystyle\frac{\alpha_{\mu}}{\beta_{\mu}}$ can be
found which differs from the terms following it by less than any
assignable number; but rather, by taking $\mu$ great enough,
$\displaystyle\frac{\alpha_{\mu}}{\beta_{\mu}}$ can be made
greater than any assignable number whatsoever.

Though not regular and though they do not define numbers, such
sequences are found useful in the higher mathematics. They may be
said to define \textit{infinity}. Their usefulness is due to their
determinate form, which makes it possible to bring them into
combination with other sequences of like character or even with
regular sequences.

Thus the quotient of any regular sequence
$\gamma_1,\gamma_2,\ldots$ by
$\displaystyle\frac{\alpha_1}{\beta_1}, \frac{\alpha_2}{\beta_2},
\ldots$  is a regular  sequence  and defines 0;   and the quotient
of $\displaystyle\frac{\alpha_1}{\beta_1},
\frac{\alpha_2}{\beta_2},\ldots$ by a similar sequence
$\displaystyle\frac{\gamma_1}{\delta_1},
\frac{\gamma_2}{\delta_2}, \ldots$ may also be regular and
serve---if $\alpha_i$, $\beta_i$, $\gamma_i$, $\delta_i$ ($i = 1,
2,\ldots$) be properly chosen---to define any number whatsoever.

The term $\displaystyle\frac{\alpha_\mu}{\beta_\mu}$ ``approaches
infinity'' (\textit{i.~e.} increases without limit) as $\mu$ is
indefinitely increased, in a definite or determinate manner;  so
that the infinity which
$\displaystyle\frac{\alpha_1}{\beta_1},\frac{\alpha_2}{\beta_2},
\ldots$ defines is not indeterminate like the mere symbol
$\displaystyle\frac{a}{0}$ of \S~22.

But here again it is to be said that this determinateness is not
due to the mere fact that $\beta_1, \beta_2 \ldots$ defines 0,
which is all that the unqualified symbol
$\displaystyle\frac{a}{0}$ expresses. For there is an indefinite
number of different sequences which like $\beta_1, \beta_2,
\ldots$ define 0, and $\displaystyle\frac{a}{0}$ is a symbol for
the quotient of $a$ by any one of them.

\addcontentsline{toc}{section}{\numberline{}The number-system
defined by regular sequences of rationals a closed and continuous
system }

\textbf{33. The System defined by Regular Sequences of Rationals,
Closed and Continuous.} \textit{A regular sequence of irrationals
\[
a_1, a_{2},\ldots a_m, a_{m+1},\ldots a_{m+n},  \ldots
\]
(in which the differences $a_{m+n}-a_{m}$ may be made numerically
less than any assignable number by taking $m$ great enough)
defines a number, but never a number which may not also be defined
by a sequence of rational numbers.}

For $\beta_1, \beta_2, \ldots$ being any sequence of rationals
which defines 0, construct a sequence of rationals $\alpha_1,
\alpha_2,\ldots$ such that $a_1-\alpha_1$ is numerically less than
$\beta_1$ (\S~30), and in the same sense $a_2-\alpha_2<\beta_2$,
$a_3-\alpha_3<\beta_3$ etc. Then limit $(a_m-\alpha_m) = 0$
(\S\S~28, 31), or limit $(a_m) = \text{limit}(\alpha_m)$.

This theorem justifies the use of regular sequences of irrationals
for defining numbers, and so makes possible a simple expression of
the results of some very complex operations. Thus $a^m$, where $m$
is irrational, is a number; the number, namely, which the sequence
$a^{\alpha_1},a^{\alpha_2},\ldots$ defines, when
$\alpha_1,\alpha_2,\ldots$ is any sequence of rationals defining
$m$.

But the importance of the theorem in the present discussion lies
in its declaration that the number-system defined by regular
sequences of rationals contains all numbers which result from the
operations of regular sequence-building in general. It is a
\textit{closed} system with respect to the four fundamental
operations and this new operation, exactly as the rational numbers
constitute a closed system with respect to the four fundamental
operations only (cf. \S~25).

The system of numbers defined by regular sequences of
rationals---\textit{real} numbers, as they are called---therefore
possesses the following two properties: (1) between every two
unequal, real numbers there are other real numbers; (2) a variable
which runs through any regular sequence of real numbers, rational
or irrational, will approach a real number as limit. We indicate
all this by saying that the system of real numbers is
\textbf{continuous}.


\chapter{THE IMAGINARY\@.   COMPLEX NUMBERS.}

\addcontentsline{toc}{section}{\numberline{}The pure imaginary  }

\textbf{34. The Pure Imaginary.} The other symbol which is needed
to complete the number-system of algebra, unlike the irrational
but like the negative and the fraction, admits of definition by a
single equation of a very simple form, viz.,
\[
x^2+1=0
\]

It is the symbol whose square is $-1$, the symbol $\sqrt{-1}$, now
commonly written $i$.\footnote{Gauss introduced the use of $i$ to
represent $\sqrt{-1}$.} It is called the \textit{unit of
imaginaries}.

In contradistinction to $i$ all the forms of number hitherto
considered are called \textit{real}. These names, ``real'' and ``imaginary,'' \, are unfortunate, for they suggest an opposition
which does not exist. Judged by the only standards which are
admissible in a pure doctrine of numbers $i$ is imaginary in the
same sense as the negative, the fraction, and the irrational, but
in no other sense; all are alike mere symbols devised for the sake
of representing the results of operations even when these results
are not numbers (positive integers). $i$ got the name imaginary
from the difficulty once found in discovering some
extra-arithmetical reality to correspond to it.

As the only property attached to $i$ by definition is that its
square is $-1$, nothing stands in the way of its being ``multiplied'' \, by any real number $a$; the product, $ia$, is
called a \textit{pure imaginary}.

An entire new system of numbers is thus created, coextensive with
the system of real numbers, but distinct from it. Except $0$,
there is no number in the one which is at the same time contained
in the other.\footnote{Throughout this discussion $\infty$ is not
regarded as belonging to the number-system, but as a limit of the
system, lying without it, a symbol for something greater than any
number of the system.} Numbers in either system may be compared
with each other by the definitions of equality and greater and
lesser inequality (\S~30), $ia$ being called
$\displaystyle\gtreqqless ib$, as $\displaystyle a \gtreqqless b$;
but a number in one system cannot be said to be either greater
than, equal to or less than a number in the other system.

\addcontentsline{toc}{section}{\numberline{}Complex numbers}

\textbf{35. Complex Numbers.} The sum $a + ib$ is called a
\textit{complex number}. Its terms belong to two distinct systems,
of which the fundamental units are $1$ and $i$.

The \textit{general} complex number $a + ib$ is defined by a
\textit{complex sequence}
\[
\alpha_1+i\beta_1, \, \alpha_2+i\beta_2, \ldots,
\alpha_\mu+i\beta_\mu, \ldots,
\]
where $\alpha_1, \alpha_2, \ldots $; $\beta_1, \beta_2, \ldots $
are regular sequences.

Since $a=a+i0$ (\S~36, 3, Cor.) and $ib=0+ib$, all real numbers,
$a$, and pure imaginaries, $ib$, are contained in the system of
complex numbers $a+ib$.

$a+ib$ can vanish only when both $a=0$ and $b=0$.

\addcontentsline{toc}{section}{\numberline{}The fundamental
operations on complex numbers}

\textbf{36. The Four Fundamental Operations on Complex Numbers.}
The assumption of the permanence of the fundamental laws leads
immediately to the following definitions of the addition,
subtraction, multiplication, and division of complex numbers.

\begin{equation*}
\begin{aligned}
 1. \qquad (a+ib)+(a'+ib') = \, & a+a'+i(b+b'). \\
 \text{For} \quad (a+ib)+(a'+ib') = \, & a+ib+a'+ib', \qquad \text{Law II}.\\
 = \, & a+a'+ib+ib', \qquad \text{Law I}.\\
 = \, & a+a'+i(b+b'). \qquad \text{Laws II, V}.\\
 2. \qquad  (a+ib)-(a'+ib')  = \, & a-a'+i(b-b').\\
\end{aligned}
\end{equation*}

By definition of subtraction (VI) and \S~36, 1.

COR. \textit{The necessary as well as the sufficient condition for
the equality of two complex numbers $a+ib$, $a'+ib'$ is that
$a=a'$ and $b=b'$.}

\begin{equation*}
\begin{aligned}
\text{For if} \quad (a+ib)-(a'+ib')= \, & a-a'+i(b-b')=0,\\
a-a'=0, b-b'= \, & 0 \; (\S~35), \; \text{or} \; a=a', b=b'.\\
3. \qquad (a+ib)(a'+ib')= \, & aa'-bb'+i(ab'+ba').\\
\end{aligned}
\end{equation*}

\begin{equation*}
\begin{aligned}
\text{For} \quad (a+ib)(a'+ib')= \, & (a+ib)a'+(a+ib)ib', \qquad \qquad \text{Law V}.\\
= \, & aa'+ib\cdot a'+a\cdot ib'+ib\cdot ib', \qquad \text{Law V}.\\
=\, & (aa'-bb')+i(ab'+ba'). \qquad \qquad \text{Laws I--V}.\\
\end{aligned}
\end{equation*}

COR. \textit{If either factor of a product vanish, the product
vanishes.}

\[ \text{For} \quad i\times 0=i(b-b)=ib-ib \; (\S~10, 5), =0 \; (\S~14, 1). \]
\[ \text{Hence} \quad  (a+ib)0=a\times 0+ib\times 0=a\times 0+i(b\times 0)=0.\]
\begin{flushright}
Laws V, IV, \S~28, \S~29, 3.
\end{flushright}
\[4. \qquad \frac{a+ib}{a'+ib'}=\frac{aa'+bb'}{a'^2+b'^2}+i\frac{ba'-ab'}{a'^2+b'^2}.\]

For let the quotient of $a+ib$ by $a'+ib'$ be $x+iy$.

By the definition of division (VIII),
\begin{align*}
& (x+iy)(a'+ib')=a+ib. \\
\therefore \quad & xa'-yb'+i(xb'+ya')=a+ib. \qquad \S~36, 3\\
\therefore \quad & xa'-yb'=a, \; xb'+ya'=b. \qquad \S~36, 2, Cor. \\
\end{align*}

Hence, solving for $x$ and $y$ between these two equations,

\[ x=\frac{aa'+bb'}{a'^2+b'^2}, \quad y=\frac{ba'-ab'}{a'^2+b'^2}.\]

Therefore, as in the case of real numbers, division is a
determinate operation, except when the divisor is 0; it is then
indeterminate. For $x$ and $y$ are determinate (by IX) unless
$a'^2+b'^2=0$, that is, unless $a'=b'=0$, or $a'+ib'=0$; for $a'$
and $b'$ being real, $a'^2$ and $b'^2$ are both positive, and one
cannot destroy the other.\footnote{What is here proven is that in
the system of complex numbers formed from the fundamental units 1
and $i$ there is one, and but one, number which is the quotient of
$a+ib$ by $a'+ib'$; this being a consequence of the
determinateness of the division of real numbers and the peculiar
relation ($i^2=-1$) holding between the fundamental units. For the
sake of the permanence of IX we make the assumption, otherwise
irrelevant, that this is the only value of the quotient whether
within or without the system formed from the units 1 and $i$.}
Hence, by the reasoning in \S~24,

COR. \textit{If a product of two complex numbers vanish, one of
the factors must vanish.}

\addcontentsline{toc}{section}{\numberline{}Numerical comparison
of complex numbers}

\textbf{37. Numerical Comparison of Complex Numbers.} Two complex
numbers, $a+ib$, $a'+ib'$, do not, generally speaking, admit of
direct comparison with each other, as do two real numbers or two
pure imaginaries; for $a$ may be greater than $a'$, while $b$ is
less than $b'$.

They are compared \textit{numerically}, however, by means of their
\textit{moduli} $\sqrt{a^2+b^2}$, $\sqrt{a'^2+b'^2}$; $a+ib$ being
said to be numerically greater than, equal to or less than
$a'+ib'$ according as $\sqrt{a^2+b^2}$ is greater than, equal to
or less than $\sqrt{a'^2+b'^2}$. Compare \S~47.

\addcontentsline{toc}{section}{\numberline{}Adequateness of the
system of complex number}

\textbf{38. The Complex System Adequate.} The system $a+ib$ is an
adequate number-system for algebra. For, as will be shown (Chapter
VII), all roots of algebraic equations are contained in this
system.

But more than this, the system $a+ib$ is a closed system with
respect to all existing mathematical operations, as are the
rational system with respect to all finite combinations of the
four fundamental operations and the real system with respect to
these operations and regular sequence-building. For the results of
the four fundamental operations on complex numbers are complex
numbers (\S~36, 1, 2, 3, 4). Any other operation may be resolved
into either a finite combination of additions, subtractions,
multiplications, divisions or such combinations indefinitely
repeated. In either case the result, if determinate, is a complex
number, as follows from the definitions 1, 2, 3, 4 of \S~36, and
the nature of the real number-system as developed in the preceding
chapter (see Chapter VIII).

The most important class of these higher operations, and the class
to which the rest may be reduced, consists of those operations
which result in infinite series (Chapter VIII); among which are
involution, evolution, and the taking of logarithms (Chapter IX),
sometimes included among the fundamental operations of algebra.

\addcontentsline{toc}{section}{\numberline{}Fundamental
characteristics of the algebra of number}

\textbf{39. Fundamental Characteristics of the Algebra of Number.}
The algebra of number is completely characterized, formally
considered, by the laws and definitions I--IX and the fact that its
numbers are expressible linearly in terms of two fundamental
units.\footnote{That is, in terms of the first powers of these
units.} It is a linear, associative, distributive, commutative
algebra. Moreover, the most general linear, associative,
distributive, commutative algebra, whose numbers are complex
numbers of the form $x_1e_1+x_2e_2+\cdots+x_ne_n$, built from $n$
fundamental units $e_1, e_2,\ldots, e_n$, is reducible to the
algebra of the complex number $a+ib$. For Weierstrass\footnote{Zur Theorie der aus $n$ Haupteinheiten gebildeten
complexen Gr\"{o}ssen. G\"{o}ttinger Nachrichten Nr. 10, 1884.

Weierstrass finds that these general complex numbers differ in
only one important respect from the complex number $a+ib$. If the
number of fundamental units be greater than 2, there always exist
numbers, different from 0, the product of which by certain other
numbers is 0. Weierstrass calls them divisors of 0. The number of
exceptions to the determinateness of division is infinite instead
of one.} has shown that any two complex numbers $a$ and $b$ of the
form $x_1e_1+x_2e_2+ \cdots +x_ne_n$, whose sum, difference,
product, and quotient are numbers of this same form, and for which
the laws and definitions I--IX hold good, may by suitable
transformations be resolved into components $a_1, a_2,\ldots a_r$;
$b_1, b_2,\ldots b_r$, such that

\begin{align*}
a= \, & a_1+a_2+ \cdots +a_r,\\
b= \, & b_1+b_2+\cdots+b_r,\\
a \pm b= \, & a_1 \pm b_1 + a_2 \pm b_2+\cdots+a_r \pm b_r,\\
ab= \, & a_1b_1+a_2b_2+\cdots+a_rb_r,\\
\frac{a}{b}= \, & \frac{a_1}{b_1}+\frac{a_2}{b_2}+\cdots+\frac{a_r}{b_r}.\\
\end{align*}

\noindent The components $a_i$, $b_i$ are constructed either from
one fundamental unit $g_i$ or from two fundamental units $g_i$,
$k_i$.\footnote{These units are, generally speaking, not
$e_1, e_2,\ldots, e_n$, but linear combinations of them, as
$\gamma_1e_1+\gamma_2e_2+\cdots+\gamma_ne_n$,
$\kappa_1e_1+\kappa_2e_2+\cdots+\kappa_ne_n$. Any set of $n$
independent linear combinations of the units $e_1, e_2,\ldots, e_n$
may be regarded as constituting a set of fundamental units, since
all numbers of the form
$\alpha_1e_1+\alpha_2e_2+\cdots+\alpha_ne_n$ may be expressed
linearly in terms of them.}

For components of the first kind the multiplication formula is
\[(\alpha g_i)(\beta g_i)=(\alpha\beta)g_i.\]

For components of the second kind the multiplication formula is
\[ (\alpha g_i+\beta k_i)(\alpha'g_i+\beta'k_i)
=(\alpha\alpha'-\beta\beta')g_i+(\alpha\beta'+\beta\alpha')k_i.\]

And these formulas are evidently identical with the multiplication
formulas
\begin{align*}
(\alpha1)(\beta1)= \, & (\alpha\beta)1,\\
(\alpha1+\beta i)(\alpha'1+\beta'i) = \, & (\alpha\alpha'-\beta\beta')1+(\alpha\beta'+\beta\alpha')i\\
\end{align*}

\noindent of common algebra.


\chapter{GRAPHICAL REPRESENTATION OF NUMBERS\@. THE VARIABLE.}

\addcontentsline{toc}{section}{\numberline{}Correspondence between
the real number-system and the points of a line }

\textbf{40. Correspondence between the Real Number-System and the
Points of a Line.} Let a right line be chosen, and on it a fixed
point, to be called the null-point; also a fixed unit for the
measurement of lengths.

Lengths may be measured on this line either from left to right or
from right to left, and equal lengths measured in opposite
directions, when added, annul each other; opposite algebraic signs
may, therefore, be properly attached to them. Let the sign {\Large
$+$} be attached to lengths measured to the right, the sign
{\Large $-$} to lengths measured to the left.

\textit{The entire system of real numbers may be represented by
the points of the line}, by taking to correspond to each number
that point whose distance from the null-point is represented by
the number. For, as we proceed to demonstrate, the distance of
every point of the line from the null-point, measured in terms of
the fixed unit, is a real number; and we may assume that for each
real number there is such a point.

1. \textit{The distance of any point on the line from the
null-point is a real number.}

Let any point on the line be taken, and suppose the segment of the
line lying between this point and the null-point to contain the
unit line $\alpha$ times, with a remainder $d_1$, this remainder
to contain the tenth part of the unit line $\beta$ times, with a
remainder $d_2$, $d_2$ to contain the hundredth part of the unit
line $\gamma$ times, with a remainder $d_3$, etc.

The sequence of rational numbers thus constructed, viz.,
$\alpha,\alpha.\beta,\alpha.\beta\gamma,\ldots$ (adopting the
decimal notation) is regular; for the difference between its
$\mu$th term and each succeeding term is less than
$\displaystyle\frac{1}{10^{\mu-1}}$, a fraction which may be made
less than any assignable number by taking $\mu$ great enough; and,
by construction, this number represents the distance of the point
under consideration from the null-point.

By the convention made respecting the algebraic signs of lengths
this number will be positive when the point lies to the right of
the null-point, negative when it lies to the left.

2. \textit{Corresponding to every real number there is a point on
the line, whose distance and direction from the null-point are
indicated by the number.}

($a$) If the number is rational, we can construct the point.

For every rational number can be reduced to the form of a simple
fraction.  And if $\displaystyle\frac{\alpha}{\beta}$ denote the
given number, when thus expressed, to find the corresponding point
we have only to lay off the $\beta$th part of the unit segment
$\alpha$ times along the line, from the null-point to the right,
if $\displaystyle\frac{\alpha}{\beta}$ is positive, from the
null-point to the left, if $\displaystyle\frac{\alpha}{\beta}$ is
negative.

($b$) If the number is irrational, we usually cannot construct the
point, or even prove that it exists.

But let \textbf{a} denote the number, and
$\alpha_1,\alpha_2,\ldots,\alpha_n,\ldots$ any regular sequence of
rationals which defines it, so that $\alpha_n$ will approach
\textbf{a} as limit when $n$ is indefinitely increased.

Then, by ($a$), there is a sequence of points on the line
corresponding to this sequence of rationals. Call this sequence of
points $A_1, A_2,\cdots, A_n,\cdots$. It has the property that the
length of the segment $A_nA_{n+m}$ will approach 0 as limit when
$n$ is indefinitely increased.

When $\alpha_n$ is made to run through the sequence of values
$\alpha_1,\alpha_2,\ldots$, the corresponding point $A_n$ will run
through the sequence of positions $A_1, A_2,\cdots$. And we
\textit{assume} that just as there is in the real system a
definite number \textbf{a} which $\alpha_n$ is approaching as a
limit, so also is there on the line a definite point \textbf{A}
which $A_n$ approaches as limit. It is this point \textbf{A} which
we make correspond to \textbf{a}.

Of course there are infinitely many regular sequences of rationals
$\alpha_1,\alpha_2,\ldots$ defining \textbf{a}, and as many
sequences of corresponding points $A_1, A_2,\cdots$. We assume that
the limit point \textbf{A} is the same for all these sequences.

\addcontentsline{toc}{section}{\numberline{}The continuous
variable}

\textbf{41. The Continuous Variable.} The relation of one-to-one
correspondence between the system of real numbers and the points
of a line is of great importance both to geometry and to algebra.
It enables us, on the one hand, to express geometrical relations
numerically, on the other, to picture complicated numerical
relations geometrically. In particular, algebra is indebted to it
for the very useful notion of the continuous variable.

One of our most familiar intuitions is that of continuous motion.

\begin{figure*}[htbp]
\centering \includegraphics[scale=0.75]{images/figa.eps}\\
\end{figure*}

Suppose the point $P$ to be moving continuously from $A$ to $B$
along the line $OAB$; and let \textbf{a}, \textbf{b}, and
\textbf{x} denote the lengths of the segments $OA$, $OB$, and $OP$
respectively, $O$ being the null-point.

It will then follow from our assumption that the segment $AB$
contains a point for every number between \textbf{a} and
\textbf{b}, that as $P$ moves continuously from $A$ to $B$,
\textbf{x} may be regarded as increasing from the value \textbf{a}
to the value \textbf{b} through all intermediate values. To
indicate this we call \textbf{x} a \textit{continuous variable}.

\addcontentsline{toc}{section}{\numberline{}Correspondence between
the complex number-system and the points of a plane}

\textbf{42. Correspondence between the Complex Number-System and
the Points of a Plane.} The entire system of complex numbers may
be represented by the points of a plane, as follows:

In the plane let two right lines $X'OX$ and $Y'OY$ be drawn
intersecting at right angles at the point $O$.

\begin{figure}[htbp]
\centering \includegraphics[scale=0.75]{images/fig1.eps}\\
\textsc{Fig. 1.}
\end{figure}

Make $X'OX$ the ``axis'' of real numbers, using its points to
represent real numbers, after the manner described in \S~40, and
make $Y'OY$ the axis of pure imaginaries, representing $ib$ by the
point of $OY$ whose distance from $O$ is $b$ when $b$ is positive,
and by the corresponding point of $OY'$ when $b$ is negative.

The point taken to represent the complex number $a+ib$ is $P$,
constructed by drawing through $A$ and $B$, the points which
represent $a$ and $ib$, parallels to $Y'OY$ and $X'OX$,
respectively.

The correspondence between the complex numbers and the points of
the plane is a one-to-one correspondence. To every point of the
plane there is a complex number corresponding, and but one, while
to each number there corresponds a single point of the
plane.\footnote{A reality has thus been found to correspond to the
hitherto uninterpreted symbol $a+ib$. But this reality has no
connection with the reality which gave rise to arithmetic, the
number of things in a group of distinct things, and does not at
all lessen the purely symbolic character of $a+ib$ when regarded
from the standpoint of that reality, the standpoint which must be
taken in a purely arithmetical study of the origin and nature of
the number concept.

The connection between the numbers $a+ib$ and the points of a
plane is purely artificial. The tangible geometrical pictures of
the relations among complex numbers to which it leads are
nevertheless a valuable aid in the study of these relations.}

\addcontentsline{toc}{section}{\numberline{}The complex variable}

If the point $P$ be made to move along any curve in its plane, the
corresponding number $x$ may be regarded as changing through a
continuous system of complex values, and is called a
\emph{continuous complex variable}. (Compare \S~41.)

\addcontentsline{toc}{section}{\numberline{}Definitions of modulus
and argument of a complex number and of sine, cosine, and circular
measure of an angle}

\textbf{43. Modulus.} The length of the line $OP$ (Fig.~1),
\textit{i.~e.}\ $\sqrt{a^2+b^2}$, is called the \emph{modulus} of
$a+ib$. Let it be represented by $\rho$.

\textbf{44. Argument.} The angle $XOP$ made by $OP$ with the
positive half of the axis of real numbers is called the
\emph{angle} of $a+ib$, or its \emph{argument}. Let its numerical
measure be represented by $\theta$.

The angle is always to be measured ``counter-clockwise'' from the
positive half of the axis of real numbers to the modulus line.

\textbf{45. Sine.} The ratio of $PA$, the perpendicular from $P$
to the axis of real numbers, to $OP$, \textit{i.~e.}
$\frac{b}{\rho}$, is called the \emph{sine} of $\theta$, written
$\sin\theta$.

$\sin\theta$ is by this definition positive when $P$ lies above
the axis of real numbers, negative when $P$ lies below this line.

\textbf{46. Cosine.} The ratio of $PB$, the perpendicular from $P$
to the axis of imaginaries, to $OP$, \textit{i.~e.}\
$\frac{a}{\rho}$, is called the \emph{cosine} of $theta$, written
$\cos\theta$.

$\cos\theta$ is positive or negative according as $P$ lies to the
right or the left of the axis of imaginaries.

\addcontentsline{toc}{section}{\numberline{}Demonstration that $a
+ ib = \rho (\cos \theta + i \sin \theta) = \rho e^{i\theta}$}

\textbf{47. Theorem.} \emph{The expression of $a+ib$ in terms of
its modulus and angle is $\rho(\cos\theta+i\sin\theta)$.}

\begin{flalign*}
&\text{\indent For by \S~46 }&
  \frac{a}{\rho} &= \cos\theta, \therefore a = \rho\cos\theta;  &&
\\
&\text{and by \S~45, }&
  \frac{b}{\rho} &= \sin\theta, \therefore b = \rho\sin\theta.  &&
\\
&\text{\indent Therefore }&
  a+ib &= \rho(\cos\theta+i\sin\theta).  &&
\end{flalign*}

The factor $\cos\theta + i\sin\theta$ has the same sort of
geometrical meaning as the algebraic signs $+$ and $-$, which are
indeed but particular cases of it: it indicates the
\emph{direction} of the point which represents the number from the
null-point.

It is the other factor, the modulus $\rho$, the distance from the
null-point of the point which corresponds to the number, which
indicates the ``absolute value'' of the number, and may represent
it when compared numerically with other numbers (\S~37),---that
one of two numbers being numerically the greater whose
corresponding point is the more distant from the null-point.

\addcontentsline{toc}{section}{\numberline{}Construction of the
points which represent the sum, difference, product, and quotient
of two complex numbers}

\textbf{48. Problem I.} \textit{Given the points $P$ and $P'$,
representing $a + ib$ and $a' + ib'$ respectively; required the
point representing $a + a' + i(b + b')$.}

The point required is $P''$, the intersection of the parallel to
$OP$ through $P'$ with the parallel to $OP'$ through $P$.

For completing the construction indicated by the figure, we have
$OD' = PE = DD''$, and therefore $OD'' = OD + OD'$; and similarly
$P''D'' = PD + P'D'$.

\textsc{Cor.}~I.  To get the point corresponding to $a-a' +
i(b-b')$, produce $OP'$ to $P'''$, making $OP''' = OP'$, and
complete the parallelogram $OP$, $OP'''$.

\begin{figure}[htbp]
\centering \includegraphics[scale=0.5]{images/fig2.eps}\\
\textsc{Fig. 2.}
\end{figure}

\textsc{Cor.}~II. \textit{The modulus of the sum or difference of
two complex numbers is less than (at greatest equal to) the sum of
their moduli.}

For $OP''$ is less than $OP + PP''$ and, therefore, than $OP +
OP$, unless $O$, $P$, $P'$ are in the same straight line, when
$OP'' = OP + OP'$. Similarly, $PP'$, which is equal to the modulus
of the difference of the numbers represented by $P$ and $P'$, is
less than, at greatest equal to, $OP + OP'$.

\textbf{49. Problem II.} \textit{Given $P$ and $P'$, representing
$a+ib$ and $a'+ib'$ respectively; required the point representing
$(a+ib)(a'+ib')$.}

\[
\begin{array}{rlr}
\text{\indent Let } \quad  a+ib &= \rho(\cos\theta + i\sin\theta),
&\S~47\\
\text{and  } \quad a'+ib' &= \rho'(\cos\theta' + i\sin\theta');\\
\text{then }\quad (a+ib)&(a'+ib')\\
&= \rho\rho'(\cos\theta+i\sin\theta) (\cos\theta'+i\sin\theta')  \\
&= \rho\rho'[(\cos\theta\cos\theta' - \sin\theta\sin\theta')  \\
&\mspace{80mu} +i(\sin\theta\cos\theta' +
\cos\theta\sin\theta')].\\

\text{\indent But } \quad \cos\theta\cos\theta' &
-\sin\theta\sin\theta' = \cos(\theta+\theta'),\footnotemark[1] \\
\text{and } \quad \sin\theta\cos\theta' &+ \cos\theta\sin\theta' =
\sin(\theta+\theta').\footnotemark[1]
\end{array}
\]

\footnotetext[1]{For the demonstration of these, the so-called addition theorems of
  trigonometry, see Wells' Trigonometry, \S~65, or any other text-book
  of trigonometry.}

Therefore $(a+ib)(a'+ib') =
\rho\rho'[\cos(\theta+\theta')+i\sin(\theta+\theta')]$; or,
\emph{The modulus of the product of two complex numbers is the
product of their moduli, its argument the sum of their arguments}.

The required construction is, therefore, made by drawing through
$O$ a line making an angle $\theta+\theta'$ with $OX$, and laying
off on this line the length $\rho\rho'$.

\textsc{Cor.}~I. Similarly the product of $n$ numbers having
moduli $\rho$, $\rho'$, $\rho''$, $\dotsb$ $\rho^{(n)}$
respectively, and arguments $\theta$, $\theta'$, $\theta''$,
$\dotsc$ $theta^{(n)}$, is the number
\[
\begin{split}
  \rho\rho'\rho''\dotsm\rho^{(n)}
  [\cos(\theta+\theta'+\theta''+\dotsb\theta^{(n)})  \\
+ i\sin(\theta+\theta'+\theta''+\dotsb\theta^{(n)})].
\end{split}
\]

In particular, therefore, by supposing the $n$ numbers equal, we
may infer the theorem
\[
  [ \rho(\cos\theta + i\sin\theta) ]^n
= \rho^n (\cos n\theta + i\sin n\theta),
\]
which is known as Demoivre's Theorem.

\textsc{Cor.}~II\@. From the definition of division and the
preceding demonstration it follows that
\[
  \frac{a+ib}{a'+ib'}
= \frac{\rho}{\rho'} [\cos(\theta-\theta') +
i\sin(\theta-\theta')];
\]
the construction for the point representing $\dfrac{a+ib}{a'+ib'}$
is, therefore, obvious.

\textbf{50. Circular Measure of Angle.} Let a circle of unit
radius be constructed with the vertex of any angle for centre. The
length of the arc of this circle which is intercepted between the
legs of the angle is called the \emph{circular measure} of the
angle.

\textbf{51. Theorem.} \textit{Any complex number may be expressed
in the form $\rho e^{i\theta}$; where $\rho$ is its modulus and
$\theta$ the circular measure of its angle.}

It has already been proven that a complex number may be written in
the form $\rho(\cos\theta+i\sin\theta)$, where $\rho$ and $\theta$
have the meanings just given them. The theorem will be
demonstrated, therefore, when it shall have been shown that
\[
e^{i\theta}=\cos\theta+i\sin\theta.
\]

If $n$ be any positive integer, we have, by \S~36 and the binomial
theorem,
\begin{align*}
\left( 1 + \frac{i\theta}{n} \right)^n &= 1 + n\frac{i\theta}{n} +
\frac{n(n-1)}{2!}\frac{(i\theta)^2}{n^2}
\\
&\phantom{= 1 + n\frac{i\theta}{n}} +
\frac{n(n-1)(n-2)}{3!}\frac{(i\theta)^3}{n^3} + \dotsb
\\
&= 1 + i\theta + \frac{1-\frac{1}{n}}{2!}(i\theta)^2
\\
&\phantom{= 1 + i\theta} + \frac{\left(1-\frac{1}{n}\right)
        \left(1-\frac{2}{n}\right)}{3!} (i\theta)^3 + \dotsb.
\end{align*}

Let $n$ be indefinitely increased; the limit of the right side of
this equation will be the same as that of the left.

But the limit of the right side is

\[ 1+i\theta+\frac{(i\theta)^2}{2!}+\frac{(i\theta)^3}{3!}+\ldots; \; \text{i.~e.} \; e^{i\theta}.\footnote{This use of the symbol $\displaystyle e^{i\theta}$ will be fully justified in \S~73.}\]

Therefore $\displaystyle e^{i\theta}$ is the limit of
$\displaystyle\left(1+\frac{i\theta}{n}\right)^n$ as $n$
approaches $\infty$.

To construct the point representing
$\displaystyle\left(1+\frac{i\theta}{n}\right)^n$:

\begin{figure}[htbp]
\centering \includegraphics[scale=0.5]{images/fig3.eps}\\
\textsc{Fig. 3.}
\end{figure}

On the axis of real numbers lay off $OA=1$.

Draw $AP$ equal to $\theta$ and parallel to $OB$, and divide it
into $n$ equal parts. Let $AA_1$ be one of these parts. Then $A_1$
is the point $\displaystyle 1+\frac{i\theta}{n}$.

Through $A_1$ draw $A_1A_2$ at right angles to $OA_1$ and
construct the triangle $OA_1A_2$ similar to $OAA_1$.

$A_2$ is then the point
$\displaystyle\left(1+\frac{i\theta}{n}\right)^2$.

\begin{align*}
\text{For} \qquad & AOA_2=2AOA_1;\\
\text{and since} \quad & OA_2:OA_1::OA_1:OA, \; \text{and} \; OA=1,\\
\text{the length} \quad & OA_2= \; \text{the square of length} \; OA_1. \qquad (see \S~49)\\
\end{align*}

In like manner construct $A_3$ to represent
$\displaystyle\left(1+\frac{i\theta}{n}\right)^3$, $A_4$ for
$\displaystyle\left(1+\frac{i\theta}{n}\right)^4, \;\\
 \cdots A_n \; \text{for} \; \left(1+\frac{i\theta}{n}\right)^n$.

Let $n$ be indefinitely increased. The broken line $AA_1A_2 \cdots
A_n$ will approach as limit an arc of length $\theta$ of the
circle of radius $OA$ and, therefore, its extremity, $A_n$, will
approach as limit the point representing $\cos\theta+i\sin\theta$
(\S~47).

Therefore the limit of $\displaystyle\left(1 +
\frac{i\theta}{n}\right)^n$ as $n$ is indefinitely increased is
$\cos\theta + i\sin\theta$.

But this same limit has already been proved to be $e^{i\theta}$.

\[\text{Hence } \qquad e^{i\theta} = \cos\theta + i\sin\theta.\footnote{Dr. F. Franklin, American Journal of Mathematics, Vol. VII,
p.~376.    Also M\"obius, Collected Works, Vol. IV, p.~726.}\]

\chapter{THE FUNDAMENTAL THEOREM OF ALGEBRA.}


\addcontentsline{toc}{section}{\numberline{}Definitions of the
algebraic equation and its roots}

\textbf{52. The General Theorem.} If

\[w = a_0z^n + a_1z^{n-1} + a_2z^{n-2} + \cdots + a_{n-1}z + a_n,\]

where $n$ is a positive integer, and $a_0, a_1, \ldots, a_n$ any
numbers, real or complex, independent of $z$, to each value of $z$
corresponds a single value of $w$.

We proceed to demonstrate that conversely to each value of $w$
corresponds a set of $n$ values of $z$, \textit{i.~e.} that there
are $n$ numbers which, substituted for $z$ in the polynomial
$\displaystyle a_0z^n + a_1z^{n-1} +\cdots + a_n$, will give this
polynomial any value, $w_0$, which may be assigned.

It will be sufficient to prove that there are $n$ values of $z$
which render $\displaystyle a_0z^n + a_1z^{n-1} +\cdots + a_n$
equal to 0, inasmuch as from this it would immediately follow that
the polynomial takes any other value, $w_0$, for $n$ values of
$z$; viz., for the values which render the polynomial of the same
degree, $\displaystyle a_0z^n + a_1z^{n-1} +\cdots + (a_n - w_0)$,
equal to 0.

\textbf{53. Root of an Equation.} A value of $z$ for which
$\displaystyle a_0z^n + a_1z^{n-1} +\cdots + a_n$ is 0 is called a
root of this polynomial, or more commonly a root of the
\textit{algebraic equation}

\[ a_0z^n + a_1z^{n-1} +\cdots + a_n = 0.\]

\textbf{54. Theorem.} \textit{Every algebraic equation has a
root.}

Given $w=a_0z^n+a_1z^{n-1}+\dotsb+a_n$.

Let $\lvert w\rvert$ denote the modulus of $w$. We shall assume,
though this can be proved, that among the values of $\lvert
w\rvert$ corresponding to all possible values of $z$ there is a
\emph{least} value, and that this least value corresponds to a
finite value of $z$.


\begin{figure}[htbp]
\centering \includegraphics[scale=0.5]{images/fig4.eps}\\
\textsc{Fig. 4.}
\end{figure}

Let $\lvert w_0 \rvert$ denote this least value of $\lvert w
\rvert$, and $z_0$ the value of $z$ to which it corresponds. Then
$\lvert w_0 \rvert = 0$.

For if not, $w_0$ will be represented in the plane of complex
numbers by some point $P$ distinct from the null-point $O$.

Through $P$ draw a circle having its centre in the null-point $O$.
Then, by the hypothesis made, no value can be given $z$ which will
bring the corresponding $w$-point within this circle.

But the $w$-point \emph{can be brought within this circle}.

For, $z_0$ and $w_0$ being the values of $z$ and $w$ which
correspond to $P$, change $z$ by adding to $z_0$ a small increment
$\delta$, and let $\Delta$ represent the consequent change in $w$.
Then $\Delta$ is defined by the equation
\[
\begin{split}
(w_0 &+ \Delta) = a_0(z_0+\delta)^n + a_1(z_0+\delta)^{n-1}  \\
     &+ a_2(z_0+\delta)^{n-2} + \dotsb + a_{n-1}(z_0+\delta) + a_n.
\end{split}
\]

On applying the binominal theorem and arranging the terms with
reference to powers of $\delta$, the right member of this equation
becomes
\[
\begin{split}
a_0z_0^n
&+ a_1z_0^{n-1} + \dotsb + a_{n-1}z_0 + a_n  \\
&+ [na_0z_0^{n-1} + (n-1)a_1z_0^{n-2} + \dotsb + a_{n-1}]\delta  \\
&+ \text{ terms involving $\delta^2$, $\delta^3$, etc.}
\end{split}
\]

\begin{flalign*}
&\text{\indent But }&
  w_0 &= a_0z_0^n + a_1z_0^{n-1} + \dotsb + a_{n-1}z_0 + a_n.  &&
\\
&&  \therefore \Delta &= [na_0z_0^{n-1} + (n-1)a_1z_0^{n-2} +
\dotsb + a_{n-1}]\delta  &&
\\
&&  &\quad + \text{ terms involving $\delta^2$, $\delta^3$, etc.}
&&
\end{flalign*}

Let $\rho'(\cos\theta'+i\sin\theta')$ be the complex number
\[na_0z_0^{n-1}+(n-1)a_1z_0^{n-2}+ \dotsb +a_{n-1},\]
expressed in terms of its modulus and angle, and
\[\rho(\cos\theta+i\sin\theta)\]
the corresponding expression for $\delta$. Then
\begin{align*}
  \Delta &= \rho'(\cos\theta'+i\sin\theta') \times
            \rho (\cos\theta +i\sin\theta )
\\
&\phantom{= \rho'(\cos\theta'} + \text{ terms involving $\rho^2$,
$\rho^3$, etc.}
\\
&= \rho\rho'[\cos(\theta+\theta') + i\sin(\theta+\theta')]
\\
&\phantom{= \rho'(\cos\theta'} + \text{ terms involving $\rho^2$,
$\rho^3$, etc. \qquad \S~49.}
\end{align*}

The point which represents
$\rho\rho'[\cos(\theta+\theta')+i\sin(\theta+\theta')]$ for any
particular value of $\rho$ can be made to describe a circle of
radius $\rho\rho'$ about the null-point by causing $\theta$ to
increase continuously from 0 to 4 right angles.

In the same circumstances the point representing
\[w_0+\rho\rho'[\cos(\theta+\theta')+i\sin(\theta+\theta')]\]
will describe an equal circle about the point $P$ and, therefore,
come within the circle $OP$.

But by taking $\rho$ small enough, $\Delta$ may be made to differ
as little as we please from $\rho\rho'[\cos(\theta+\theta') +
i\sin(\theta+\theta')]$,\footnotemark[1] and, therefore, the curve
traced out by $P'$ (which represents $w_0+\Delta$, as $\theta$
runs through its cycle of values), to differ as little as we
please from the circle of centre $P$ and radius $\rho\rho'$.

Therefore by assigning proper values to $\rho$ and $\theta$, the
$w$-point ($P'$) may be brought within the circle $OP$.

\footnotetext[1]{ In the series $A\rho+B\rho^2+C\rho^3+$ etc., the
ratio of all the terms following the first to the first,
\textit{i.~e.}
\[
\frac{B\rho^2+c\rho^3+\text{ etc.}}{A\rho},
= \rho\times \frac{B+C\rho+\text{ etc.}}{A};
\]
which by taking $\rho$ small enough may evidently be made as small
as we please.}

The $w$-point nearest the null-point must therefore be the
null-point itself.\footnotemark[2]

\footnotetext[2]{ In the above demonstration it is assumed that
the coefficient of $\delta$ is not 0. If it be 0, let $A\delta^r$
denote the first term of $\Delta$ which is not 0. If
$A=\rho''(\cos\theta''+i\sin\theta'')$, we then have
\[
\Delta = \rho''\rho^r[ \cos(r\theta+\theta'')
                       + i\sin(r\theta+\theta'') ]
         + \text{ terms in }\theta^{r+1}, \dots b,
\]
from which the same conclusion follows as above.}


\textbf{55. Theorem.} \textit{If $\alpha$ be a root of
$a_0z^n+a_1z^{n-1}+\dotsb+a_n$, this polynomial is divisible by
$z-a$.}

For divide $a_0z^n+a_1z^{n-1}+\dotsb+a_n$ by $z-a$, continuing the
division until $z$ disappears from the remainder, and call this
remainder $R$, the quotient $Q$, and, for convenience, the
polynomial $f(z)$.

Then we have immediately
\[f(z)=(z-\alpha)Q+R,\]
holding for all values of $z$.

Let $z$ take the value $\alpha$; then $f(z)$ vanishes, as also the
product $(z-\alpha)Q$.

Therefore when $z=\alpha$, $R=0$, and being independent of $z$ it
is hence always 0.

\addcontentsline{toc}{section}{\numberline{}Demonstration that an
algebraic equation of the $n$th degree has $n$ roots}

\textbf{56. The Fundamental Theorem.} \textit{The number of the
roots of the polynomial $a_0z^n+a_1z^{n-1}+\dotsb+a_n$ is $n$.}

For, by \S~54, it has at least one root; call this $\alpha$; then,
by \S~55, it is divisible by $z-\alpha$, the degree of the
quotient being $n-1$.

Therefore we have
\[
  a_0z^n + a_1z^{n-1} + \dotsb + a_n
= (z-\alpha) (a_0z^{n-1} + b_1z^{n-2} + \dotsb + b_{n-1}).
\]

Again, by \S~54, the polynomial
$a_0z^{n-1}+b_1z^{n-2}+\dotsb+b_{n-1}$ has a root; call this
$\beta$, and dividing as before, we have
\[
  a_0z^n + a_1z^{n-1} + \dotsb + a_n
= (z-\alpha)(z-\beta)(\alpha_1z^{n-2} + c_1z^{n-3} + \dotsb
+c_{n-2}).
\]

Since the degree of the quotient is lowered by 1 by each
repetition of this process, $n-1$ repetitions reduce it to the
first degree, or we have

\[
a_0z^n + a_1z^{n-1} + \cdots + a_n =
a_0(z-\alpha)(z-\beta)(z-\gamma) \cdots (z-\nu),
\]

\noindent a product of $n$ factors, each of the first degree.

Now a product vanishes when one of its factors vanishes (\S~36, 3,
Cor.), and the factor $z-\alpha$ vanishes when $z=\alpha$,
$z-\beta$ when $z=\beta, \ldots , z-\nu$ when $z=\nu$. Therefore
$a_0z^n + a_0z^{n-1} + \cdots + a_n$ vanishes for the $n$ values,
$\alpha, \beta, \gamma, \cdots \nu$, of $z$.

Furthermore, a product cannot vanish unless one of its factors
vanishes (\S~36, 4, Cor.), and not one of the factors $z-\alpha,
z-\beta, \ldots , z-\nu$, vanishes unless $z$ equals one of the
numbers $\alpha, \beta, \cdots \nu$.

The polynomial has therefore $n$ and but $n$ roots.

The theorem that the number of roots of an algebraic equation is
the same as its degree is called the fundamental theorem of
algebra.

\chapter{INFINITE SERIES.}


\textbf{57.   Definition.}    Any operation which is the limit of
additions indefinitely repeated produces an infinite series.   We
are to determine the conditions which an infinite  series must
fulfil to represent a number.

If the terms of a series are real numbers, it is called a
\textit{real series}; if complex, a \textit{complex series.}


\section{REAL SERIES.}

\addcontentsline{toc}{section}{\numberline{}Definitions of sum,
convergence, and divergence}

 \textbf{58.  Sum.    Convergence. Divergence.}   An infinite series

\[
a_1 + a_2 + a_3 + \cdots +a_n + \cdots
\]
\noindent represents a number or not, according as the sequence

\[
s_1, s_2, s_3, \ldots s_m, s_{m+1}, \ldots s_{m+n}, \ldots ,
\]

\[
\text{where } \qquad        s_1=a_1, s_2=a_1 + a_2, \cdots , s_i=a_1 + a_2 + \cdots a_i,
\]
is \textit{regular} or not.

If $s_{1}, s_{2}, \cdots,$ be a regular sequence, the number which
it defines, or $\lim_{n \doteq \infty}(s_{n})$, is called the
\textit{sum} of the infinite series

\[a_{1}+a_{2}+a_{3}+\cdots+a_{n}+\cdots,\]

\noindent and the series is said to be \textit{convergent}.

If $s_{1}, s_{2},$ be not a regular sequence, $s_{n}$ either
transcends any finite value whatsoever, as $n$ is indefinitely
increased, or while remaining finite becomes altogether
indeterminate. The infinite series then has no sum, and is said to
be \textit{divergent}.

The series $1+1+1+\cdots$ and $1-1+1-1+\cdots$ are examples of
these two classes of divergent series.

A divergent series cannot represent a number.

\addcontentsline{toc}{section}{\numberline{}General test of
convergence}

\textbf{59. General Test of Convergence.} From these definitions
and \S~27 it immediately follows that:

\textit{The infinite series $a_{1}+a_{2}+\cdots+a_{m}+\cdots$ is
convergent when $m$ may be so taken that the differences
$s_{m+n}-s_{m}$ are numerically less than any assignable number
$\delta$ for all values of $n$,} where $s_{m}$ and $s_{m+n}$ are
the sum of the first $m$ and of the first $m+n$ terms of the
series respectively.

\textit{If these conditions be not fulfilled, the series is
divergent.}

The limit of the $n$th term of a convergent series is 0; for the
condition of convergence requires that by taking $m$ great enough,
$s_{m+1}-s_{m}$, \textit{i.~e.} $a_{m+1},$ may be found less than
any assignable number. But it is not to be assumed conversely that
a series is convergent, if the limit of its $n$th term is 0; other
conditions have also to be fulfilled, $s_{m+n}-s_{m}$ must be less
than $\delta$ for \textit{all} values of $n$.

Thus the limit of the $n$th term of the series $\displaystyle
1+\frac{1}{2}+\frac{1}{3}+\cdots$ is 0; but, as will presently be
shown, this is a divergent series.

\addcontentsline{toc}{section}{\numberline{}Absolute and
conditional convergence}

\textbf{60. Absolute Convergence.} It is important to distinguish
between convergent series which remain convergent when all the
terms are given the same algebraic signs and convergent series
which become divergent on this change of signs. Series of the
first class are said to be \textit{absolutely} convergent; those
of the second class, only \textit{conditionally} convergent.

\textit{Absolutely convergent series have the character of
ordinary sums; i.~e.\ the order of the terms may be changed without
altering the sum of the series.}

For consider the series  $a_1 + a_{2} + a_{3} +\cdots$ supposed to
be absolutely convergent and to have the sum $S$, when the terms
are in the normal order of the indices.

It is immediately obvious that no change can be made in the sum of
the series by interchanging terms with finite indices; for $n$ may
be taken greater than the index of any of the interchanged terms.
Then $S_{n}$ has not been affected by the change, since it is a
finite sum and it is immaterial in what order the terms of a
finite sum are added; and as for the rest of the series, no change
has been made in the order of its terms.

But $a_{1} + a_{2} + a_{3} +\cdots$ may be separated into a number
of infinite series, as, for instance, into the series $a_1 + a_3 +
a_{5} +\cdots$ and $a_{2} + a_{4} + a_{6} +\cdots$, and these
series summed separately. Let it be separated into $l$ such
series, the sums of which---they must all be absolutely
convergent, as being parts of an absolutely convergent
series---are $S^{(1)}, S^{(2)},\cdots S^{(l)}$, respectively; it
is to be proven that
\[
S=S^{(1)}+S^{(2)}+S^{(3)}+\cdots+S^{(l)}.
\]

Let $S_m^{(1)},S_m^{(2)},\cdots $ be the sums of the first $m$
terms of the series $S^{(1)}, S^{(2)}, \cdots $, respectively.

Then, by the hypothesis that the series $a_{1} + a_{2}+\cdots $ is
absolutely convergent, $m$ may be taken so large that the sum
\[
{S_{m+n}}^{(1)}+{S_{m+n}}^{(2)}+\cdots+{S_{m+n}}^{(l)}
\]
shall differ from $S$ by less than any assignable number $\delta$
for all values of $n$; therefore the limit of this sum is $S$.

But again, $n$ may be so taken that ${S_{m+n}}^{(1)}$ shall differ
from $S^{(1)}$ by less than $\displaystyle \frac{\delta}{l}$,
${S_{m+n}}^{(2)}$ from $S^{(2)}$ by less than
$\displaystyle\frac{\delta}{l}, \ldots$; and therefore the sum
${S_{m+n}}^{(1)}+{S_{m+n}}^{(2)}+\cdots+{S_{m+n}}^{(l)}$ from
$S^{(1)}+S^{(2)}+\cdots+S^{(l)}$ by less than
$\displaystyle\left(\frac{\delta}{l}\right)l$; \textit{i.~e.} by
less than $\delta$.   Hence the limit of this sum is
$S^{(1)}+S^{(2)}+\cdots+S^{(l)}$.

Therefore $S$ and $S^{(1)}+S^{(2)}+\cdots+S^{(l)}$ are limits of
the same finite sum and hence equal.    (We omit the proof for the
case $l$ infinite.)

\textbf{61. Conditional Convergence.} On the other hand,
\textit{the terms of a conditionally convergent series can be so
arranged that the sum of the series may take any real value
whatsoever.}

In a conditionally convergent series the positive and the negative
terms each constitute a divergent series having 0 for the limit of
its last term.

If, therefore, $C$ be any positive number, and $S_{n}$ be
constructed by first adding positive terms (beginning with the
first) until their sum is greater than $C$, to these negative
terms until their sum is again less than $C$, then positive terms
till the sum is again greater than $C$, and so on indefinitely;
the limit of $S_{n}$, as $n$ is indefinitely increased, is $C$.

\addcontentsline{toc}{section}{\numberline{}Special tests of
convergence}

\textbf{62. Special Tests of Convergence.} 1. \textit{If each of
the terms of a series $a_{1} + a_{2} + \cdots$ be numerically less
than (at greatest equal to) the corresponding term of an
absolutely convergent series, or if the ratio of each term of
$a_{1} + a_{2} + \cdots$ to the corresponding term of an
absolutely convergent series never exceed some finite number $C$,
the series $a_{1} + a_{2} + \cdots $ is absolutely convergent.}

\textit {If, on the other hand, each term of $a_{1} + a_{2} +
\cdots  $ be numerically greater than (at the lowest equal to) the
corresponding term of a divergent series, or if the ratio of each
term of $a_{1} + a_{2} + \cdots $ to the corresponding term of a
divergent series be never numerically less than some finite number
$C'$, different from 0, the series $a_{1} + a_{2} + \cdots$ is
divergent.}

2. \textit{The series $a_{1} - a_{2} + a_{3} - a_{4} + \cdots $,
the terms of which are alternately positive and negative, is
convergent, if after some term $a_{i}$ each term be numerically
less or, at least, not greater than the term which immediately
precedes it, and the limit of $a_{n}$, as $n$ is indefinitely
increased, be 0.}

For here

\[
s_{m+n} - s_{m} = (-1)^{m}[a_{m+1} - a_{m+2} + \cdots
(-1)^{n-1}a_{m+n}]
\]

The expression within brackets may be written in either of the
forms

\begin{align*}
&(a_{m+1} - a_{m+2}) + (a_{m+3} - a_{m+4}) + \cdots   \tag{1}\\
\text{or} \qquad & a_{m+1} - (a_{m+2} - a_{m+3}) - \cdots \tag{2}
\end{align*}

It is therefore positive, (1), and less than $a_{m+1}$, (2); and
hence by taking $m$ large enough, may be made numerically less
than any assignable number whatsoever.

The series $\displaystyle
1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$ is, by this theorem,
convergent.

3. \textit{The series $\displaystyle
1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$ is divergent.}

For the first $2^{\lambda}$ terms after the first may be written

\begin{align*}
\frac{1}{2}+\left(\frac{1}{2+1}+\frac{1}{2+2}\right) &
+\left(\frac{1}{2^2+1}+\frac{1}{2^2+2}+\frac{1}{2^2+3}+\frac{1}{2^2+2^2}\right)+\cdots \\
&
+\left(\frac{1}{2^{\lambda-1}+1}+\frac{1}{2^{\lambda-1}+2}+\cdots
\frac{1}{2^{\lambda-1}+2^{\lambda-1}}\right),
\end{align*}

\noindent where, obviously, each of the expressions within
parentheses is greater than $\displaystyle \frac{1}{2}$.

The sum of the first $2^{\lambda}$ terms after the first is
therefore greater than $\displaystyle\frac{\lambda}{2}$, and may
be made to exceed any finite quantity whatsoever by taking
$\lambda$ great enough.

This series is commonly called the harmonic series.

By a similar method of proof it may be shown that the series
$\displaystyle 1+\frac{1}{2^p}+\frac{1}{3^p}+\cdots$ is convergent
if $p>1$.

\[\text{Here,} \qquad \frac{1}{2^p}+\frac{1}{3^p}<\frac{2}{2^p},
\;
\frac{1}{4^p}+\frac{1}{5^p}+\frac{1}{6^p}+\frac{1}{7^p}<\frac{4}{4^p},
\; \textit{i.~e.} \; <\left(\frac{2}{2^p}\right)^2 \cdots,
\]
and the sum of the series is, therefore, less than that of the
decreasing geometric series $\displaystyle
1+\frac{2}{2^p}+\left(\frac{2}{2^p}\right)^2+\cdots$.

The series $\displaystyle 1+\frac{1}{2^p}+\frac{1}{3^p}+ \cdots $
is divergent if $p<1$, the terms being then greater than the
corresponding terms of
\[1+\frac{1}{2}+\frac{1}{3}+ \cdots.
\]

4. \textit{The series $a_1+a_2+a_3+\cdots$ is absolutely
convergent if after some term of finite index, $a_i$, the ratio of
each term to that which immediately precedes it be numerically
less than 1 and, as the index of the term is indefinitely
increased, approach a limit which is less than $1$; but divergent,
if this ratio and its limit be greater than $1$.}

For---to consider the first hypothesis---suppose that after the
term $a_i$ this ratio is always less than $\alpha$, where $\alpha$
denotes a certain positive number less than 1.

\begin{align*}
\text{Then,} \qquad  \frac{a_{i+1}}{a_i} & \leqq \alpha, \; \therefore \; a_{i+1}\leqq a_i\alpha;\\
\frac{a_{i+2}}{a_{i+1}} & \leqq \alpha, \; \therefore \; a_{i+2}\leqq a_{i+1}\alpha\leqq a_i\alpha^2.\\
\cdot \qquad & \cdot \qquad \cdot \qquad \cdot \qquad \cdot \qquad \cdot \qquad \cdot\\
\frac{a_{i+k}}{a_{i+(k-1)}} & \leqq \alpha, \; \therefore \; a_{i+k}\leqq a_{i+(k-1)}\alpha\leqq \cdots \leqq a_i\alpha^k.\\
\cdot \qquad & \cdot \qquad \cdot \qquad \cdot \qquad \cdot \qquad \cdot \qquad \cdot \\
\end{align*}

The given series is therefore $\leqq$
\[s_i+a_i[\alpha+\alpha^2+\alpha^3+\cdots \alpha^k+\cdots].\]

And this is an absolutely convergent series.

\begin{align*}
\text{For} \qquad \alpha+\alpha^2+ \cdots \alpha^k+ \cdots & =\lim_{n\doteq\infty}(\alpha+\alpha^2+ \cdots +\alpha^n) \\
& =\lim_{n\doteq\infty}\left(\frac{\alpha-\alpha^{n+1}}{1-\alpha}\right)\\
& =\frac{\alpha}{1-\alpha}, \; \text {since $\alpha$ is a
fraction.}
\end{align*}

The given series is therefore absolutely convergent, \S~62, 1.

The same course of reasoning would prove that the series is
divergent when after some term $\alpha_i$ the ratio of each term
to that which precedes it is never less than some quantity,
$\alpha$, which is itself greater than 1.

When the limit of the ratio of each term of the series to the term
immediately preceding it is 1, the series is sometimes convergent,
sometimes divergent. The series considered in \S~62, 3 are
illustrations of this statement.

\addcontentsline{toc}{section}{\numberline{}Limits of convergence}

\textbf{63. Limits of Convergence.} An important application of
the theorem just demonstrated is in determining what are called
the limits of convergence of infinite series of the form
\[a_0+a_1x+a_2x^2+a_3x^3+\cdots ,\]
where $x$ is supposed variable, but the coefficients $a_0$, $a_1$,
etc., constants as in the preceding discussion. Such a series will
be convergent for very small values of $x$, if the coefficients be
all finite, as will be supposed, and generally divergent for very
great values of $x$; and by the limits of convergence of the
series are meant the values of $x$ for which it ceases to be
convergent and becomes divergent.

By the preceding theorem the series will be \textit{convergent} if
the limit of the ratio of any term to that which precedes it be
numerically less than 1; \textit{i.~e.} if
\[
\lim_{n\doteq\infty}\left(\frac{a_{n+1}x^{n+1}}{a_nx^n}\right), \;
\text{ or} \lim_{n\doteq\infty}\left(\frac{a_{n+1}}{a_n}x\right),
\; <1;
\]
that is, \textit{if $x$ be numerically } $\displaystyle
<\lim_{n\doteq\infty}\left(\frac{a_n}{a_{n+1}}\right)$; and
\textit{divergent, if $x$ be numerically} $\displaystyle
>\lim_{n\doteq\infty}\left(\frac{a_n}{a_{n+1}}\right)$.

1. Thus the infinite series
\[a^m+ma^{m-1}x+\frac{m(m-1)}{2!}a^{m-2}x^2+\cdots,\]
which is the expansion, by the binomial theorem, of $(a+x)^m$ for
other than positive integral values of $m$, is convergent for
values of $x$ numerically less than $a$, divergent for values of
$x$ numerically greater than $a$.

For in this case
\begin{align*}
\lim_{n\doteq\infty}\left(\frac{a_n}{a_{n+1}}\right) &
=\lim_{n\doteq\infty}
\left[a\times\frac{\frac{m(m-1)\cdots(m-n+1)}{(n)!}}{\frac{m(m-1)\cdots(m-n)}{(n+1)!}}\right] \\
& =\lim_{n\doteq\infty}\left(a\times\frac{n+1}{m-n}\right)\\
& =\lim_{n\doteq\infty}\left(\frac{a\left(1+\frac{1}{n}\right)}{-1+\frac{m}{n}}\right)=-a.\\
\end{align*}

2. Again, the expansion of $e^{x}$, i.~e. $\displaystyle
1+x+\frac{x^{2}}{2!}+\cdots$, is convergent for all finite values
of $x$.

\[
\text{For here} \quad
\lim_{n\doteq\infty}\left(\frac{a_{n}}{a_{n+1}}\right)=
\lim_{n\doteq\infty}\left(\frac{\frac{1}{(n)!}}{\frac{1}{(n+1)!}}\right)=
\lim_{n\doteq\infty}(n+1)=\infty.
\]

The same is true for the series which is the expansion of $a^{x}$.

\addcontentsline{toc}{section}{\numberline{}The fundamental
operations on infinite series}

\textbf{64. Operations on Infinite Series.} 1. \textit{The sum of
two convergent series, $a_{1}+a_{2}+\cdots$ and
$b_{1}+b_{2}+\cdots$, is the series
$(a_{1}+b_{1})+(a_{2}+b_{2})+\cdots$; and their difference is the
series $(a_{1}-b_{1})+(a_{2}-b_{2})+\cdots$.}

The sum of the series $a_{1}+a_{2}+\cdots$ is the number defined
by $s_{1},s_{2},\cdots$, and the sum of the series
$b_{1}+b_{2}+\cdots$ is the number defined by
$t_{1},t_{2},\cdots$, where $s_{i}=a_{1}+a_{2}+\cdots+a_{i}$ and
$t_{i}=b_{1}+b_{2}+\cdots+b_{i}$. The sum of the two series is
therefore the number defined by $s_{1}+t_{1},s_{2}+t_{2},\cdots$,
\S~29, (1).

But if $S_{i}=(a_{1}+b_{1})+(a_{2}+b_{2})+\cdots+(a_{i}+b_{i})$,
we have $S_{i}=s_{i}+t_{i}$ for all values of $i$. This is
immediately obvious for finite values of $i$, and there can be no
difference between $S_{i}$ and $s_{i}+t_{i}$ as $i$ approaches
$\infty$, since it would be a difference having 0 for its limit.

Therefore the number defined by $s_{1}+t_{1},s_{2}+t_{2},\cdots $,
is the sum of the series $(a_{1}+b_{1})+(a_{2}+b_{2})+\cdots$.

2. \textit{The product of two absolutely convergent series}
\begin{align*}
& a_{1}+a_{2}+\cdots \; \textit{and} \; b_{1}+b_{2}+\cdots \\
\textit{is the series} \quad a_{1}b_{1} & +(a_{1}b_{2}+a_{2}b_{1})+(a_{1}b_{3}+a_{2}b_{2}+a_{3}b_{1})+\cdots \\
&
+(a_{1}b_{n}+a_{2}b_{n-1}+\cdots+a_{n-1}b_{2}+a_{n}b_{1})+\cdots.
\end{align*}

Each set of terms within parentheses is to be regarded as
constituting a single term of the product; and it will be noticed
that the first of them consists of the one partial product in
which the sum of the indices is 2, the second of all in which the
sum of the indices is 3, etc.

By \S~29, (3), the product of $a_{1}+a_{2}+\cdots$ by
$b_{1}+b_{2}+\cdots$ is $\displaystyle
\lim_{n\doteq\infty}(s_{n}t_{n})$, where $s_{n}$ and $t_{n}$
represent the sums of the first $n$ terms of $a_{1}+a_{2}+\cdots$,
$b_{1}+b_{2}+\cdots$, respectively.

Suppose first that the terms of $a_{1}+a_{2}+\cdots$ and
$b_{1}+b_{2}+\cdots$ are all positive. Then if $S_{n}$ be the sum
of the first $n$ terms of $a_1b_1 + (a_1b_2 + a_2b_1) + \cdots$,
and $m$ represent $\displaystyle \frac{n}{2}$ when $n$ is even and
$\displaystyle \frac{n-1}{2}$ when $n$ is odd,

\begin{align*}
\text{evidently} \qquad s_nt_n > & S_n > s_mt_m.\\
\text{But} \qquad \lim_{n \doteq \infty}(s_nt_n) & = \lim_{n \doteq \infty}(s_mt_m).\\
\text{Therefore} \qquad \lim_{n \doteq \infty}(S_n) & = \lim_{n
\doteq \infty}(s_nt_n).
\end{align*}

If the terms of $a_1 + a_2 + \cdots$, $b_1 + b_2 + \cdots$ be not
all of the same sign, call the sums of the first $n$ terms of the
series got by making all the signs plus, $s_n'$ and $t_n'$
respectively; also $S_n'$, the sum of the first $n$ terms of the
series which is their product.

Then by the demonstration just given
\[
\lim_{n \doteq \infty}(S'_n) = \lim_{n \doteq \infty}(s'_nt'_n);
\]
but $S_n$ always differs from $s_nt_n$ by less than (at greatest
by as much as) $S'_n$ from $s'_nt'_n$; therefore, as before,
\[
\lim_{n \doteq \infty}(S_n) = \lim_{n \doteq \infty}(s_nt_n).
\]

3. The \textit{quotient} of the series $a_0 + a_1x + \cdots$ by
the series $b_0 + b_1x + \cdots$ ($b_0$ not 0) is a series of a
similar form, as $c_0 + c_1x + \cdots$, which converges when $a_0
+ a_1x + \cdots$ is absolutely convergent and $b_1x + \cdots$ is
numerically less than $b_0$.

\section{COMPLEX SERIES.}

The terms \textit{sum, convergent, divergent}, have the same
meanings in connection with complex as in connection with real
series.

\addcontentsline{toc}{section}{\numberline{}General test of
convergence}

\textbf{65. General Test of Convergence.} \textit{A  complex
series, $a_1 + a_2 + \cdots$, is convergent when the modulus of
$s_{m+n} - s_m$ may be made less than any assignable number
$\delta$ by taking $m$ great enough, and that for all values of
$n$; divergent, when this condition is not satisfied.} See \S~48,
Cor. II; \S~59.

\addcontentsline{toc}{section}{\numberline{}Absolute and
conditional convergence}

\textbf{66. Of Absolute Convergence.}   Let
\begin{align*}
& a_1 + a_2 + \cdots \; \text{ be a complex series,}\\
\text{and} \qquad & A_1 + A_2 + \cdots, \; \text{ the series of
the moduli of its terms}
\end{align*}


\textit{If the series $A_{1}+A_{2}+\cdots$, be convergent, the
series $a_{1}+a_{2}+\cdots$ will be convergent also.}

For the modulus of the sum of a set of complex numbers is less
than (at greatest equal to) the sum of their moduli (\S~48, Cor.
II). By hypothesis, $S_{m+n}-S_{m}$ is less than any assignable
number $\delta$, when $S_{m}=A_{1}+A_{2}+\cdots+A_{m}$, etc.; much
more must the modulus of $s_{m+n}-s_{m}$ be less than $\delta$.

The converse of this theorem is not necessarily true; and a
convergent series, $a_{1}+a_{2}+\cdots$, is said to be
\textit{absolutely} or only \textit{conditionally} convergent,
according as the series $A_{1}+A_{2}+\cdots$ is convergent or
divergent.

\addcontentsline{toc}{section}{\numberline{}The region of
convergence}

\textbf{67. The Region of Convergence of a Complex Series.}
\textit{If the complex series $a_{0}+a_{1}z+a_{2}z^{2}+\cdots$ is
convergent when $z=Z$,
%[*Transcriber's note: corrected a_{1}z^{2} to a_{2}z^{2}*]
it is absolutely convergent for every value of $z$ which is
numerically less than $Z$, that is, it converges absolutely at
every point within that circle in the plane of complex numbers
which has the null-point for centre and passes through the point
$Z$.}

For since the series $a_{0}+a_{1}Z+a_{2}Z^{2}+\cdots$ is
convergent, its term $a_{n}Z^{n}$ approaches 0 as limit when $n$
is indefinitely increased. It is therefore possible to find a real
number $M$ which is numerically greater than every term of this
series.

Assign to $z$ any value which is numerically less than $Z$, whose
corresponding point, therefore, lies within the circle through the
point $Z$.

For this value of $z$ the terms of the series
$a_{0}+a_{1}z+a_{2}z^{2}+\cdots$ will be numerically less than the
corresponding terms of the series
\[
M+M\frac{z}{Z}+M\left(\frac{z}{Z}\right)^{2}+\cdots. \tag{1}
\]

\noindent For, since $a_{n}Z^{n}<M$, we have $\displaystyle
a_{n}z^{n}<M\left(\frac{z}{Z}\right)^n$ numerically.

But the series (1) is absolutely convergent (\S~62, 4).

Therefore the given series $a_{0}+a_{1}z+a_{2}z^{2}+\cdots$ also
is absolutely convergent for the value of $z$ under consideration,
that is, for all values of $z$ whose corresponding points lie
within the circle through the point $Z$.

\textsc{Note}. For other points than $Z$ on the
\emph{circumference} of this circle through $Z$ the series is not
necessarily convergent.

Thus the series $1+\frac{z}{2}+\frac{z^2}{3}+\dotsb$ converges
when $z=Z=-1$. But on the circle through the point $-1$, the point
$1$ also lies; and the series diverges when $z=1$.

\addcontentsline{toc}{section}{\numberline{}A theorem respecting
complex series}

\textbf{68. Theorem.} The following is a theorem on which many of
the properties of functions defined by series depend.

\textit{If the series $\qquad a{0}+a_{1}z+a_{2}z^{2}+\dotsb+a_{n}z^{n}+\dotsb$  \\
\noindent have a circle of convergence greater than the null-point
itself, and $z$ run through a regular sequence of values $z_{1}$,
$z_{2}$, $\dotsc$ defining $0$, the sum of all terms following the
first}, viz.,
\[
  a_{1}z+a_{2}z^{2}+ \dotsb +a_{n}z^{n}+ \dotsb
\]
\textit{will run through a sequence of values likewise regular and
defining $0$; or, the entire series may be made to differ as
little as one chooses from its first term $a_{0}$.}

The numbers $z_{1}$, $z_{2}$, $\dotsc$ are, of course, all
supposed to lie within the circle of convergence, and for
convenience, to be real. It will be convenient also to suppose
$z_{1}>z_{2}>z_{3}$, etc.; i.~e.\ that each is greater than the one
following it.

\begin{flalign*}
&{\indent Since }&
& a_{0} + a_{1}z + a_{2}z^{2} + \dotsb + a_{n}z^{n} + \dotsb  &&\\
\intertext{converges absolutely for $z=z_{1}$, so also does} &&
& a_{1}z + a_{2}z^{2} + \dotsb + a_{n}z^{n} + \dotsb,  &&\\
&\text{and, therefore, }&
& a_{1} + a_{2}z + \dotsb + a_{n}z^{n-1} + \dotsb.  &&\\
%[*Transcriber's note:
%The last term has been corrected from a_{n}z^{n} to a_{n}z^{n-1} and similarly in the following 2 infinite sums.]
&\text{\indent Hence }& & A_{1} + A_{2}z_{1} + \dotsb +
A_{n}z_{1}^{n-1} + \dotsb
\end{flalign*}
(where $A_{i}= \text{ modulus } a_{i}$) is convergent, and a
number $M$ can be found greater than its sum.

And since for $z=z_{2}$, $z_{3}$, $\dotsc$ the individual terms of
\[
  A_{1}+A_{2}z+ \dotsb +A_{n}z^{n-1}+ \dotsb
\]
are less than the corresponding terms of $A_{1}+A_{2}z_{1}+ \dotsb
+A_{n}z_{1}^{n-1}+ \dotsb$, this series and, therefore, $modulus
(a_{1}+a_{2}z+ \dotsb)$ remain always less than $M$ as $z$ runs
through the sequence of values $z_{2}$, $z_{3}$, $\dotsb$.

Hence the values of $modulus (a_{1}z+a_{2}z^{2}+ \dotsb)$ which
correspond to $z=z_{1}$, $z_{2} \dotsc$ constitute a regular
sequence defining $0$, each term being numerically less than the
corresponding term of the regular sequence $z_{1}M$, $z_{2}M$,
$\dotsc$ which defines $0$.

\textsc{Cor.} The same argument proves that if
\begin{flalign*}
&&            & a_{m}z^{m} + a_{m+1}z^{m+1} + \cdots,  &&\\
&\text{or }&  & z^{m} (a_{m} + a_{m+1}z + \cdots),  &&
\end{flalign*}
be the sum of all terms of the series from the $(m+1)$th on, the
series $a_{m}+a_{m+1}z+\cdots$ can be made to differ as little as
one may please from its first term $a_{m}$.

\addcontentsline{toc}{section}{\numberline{}The fundamental
operations on complex series}

\textbf{69. Operations on Complex Series.} The definitions of
\emph{sum}, \emph{difference}, and \emph{product} of two
convergent complex series are the same as those already given for
real series, viz.:

1. \emph{The sum of two convergent series, $a_{1}+a_{2}+\cdots$
and $b_{1}+b_{2}+\cdots$, is the series $(a_{1}+b_{1}) +
(a_{2}+b_{2}) + \cdots$; their difference, the series
$(a_{1}-b_{1}) + (a_{2}-b_{2}) + \cdots$.}

\begin{flalign*}
&\text{\indent For if }& & s_{i}=a_{1}+a_{2}+\cdots+a{i} \text{
and }
  t_{i}=b_{1}+b_{2}+\cdots+b{i},
&&\\
&& & \text{ modulus } [(s_{m+n}\pm t_{m+n}) - (s_{m}\pm t_{m})]
&&\\
&& &\space{60mu} \leq \text{ modulus } (s_{m+n}-s_{m})
   + \text{ modulus } (t_{m+n}-t_{m}),  &&
\end{flalign*}

and may, therefore, be made less than any assignable number by
taking $m$ great enough. The theorem therefore follows by the
reasoning of \S~64,~1.

2. \emph{The product of two absolutely convergent series,}
\[
  a_{1} + a_{2} + a_{2} + \cdots \text{ \emph{ and }}
  b_{1} + b_{2} + b_{3} + \cdots,
\]
\emph{is the series}  $a_{1}b_{1} + (a_{1}b_{2}+a_{2}b_{1}) +
(a_{1}b_{3}+a_{2}b_{2}+a_{3}b_{1}) \cdots$.

For, letting $S_{i}=A_{1}+A_{2}+\cdots+A_{i}$ and
$T_{i}=B_{1}+B_{2}+\cdots+B_{i}$, where $A_{i}$, $B_{i}$, are the
moduli of $a_{i}$, $b_{i}$, respectively, and representing by
$\sigma_{n}$ the sum of the first $n$ terms of the series
\begin{flalign*}
&&  & a_{1}b_{1} + (a_{1}b_{2}+a_{2}b_{1}) + \cdots  &&\\
\intertext{and by $\Sigma_{n}$ sum of the first $n$ terms of the
series }
&&  & A_{1}B_{1} + (A_{1}B_{2}+A_{2}B_{1}) + \cdots,  &&\\
&\text{we have }&
& \text{ modulus } (s_{n}t_{n}-\sigma_{n})\leq S_{n}T_{n}-\Sigma_{n}.  &&\\
\intertext{\indent But the limit of the right member of this
inequality (or equation) is 0 (\S~64,~2); therefore } &&  &
\lim_{n\doteq\infty}(\sigma_{n})
    = \lim_{n\doteq\infty}(s_{n}t_{n}).  &&
\end{flalign*}

\chapter{THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS\@. UNDETERMINED
COEFFICIENTS\@. INVOLUTION AND EVOLUTION\@. THE BINOMIAL THEOREM.}

\addcontentsline{toc}{section}{\numberline{}Definition of
function}

\textbf{70. Function.} A variable $w$ is said to be a
\textit{function} of a second variable $z$ for the area $A$ of the
$z$-plane (§42), when to the $z$ belonging to every point of $A$
there corresponds a determinate value or set of values of $w$.

Thus if $w=2z$, $w$ is a function of $z$. For when $z=1$, $w=2$;
when $z=2$, $w=4$; and there is in like manner a determinate value
of $w$ for every value of $z$. In this case $A$ is coextensive
with the entire $z$-plane.

Similarly $w$ is a function of $z$, if
\[w=a_0+a_1 z+a_2 z^2+\ldots+a_n z^n+\ldots,\]
so long as this infinite series is convergent, \textit{i.~e.} for
the portion of the $z$-plane bounded by a circle having the
null-point for centre, and for radius the modulus of the smallest
value of $z$ for which the series diverges.

It is customary to use for $w$ when a function of $z$ the symbol
$f(z)$, read ``function $z$.''

\addcontentsline{toc}{section}{\numberline{}Functional equation of
the exponential function}

\textbf{71. Functional Equation of the Exponential Function.} For
positive integral values of $z$ and $t$, $a^z\cdot a^t=a^{z+t}$.
The question naturally suggests itself, is there a function of $z$
which will satisfy the condition expressed by this equation, or
the ``functional equation'' $f(z)f(t)=f(z+t)$, for \textit{all}
values of $z$ and $t$?

We proceed to the investigation of this question and another which
it suggests, not only because they lead to definitions of the
important functions $a^z$ and $\log_az$ for complex values of $a$
and $z$, and so give the operations of involution, evolution, and
the taking of logarithms the perfectly general character already
secured to the four fundamental operations,---but because they
afford simple examples of a large class of mathematical
investigations.\footnote{An application of the principle of
permanence (§12) is involved in the use of functional equations to
define functions. The equation $a^za^t=a^{z+t}$, for instance,
only becomes a functional equation when its \textit{permanence is
assumed} for other values of $z$ and $t$ than those for which it
has been actually demonstrated.

In this respect the methods of definition of the negative and the
fraction on the one hand, and the functions $a^z$, $\log_az$, on
the other, are identical; but, while the equation $(a-b)+b=a$
itself served as definition of $a-b$, there being no simpler
symbols in terms of which $a-b$ could be expressed, from the
equation $a^za^t=a^{z + t}$ a series (\S~73, (4)) may be deduced
which defines $a^z$ in terms of numbers of the system $a+ib$.}

\addcontentsline{toc}{section}{\numberline{}Undetermined
coefficients}

\textbf{72. Undetermined Coefficients.} In investigations of this
sort, the method commonly used in one form or another is that of
\textit{undetermined coefficients}. This method consists in
assuming for the function sought an expression involving a series
of unknown but constant quantities---coefficients,---in
substituting this expression in the equation or equations which
embody the conditions which the function must satisfy, and in so
determining these unknown constants that these equations shall be
\textit{identically} satisfied, that is to say, satisfied for all
values of the variable or variables.

The method is based on the following theorem, called ``the
theorem of undetermined coefficients,'' \; viz.:

\textit{If the series $A+Bz+Cz^2+\cdots$ be equal to the series
$A'+B'z+C'z^2+\cdots$ for all values of $z$ which make both
convergent, and the coefficients be independent of $z$, the
coefficients of like powers of $z$ in the two are equal.}

For, since
\[A+Bz+Cz^2+\cdots =A'+B'z+C'z^2+\cdots,\]
\[A-A'+(B-B')z+(C-C')z^2+\cdots=0\]
throughout the circle of convergence common to the two given
series (\S\S~67, 69, 1).

And being convergent within this circle, the series
\[A-A'+(B-B')z+(C-C')z^2+\cdots\]
can be made to differ as little as we please from its first term,
$A - A'$ (\S~68).
\[
\therefore A - A' = 0 \; \text{(\S~30, Cor.), or} \; A = A'.
\]

Therefore
\[
(B - B')z + (C - C')z^2 + \cdots = 0
\]
throughout the common circle of convergence, and hence (at least,
for values of $z$ different from 0)
\[
B - B' + (C - C')z + \cdots = 0
\]

Therefore by the reasoning which proved that
\[
A - A' = 0, \; B - B' = 0, \; \text{or} \; B = B'.
\]

In like manner it may be proved that $C = C'$, $D = D'$, etc.
\begin{align*}
\text{COR. \textit{If}} & \quad A + Bz + Ct+Dz^2 + Ezt + Ft^2 + \cdots \\
& = A' + B'z + C't +D'z^2 + E'zt + F't^2 + \cdots
\end{align*}
\noindent \textit{for all values of z and t which make both series
convergent, and z be independent of t, and the coefficients
independent of both z and t, the coefficients of like powers of z
and t in the two series are equal.}

For, arrange both series with reference to the powers of either
variable. The coefficients of like powers of this variable are
then equal, by the preceding theorem. These coefficients are
series in the other variable, and by applying the theorem to each
equation between them the corollary is demonstrated.

\addcontentsline{toc}{section}{\numberline{}The exponential
function}

\textbf{73. The Exponential Function.} To apply this method to the
case in hand, assume
\[
f(z) = A_0 + A_1z + A_2z^2 + \cdots + A_nz^n + \cdots,
\]
and determine whether values of the coefficients $A_i$ can be
found capable of satisfying the ``functional equation,''
\[
f(z)f(t) = f(z + t), \tag{1}
\]
for all values of $z$ and $t$.

On substituting in this equation, we have, for all values of $z$
and $t$ for which the series converge,
\[
\begin{split}
(A_{0} + A_{1}z + A_{2}z^{2} + \cdots A_{n}z^{n} + \cdots) (A_{0} + A_{1}t + A_{2}t^{2} + \cdots A_{n}t^{n} + \cdots) \\
= A_{0} + A_{1}(z+t) + A_{2}(z+t)^{2} + \cdots A_{n}(z+t)^{n} +
\cdots;
\end{split}
\]
or, expanding and arranging the terms with reference to the powers
of $z$ and $t$,

\begin{align*}
A_{0}A_{0} & + A_{1}A_{0}z + A_{0}A_{1}t + A_{2}A_{0}z^{2} + A_{1}A_{1}zt + A_{0}A_{2}t^{2} + \cdots\\
& + A_{n}A_{0}z^{n} + A_{n-1}A_{1}z^{n-1}t + \cdots + A_{n-k}A_{k}z^{n-k}t^{k} + \cdots + A_{0}A_{n}t^{n} \\
& + \cdots \\
& = A_{0} + A_{1}z + A_{1}t + A_{2}z^{2} + 2A_{2}zt + A_{2}t^{2} + \cdots \\
& + A_{n}z^{n} + A_{n}nz^{n-1}t + \cdots +A_{n}n_{k}z^{n-k}t^{k} +\cdots+ A_{n}t^{n} + \cdots,\\
\text{where} & \qquad n_{k} = \frac{(n(n-1) \cdots (n-k+1)}{k!}
\end{align*}

Equating the coefficients of like powers of $z$ and $t$ in the two
members of this equation, we get
\begin{align*}
& A_{n-1}A_{k} \; \text{ equal always to} \; A_{n}n_{k}.\\
\text{In particular} \; & A_{0}A_{0} = A_{0}, \text{therefore} \; A_{0} = 1. \quad \text{Also}\\
& A_{1}A_{1} = 2A_{2}, \quad A_{2}A_{1} = 3A_{3}, \\
& A_{3}A_{1} = 4A_{4}, \; \cdots , \; A_{n-1}A_{1} = nA_{n};
\end{align*}
or, multiplying these equations together member by member,

\[
A_{1}^{n} = A_{n}n!, \;  \text{or} \;  A_{n} = \frac{A_{1}^{n}}{n!}.
\]

A part of the equations among the coefficients are, therefore,
sufficient to determine the values of all of them in terms of the
one coefficient $A_{1}$. But these values will satisfy the
remaining equations; for substituting them in the general equation
\begin{align*}
& A_{n-k}A_{k} = A_{n}n_{k},\\
\text{we get} \qquad & \frac{A_{1}^{n-k}}{(n-k)!} \times
\frac{A_{1}^{k}}{k!} = \frac{A_{1}^{n}}{n!} \times \frac{n(n-1)
\cdots (n-k+1)}{k!},
\end{align*}
which is obviously an identical equation.

The coefficient $A_{1}$ or, more simply written, $A$, remains
undetermined.

It has been demonstrated, therefore, that to satisfy equation (1),
it is only necessary that, $f(z)$ be the sum of an infinite series
of the form
\[
1 + Az + \frac{A^{2}}{2!}z^{2} + \frac{A^{3}}{3!}z^{3} + \cdots,
\tag{2}
\]
where $A$ is undetermined; a series which has a sum, i.~e.\ is
convergent, for all finite values of $z$ and $A$. (\S~63, 2,
\S~66.)

By properly determining $A$, $f(z)$ may be identified with
$a^{z}$, for any particular value of $a$.

If $a^{z}$ is to be identically equal to the series (2), $A$ must
have such a value that

\begin{align*}
& a = 1 + A + \frac{A^{2}}{2!} + \frac{A^{3}}{3!} + \cdots. \\
\text{Let} \qquad & e^{z} = 1 + z + \frac{z^{2}}{2!} + \frac{z^{3}}{3!} + \cdots , \qquad \qquad (3) \\
\text{where} \qquad & e = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} +
\cdots;\footnotemark \\
\text{Then} \qquad & e^{A} = 1 + A + \frac{A^{2}}{2!} + \frac{A^{3}}{3!} + \cdots. \\
\text{Therefore} \qquad & a = e^{A};
\end{align*}
or, calling any number which satisfies the equation

\[e^{z} = a\]

\noindent the \textit{logarithm} of a to the base $e$ and writing
it $\log_{e}a$,

\[ A = \log_{e}a.\]

\footnotetext{\label{irrationality}This number $e$, the base of
the Naperian system of logarithms, is a ``transcendental'' \,
irrational, transcendental in the sense that there is no algebraic
equation with integral coefficients of which it can be a root (see
Hermite, Comptes Rendus, LXXVII). $\pi$ has the same character, as
Lindemann proved in 1882, deducing at the same time the first
actual demonstration of the impossibility of the famous old
problem of squaring the circle by aid of the straight edge and
compasses only (see Mathematische Annalen, XX).}

Whence finally,

\[ a^z = 1 + (\log_{e}a)z + \frac{(\log_{e}a)^2z^2}{2!} +
\frac{(\log_{e}a)^3z^3}{3!} + \cdots, \tag{4}
\]
a definition of $a^z$, valid for all finite complex values of $a$
and $z$, if it may be assumed that $\log_e a$ is a number,
whatever the value of $a$.

The series (3) is commonly called the \emph{exponential series},
and its sum $e^z$ the \emph{exponential function}. It is much more
useful than the more general series (2), or (4), because of its
greater simplicity; its coefficients do not involve the logarithm,
a function not yet fully justified and, as will be shown, to a
certain extent indeterminate. Inasmuch, however, as $e^z$ is a
particular function of the class $a^z$, $a^z$ is sometimes called
the general exponential function, and series (4) the general
exponential series.

\addcontentsline{toc}{section}{\numberline{}The functions sine and
cosine}

\textbf{74. The Functions Sine and Cosine.} It was shown in \S~51
that when $\theta$ is a real number,

\begin{align*}
e^{i\theta} & = \cos\theta + i\sin\theta. \\
\text{But} \qquad e^{i\theta} & = 1 + i\theta +
\frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} +
\frac{(i\theta)^4}{4!} + \cdots \\
&= 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots \\
& + i\left(\theta - \frac{\theta^3}{3!} + \cdots\right).
\end{align*}

Therefore (by \S~36, 2, Cor.), for real values of $\theta$
\begin{equation}
\cos\theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} -
\cdots,
\end{equation}
and
\begin{equation}
\sin\theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} -
\cdots,
\end{equation}
series which both converge for all finite values of $\theta$.
Though $\cos\theta$ and $\sin\theta$ only admit of geometrical
interpretation when $\theta$ is real, it is convenient to continue
to use these names for the sums of the series (5) and (6) when
$\theta$ is complex.

\addcontentsline{toc}{section}{\numberline{}Periodicity of these
functions}

\textbf{75. Periodicity.} When $\theta$ is real, evidently neither
its sine nor its cosine will be changed if it be increased or
diminished by any multiple of four right angles, or $2\pi$; or, if
$n$ be any positive integer,
\[
\cos (\theta \pm 2n\pi) = \cos \theta, \; \sin (\theta \pm 2n\pi)
= \sin \theta,
\]
and hence
\[e^{i(\theta \pm 2n\pi)} = e^{i\theta}.\]

The functions $e^{i\theta}$, $\cos \theta$, $\sin \theta$, are on
this account called \emph{periodic} functions, with the
\emph{modulus of periodicity $2\pi$}.

\addcontentsline{toc}{section}{\numberline{}The logarithmic
function}

\textbf{76. The Logarithmic Function.} If $z = e^z$ and $t = e^T$,
\[
zt = e^z e^T = e^{Z + T}, \qquad \qquad \text{\S~73}\] or
\[
\log_e zt = \log_e z + \log_e t. \tag{7}
\]

The question again is whether a function exists capable of
satisfying this equation, or, more generally, the ``functional
equation,''
\[
f(zt) = f(z) + f(t), \tag{8}
\]
for complex values of $z$ and $t$.

When $z = 0$, (7) becomes
\[
\log_e 0 = \log_e 0 + \log_e t,
\]
an equation which cannot hold for any value of $t$ for which
$\log_e t$ is not zero unless $\log_e 0$ is numerically greater
than any finite number whatever. Therefore $\log_e 0$ is infinite.

On the other hand, when $z = 1$, (7) becomes
\[
\log_e t = \log_e 1 + \log_e t,
\]
so that $\log_e 1$ is zero.

Instead, therefore, of assuming a series with undetermined
coefficients for $f(z)$ itself, we assume one for $f(1 +z)$,
setting
\[
f(1 + z) = A_1 z + A_2 z^2 + \cdots + A_n z^n + \cdots,
\]
and inquire whether the coefficients $A_i$ admit of values which
satisfy the functional equation (8) for complex values of $z$ and
$t$.

Now
\[1+z+t=(1+z)\left(1+\frac{t}{1+z}\right), \; \text{ identically}.\]

\[\therefore f\left[1+(z+t)\right]=f(1+z)+f\left(1+\frac{t}{1+z}\right),\]

\noindent or
\begin{align*}
&A_1(z+t)+A_2(z+t)^2+\cdots +A_n(z+t)^n+\cdots\\
=&A_1z+A_2z^2+\cdots +A_nz^n+\cdots\\
+&A_1(1+z)^{-1}t+A_2(1+z)^{-2}t^2+\cdots +A_n(1+z)^{-n}t^n+\cdots
\end{align*}

Equating the coefficients of the first power of $t$ (\S~72) in the
two members of this equation,
\begin{align*}
& A_1+2A_2z+3A_3z^2+\cdots +(n+1)A_{n+1}z^n+\cdots\\
= \, & A_1(1-z+z^2-z^3+\cdots +(-1)^nz^n+\cdots );
\end{align*}
whence, equating the coefficients of like powers of z,
\begin{align*}
& A_1=A_1, 2A_2=-A_1,\cdots,nA_n=(-1)^{n-1}A_1,\cdots,\\
\text{or} \qquad &
A_2=-\frac{A_1}{2},\cdots, A_n=(-1)^{n-1}\frac{A_1}{n},\cdots.
\end{align*}

As in the case of the exponential function, a part of the
equations among the coefficients are sufficient to determine them
all in terms of the one coefficient $A_1$.  But as in that case
(by assuming the truth of the binomial theorem for negative
integral values of the exponent) it can be readily shown that
these values will satisfy the remaining equations also.

The series $\displaystyle \qquad
z-\frac{z^2}{2}+\frac{z^3}{3}-\cdots
+(-1)^{n-1}\frac{z^n}{n}+\cdots$

\noindent converges for all values of $z$ whose moduli are less
than 1 (\S~62, 3)

For such values, therefore, the function
\[
A\left(z-\frac{z^2}{2}+\cdots +(-1)^{n-1}\frac{z^n}{n}+\cdots
\right) \tag{9}
\]
satisfies the functional equation
\[
f\left[(1+z)(1+t)\right]=f(1+z)+f(1+t).
\]

\begin{align*}
\text{And since} \qquad & z\equiv 1-(1-z) \; \text{and} \;
t\equiv1-(1-t),\\
\text{the function} \qquad & -A\left(1-z+\frac{(1-z)^2}{2}+\cdots
+\frac{(1-z)^n}{n}+\cdots \right)
\end{align*}

\noindent satisfies this equation when written in the simpler form
\begin{equation*}
f(zt)=f(z)+f(t),
\end{equation*}
for values of $1-z$ and $1-t$ whose moduli are both less than 1.

1. $Log_eb$.  To identify the general function $f(1+z)$ with the
particular function $\log_e(1+z)$ it is only necessary to give the
undetermined coefficient $A$ the value 1.

For since $\log_e(1+z)$ belongs to the class of functions which
satisfy the equation (8),
\begin{equation*}
\log_e(1+z)=A\left(z-\frac{z^2}{2}+\cdots\right).
\end{equation*}

Therefore
\begin{align*}
e^{\log_e(1+z)}&=e^{A\left(z-\frac{z^2}{2}+\cdots \right)}\\
&=1+A\left(z-\frac{z^2}{2}+\cdots\right)+\frac{1}{2!}A^2\left(z-\frac{z^2}{2}+\cdots \right)^2+\cdots.\\
\text{But} \qquad  e^{\log_e(1+z)}&=1+z.\\
\end{align*}

Hence
\begin{equation*}
1+z=1+A\left(z-\frac{z^2}{2}+\cdots
\right)+\frac{1}{2!}A^2\left(z-\frac{z^2}{2}+\cdots
\right)^2+\cdots ;
\end{equation*}
or, equating the coefficients of the first power of $z$, $A=1$.

The coefficients of the higher powers of $z$ in the right number
are then identically 0.

It has thus been demonstrated that $\log_eb$ is a number (real or
complex), if when $b$ is written in the form $1+z$, the absolute
value of $z$ is less than 1. To prove that it is a number for
other than such values of $b$, let $b=\rho e^{i\theta }$, (\S~51),
where $\rho$, as being the modulus of $b$, is positive.

\[\text{Then} \qquad \log_eb=\log_e\rho +i\theta,\]
\noindent and it only remains to prove that $\log_e\rho$ is a
number.

Let $\rho$ be written in the form $\displaystyle e^n-(e^n-\rho )$,
where $e^n$ is the first integral power of $e$ greater than
$\rho$.


\begin{align*}
\text{Then since} \qquad & e^n - (e^n - \rho) \equiv e^n \left(1 - \frac{e^n-\rho}{e^n}\right),\\
& \log_e\rho  = \log_e e^n + \log_e \left(1 - \frac{e^n - \rho}{e^n} \right) \\
& \qquad \quad = n + \log_e\left(1 - \frac{e^n -
\rho}{e^n}\right),
\end{align*}
and $\displaystyle \log_e\left(1 - \frac{e^n - \rho}{e^n}\right)$
is a number since $\displaystyle \frac{e^n - \rho}{e^n}$ is less
than 1.

2. $Log_a b$. It having now been fully demonstrated that $a^z$ is
a number satisfying the equation $a^Z a^T = a^{Z+T}$ for all
finite values of $a$, $Z$, $T$; let $a^Z = z$, $a^T = t$, and call
$Z$ the \textit{logarithm of $z$ to the base $a$}, or $\log_a z$,
and in like manner $T$, $\log_a t$.

\begin{align*}
\text{Then, since} \quad & zt = a^Z a^T = a^{z+T},\\
& \log_a(zt) = \log_a z + \log_a t,
\end{align*}
or $\log_a z$ belongs, like $\log_e z$, to the class of functions
which satisfy the functional equation (8).

Pursuing the method followed in the case of $\log_e b$, it will be
found that $\displaystyle \log_a(1 +z)$ is equal to the series
$\displaystyle A\left(z - \frac{z^2}{2} + \cdots\right)$ when
$\displaystyle A= \frac{1}{log_e a}$. This number is called the
\textit{modulus} of the system of logarithms of which $a$ is base.

\addcontentsline{toc}{section}{\numberline{}Indeterminateness of
logarithms}

\textbf{77. Indeterminateness of $\mathbf{\log a}$.} Since any
complex number $a$ may be thrown into the form $\rho e^{i\theta}$,

\[
\log_e a = \log_e \rho + i\theta. \tag{10}
\]

This, however, is only one of an infinite series of possible
values of $\log_e a$. For, since $\displaystyle e^{i\theta} =
e^{i(\theta \pm 2n\pi)}$ (\S~75),
\[
\log_e a = \log_e \rho e^{i(\theta \pm 2n\pi)} = \log_e \rho +
i(\theta \pm 2n\pi),
\]
where $n$ may be any positive integer. Log$_e a$ is, therefore, to
a certain extent indeterminate; a fact which must be carefully
regarded in using and studying this function.\footnote{For
instance $\log_e(zt)$ is not equal to $\log_ez + \log_et$ for
arbitrarily chosen values of these logarithms, but to $\log_ez +
\log_et \pm i2n\pi$, where $n$ is some positive integer.} The
value given it in (10), for which $n=0$, is called its principal
value.

When $a$ is a positive real number, $\theta=0$, so that the
principal value of $\log_{e}a$ is real; on the other hand, when
$a$ is a negative real number, $\theta=\pi$, or the principal
value of $\log_{e}a$ is the logarithm of the positive number
corresponding to $a$, plus $i\pi$.

\addcontentsline{toc}{section}{\numberline{}Permanence of the laws
of exponents}

\textbf{78. Permanence of the Remaining Laws of Exponents.}
Besides the law $a^z a^t = a^{z+t}$ which led to its definition,
the function $a^z$ is subject to the laws:

\begin{align*}
1. \qquad \qquad (a^z)^t &= a^{zt}.\\
2.  \qquad \qquad (a b)^z &= a^z b^z.\footnotemark[1]\\
1.  \qquad \qquad (a^z)^t &= a^{zt}.\\
\text{For} \quad a^z =  \left(e^{\log_{e}a}\right)^z &=  1+(\log_{e}a)z+\frac{(\log_{e}a)^2z^2}{2!}+\cdots & \S~73,\ (4)\\
&=  1+z\log_{e}a + \frac{(z\log_{e}a)^2}{2!}+\cdots\\
&=  e^{z\log_{e}a}. & \S~73,\ (3)\\
\therefore (e^{\log_{e}a})^z &=  e^{z\log_{e}a}, \; \text{and} \;
\log_{e}a^z =  z\log_{e}a.
\end{align*}

From these results it follows that

\begin{align*}
(a^z)^t &=  e^{\log_{e}(a^z)^t}\\
&=  e^{t\log_{e}a^z}\\
&=  e^{tz\log_{e}a}\\
&=  a^{zt}. \\
2. \qquad (ab)^z &= a^z b^z.\\
\text{For} \qquad (ab)^z &= e^{\log_{e}(ab)^z}\\
&= e^{z\log_{e}ab}\\
&= e^{z\log_{e}a+z\log_{e}b}  \qquad \qquad \qquad &\S~76,\ (7)\\
&= e^{z\log_{e}a}\cdot e^{z\log_{e}b}  &\S~73,\ (1)\\
\nonumber &= a^z \cdot b^z.
\end{align*}

\footnotetext[1]{$\displaystyle \frac{a^z}{a^t}=a^{z-t}$, which is
sometimes included among the fundamental laws to which $a^z$ is
subject, follows immediately from $a^z a^t = a^{z+t}$ by the
definition of division.}

\addcontentsline{toc}{section}{\numberline{}Permanence of the laws
of logarithms}

\textbf{79. Permanence of the Remaining Law of Logarithms.} In
like manner, the function $\log_a z$ is subject not only to the
law
\begin{flalign*}
&&  \log_a(zt) &= \log_az + \log_at,  &&\\
\intertext{but also to the law } &&  \log_a z^t &= t\log_a z.  &&
\\
&\text{\indent For }&  z &= a^{\log_a z},  &&\\
&\text{and hence }&  z^t &= (a^{\log_az})^t  &&\\
&&  &= a^{t\log_a z}.  &\text{ \S~78, 1}&
\end{flalign*}

\addcontentsline{toc}{section}{\numberline{}Involution and
evolution}

\textbf{80. Evolution.} Consider three complex numbers $\zeta$,
$z$, $Z$, connected by the equation $\zeta^Z=z$.

This equation gives rise to three problems, each of which is the
inverse of the other two. For $Z$ and $\zeta$ may be given and $z$
sought; or $\zeta$ and $z$ may be given and $Z$ sought; or,
finally, $z$ and $Z$ may be given and $\zeta$ sought.

The exponential function is the general solution of the first
problem (\emph{involution}), and the logarithmic function of the
second.

For the third (\emph{evolution}) the symbol $\sqrt[Z]{z}$ has been
devised. This symbol does not represent a new function; for it is
defined by the equation $(\sqrt[Z]{z})^Z=z$, an equation which is
satisfied by the exponential function $z^{\frac{1}{Z}}$.

Like the logarithmic function, $\sqrt[Z]{z}$ is indeterminate,
though not always to the same extent. When $Z$ is a positive
integer, $\zeta^Z=z$ is an algebraic equation, and by \S~56 has
$Z$ roots for any one of which $\sqrt[Z]{z}$ is, by definition, a
symbol. From the mere fact that $z=t$, therefore, it cannot be
inferred that $\sqrt[Z]{z}=\sqrt[Z]{t}$, but only that one of the
values of $\sqrt[Z]{z}$ is equal to one of the values of
$\sqrt[Z]{t}$. The same remark, of course, applies to the
equivalent symbols $z^{\frac{1}{Z}}$, $t^{\frac{1}{Z}}$.

\addcontentsline{toc}{section}{\numberline{}The binomial theorem
for complex exponents}

\textbf{81. Permanence of the Binomial Theorem.} By aid of the
results just obtained, it may readily be demonstrated that the
binomial theorem is valid for general complex as well as for
rational values of the exponent.

For $b$ being any complex number whatsoever, and the absolute
value of $z$ being supposed less than 1,
\begin{align*}
(1 + z)^b &= e^{b\log_e(1+z)} \\
          &= e^{b\left( z - \frac{z^2}{2} + \cdots \right)} \\
          &= 1 + bz + \text{ terms involving higher powers of } z.
\end{align*}

Therefore let
\[
(1 + z)^b = 1 + bz + A_2z^2 + \cdots + A_nz^n + \cdots. \tag{11}
\]

Since, then, $(a + Z)^b = a^b\left( 1 + \frac{z}{a} \right)^b,$ \qquad \qquad \qquad \S~78,~2\\
if $\frac{z}{a}$ be substituted for $z$ in
(11), and the equation be multiplied throughout by $a^b$,
\[
(a + z)^b = a^b + ba^{b-1}z + A_2a^{b-2}z^2 + \cdots +
A_na^{b-n}z^n + \cdots. \tag{12}
\]

Starting with the identity
\[
(1 + \underline{z + t})^b = (\underline{1 + z} + t)^b,
\]
developing $(1 + \underline{z + t})^b$ by (11) and $(\underline{1
+ z} + t)^b$ by (12), equating the coefficients of the first power
of $t$ in these developments, multiplying the resultant equation
by $1 + z$, and equating the coefficients of like powers of $z$ in
this product, equations are obtained from which values may be
derived for the coefficients $A_i$ identical in form with those
occurring in the development for $(1 + z)^b$ when $b$ is a
positive integer.

It may also be shown that these values of the coefficients satisfy
the equations which result from equating the coefficients of
higher powers of $t$.

\part{HISTORICAL.}


\chapter{PRIMITIVE NUMERALS.}
\setcounter{subsection}{0}

\addcontentsline{toc}{section}{\numberline{}Gesture symbols}

\textbf{82. Gesture Symbols.} There is little doubt that primitive
counting was done on the fingers, that the earliest numeral
symbols were groups of the fingers formed by associating a single
finger with each individual thing in the group of things whose
number it was desired to represent.

Of course the most immediate method of representing the number of
things in a group---and doubtless the method first used---is by
the presentation of the things themselves or the recital of their
names. But to present the things themselves or to recite their
names is not in a proper sense to count them; for either the
things or their names represent all the properties of the group
and not simply the number of things in it. Counting was first done
when a group was used to represent the number of things in some
other group; of that group it would represent the number only and,
therefore, be a true numeral symbol, which it is the sole object
of counting to reach.

Counting ignores all the properties of a group except the
distinctness or separateness of the things in it and presupposes
whatever intelligence is required consciously or unconsciously to
abstract this from its remaining properties. On this account, that
group serves best to represent numbers, in which the individual
differences of the members are least obtrusive. The naturalness of
finger-counting, therefore, lies not only in the accessibility of
the fingers, in their being always present to the counter, but in
this: that the fingers are so similar in form and function that it
is almost easier to ignore than to take account of their
differences.

But there is other evidence than its intrinsic probability for the
priority of finger-counting over any other. Nearly every system of
numeral notation of which we have any knowledge is either quinary,
decimal, vigesimal, or a mixture of
these;\footnote{\label{Instances of quinary and vigesimal systems of notation}Pure quinary and vigesimal systems are rare, if indeed
they occur at all. As an example of the former, Tylor (Primitive
Culture, I, p.~261) instances a Polynesian number series which
runs 1, 2, 3, 4, 5, $5\cdot 1$, $5\cdot 2$,\ldots; and as an
example of the latter, Cantor (Geschichte der Mathematik, p.~8),
following Pott, cites the notation of the Mayas of Yucatan who
have special words for 20, 400, 8000, 160,000. The Hebrew
notation, like the Indo-Arabic, affords an example of a pure
decimal notation. Mixed systems are common. Thus the Roman is
mixed decimal and quinary, the Aztec mixed vigesimal and quinary.
Speaking generally, the quinary and vigesimal systems are more
frequent among the lower races, the decimal among the higher.
(Primitive Culture, I, p.~262.)} that is to say, expresses numbers
which are greater than 5 in terms of 5 and lesser numbers, or
makes a similar use of 10 or 20. These systems point to primitive
methods of reckoning with the fingers of one hand, the fingers of
both hands, all the fingers and toes, respectively.

Finger-counting, furthermore, is universal among uncivilized
tribes of the present day, even those not far enough developed to
have numeral words beyond 2 or 3 representing higher numbers by
holding up the appropriate number of fingers.\footnote{So, for
instance, the aborigines of Victoria and the Bororos of Brazil
(Primitive Culture, I, p.~244).}

\addcontentsline{toc}{section}{\numberline{}Spoken symbols}

\textbf{83. Spoken Symbols.} Numeral words---spoken
symbols---would naturally arise much later than gesture symbols.
Wherever the origin of such a word can be traced, it is found to
be either descriptive of the corresponding finger symbol or---when
there is nothing characteristic enough about the finger symbol to
suggest a word, as is particularly the case with the smaller
numbers---the name of some familiar group of things. Thus in the
languages of numerous tribes the numeral 5 is simply the word for
hand, 10 for both hands, 20 for ``an entire man'' \, (hands and
feet); while 2 is the word for the eyes, the ears, or
wings.\footnote{\label{Instances of digit numerals}In the language
of the Tamanacs on the Orinoco the word for 5 means ``a whole
hand,'' \, the word for 6, ``one of the other hand,'' \, and so
on up to 9; the word for 10 means ``both hands,'' \, 11, ``one
to the foot,'' \, and so on up to 14; 15 is ``a whole foot,'' \,
16, ``one to the other foot,'' \, and so on up to 19; 20 is ``one Indian,'' \, 40, ``two Indians,'' \, etc. Other languages
rich in digit numerals are the Cayriri, Tupi, Abipone, and Carib
of South America; the Eskimo, Aztec, and Zulu (Primitive Culture,
I, p.~247).

``Two'' \, in Chinese is a word meaning ``ears,'' \, in Thibet
``wing,'' \, in Hottentot ``hand.'' \, (Gow, Short History of
Greek Mathematics, p.~7.) See also Primitive Culture, I, pp.~252--259.}

As its original meaning is a distinct encumbrance to such a word
in its use as a numeral, it is not surprising that the numeral
words of the highly developed languages have been so modified that
it is for the most part impossible to trace their origin.

The practice of counting with numeral words probably arose much
later than the words themselves. There is an artificial element in
this sort of counting which does not appertain to primitive
counting\footnote{Were there any reason for supposing that
primitive counting was done with numeral words, it would be
probable that the ordinals, not the cardinals, were the earliest
numerals. For the normal order of the cardinals must have been
fully recognized before they could be used in counting.

In this connection, see Kronecker, Ueber den Zahlbegriff; Journal
f\"ur die reine und angewandte Mathematik, Vol. 101, p.~337.
Kronecker goes so far as to declare that he finds in the ordinal
numbers the natural point of departure for the development of the
number concept.} (see \S~5).

One fact is worth reiterating with reference to both the primitive
gesture symbols and word symbols for numbers. There is nothing in
either symbol to represent the individual characteristics of the
things counted or their arrangement. The use of such symbols,
therefore, presupposes a conviction that the number of things in a
group does not depend on the character of the things themselves or
on their collocation, but solely on their maintaining their
separateness and integrity.

\addcontentsline{toc}{section}{\numberline{}Written symbols }

\textbf{84. Written Symbols.} The earliest \textit{written}
symbols for number would naturally be mere groups of
strokes----$|$, $||$, $|||$, etc. Such symbols have a double
advantage over gesture symbols: they can be made permanent, and
are capable of indefinite extension---there being, of course, no
limit to the numbers of strokes which may be drawn.

\chapter{HISTORIC SYSTEMS OF NOTATION.}

\addcontentsline{toc}{section}{\numberline{}Egyptian and Ph\oe
nician}

\textbf{85.   Egyptian and Ph\oe nician.}    This written
symbolism did not assume the complicated character it might have
had, had counting with written strokes and not with the fingers
been the primitive method. Perhaps the written strokes were
employed in connection with counting numbers higher than 10 on the
fingers to indicate how often all the fingers had been used; or if
each stroke corresponded to an individual in the group counted,
they were arranged as they were drawn in groups of 10, so that the
number was represented by the number of these complete groups and
the strokes in a remaining group of less than 10.

At all events, the decimal idea very early found expression in
special symbols for 10, 100, and if need be, of higher powers of
10. Such signs are already at hand in the earliest known writings
of the Egyptians and Phoenicians in which numbers are represented
by unit strokes and the signs for 10, 100, 1000, 10,000, and even
100,000, each repeated up to 9 times.

\addcontentsline{toc}{section}{\numberline{}Greek}

\textbf{86. Greek.}     In two of the best known notations of
antiquity, the old Greek notation---called sometimes the
Herodianic, sometimes the Attic---and the Roman, a primitive
system of counting on the fingers of a single hand has left its
impress in special symbols for 5.

In the Herodianic notation the only symbols---apart from certain
abbreviations for products of 5 by the powers of 10---are
$\mathsf{I}$, $\Gamma$ ($\pi\acute{\epsilon}\nu\tau\epsilon$, 5),
$\Delta$ ($\delta\acute{\epsilon}\kappa\alpha$, 10), $\mathsf{H}$
($\grave{\epsilon}\kappa\alpha\tau\acute{o}\nu$, 100), $\chi$
($\chi\acute{\iota}\lambda\iota o\iota$, 1000), $\mathsf{M}$
($\mu\nu\rho\acute{\iota}o\iota$, 10,000); all of them, except
$\mathsf{I}$, it will be noticed, initial letters of numeral
words. This is the only notation, it may be added, found in any
Attic inscription of a date before Christ. The later and, for the
purposes of arithmetic, much inferior notation, in which the 24
letters of the Greek alphabet with three inserted strange letters
represent in order the numbers 1, 2, \ldots 10, 20, \ldots 100,
200, \ldots 900, was apparently first employed in Alexandria early
in the 3d century B.~C., and probably originated in that city.

\addcontentsline{toc}{section}{\numberline{}Roman}

\textbf{87. Roman.} The Roman notation is probably of Etruscan
origin. It has one very distinctive peculiarity: the subtractive
meaning of a symbol of lesser value when it precedes one of
greater value, as in $\mathsf{IV}$ = 4 and in early inscriptions
$\mathsf{IIX} = 8$. In nearly every other known system of notation
the principle is recognized that the symbol of lesser value shall
follow that of greater value and be added to it.

In this connection it is worth noticing that two of the four
fundamental operations of arithmetic---addition and
multiplication---are involved in the very use of special symbols
for 10 and 100, for the one is but a symbol for the \textit{sum}
of 10 units, the other a symbol for 10 sums of 10 units each, or
for the \textit{product} 10 $\times$ 10. Indeed, addition is
primarily only abbreviated counting; multiplication, abbreviated
addition. The representation of a number in terms of tens and
units, moreover, involves the expression of the result of a
division (by 10) in the number of its tens and the result of a
subtraction in the number of its units. It does not follow, of
course, that the inventors of the notation had any such notion of
its meaning or that these inverse operations are, like addition
and multiplication, as old as the symbolism itself. Yet the
Etrusco-Roman notation testifies to the very respectable antiquity
of one of them, subtraction.

\addcontentsline{toc}{section}{\numberline{}Indo-Arabic}

\textbf{88. Indo-Arabic.} Associated thus intimately with the four
fundamental operations of arithmetic, the character of the numeral
notation determines the simplicity or complexity of all reckonings
with numbers. An unusual interest, therefore, attaches to the
origin of the beautifully clear and simple notation which we are
fortunate enough to possess. What a boon that notation is will be
appreciated by one who attempts an exercise in division with the
Roman or, worst of all, with the later Greek numerals.

The system of notation in current use to-day may be characterized
as the positional decimal system. A number is resolved into the
sum:
\[
a_{n}10^{n} + a_{n-1}10^{n-1} + \cdots + a_{1}10 + a_{0},
\]
where $10^{n}$ is the highest power of 10 which it contains, and
$a_{n}$, $a_{n-1}$, $\ldots$ $a_{0}$ are all numbers less than 10;
and then represented by the mere sequence of numbers $a_{n}a_{n-1}
\cdots a_{0}$---it being left to the \emph{position} of any number
$a_i$ in this sequence to indicate the power of 10 with which it
is to be associated. For a system of this sort to be complete---to
be capable of representing all numbers unambiguously---a symbol
(0), which will indicate the absence of any particular power of 10
from the sum $a_{n}10^{n} + a_{n-1}10^{n-1} + \cdots + a_{1}10 +
a_{0}$, is indispensable. Thus without 0, 101 and 11 must both be
written 11. But this symbol at hand, any number may be expressed
unambiguously in terms of it and symbols for 1, 2, $\ldots$ 9.

The positional idea is very old. The ancient Babylonians commonly
employed a decimal notation similar to that of the Egyptians; but
their astronomers had besides this a very remarkable notation, a
\emph{sexagesimal} positional system. In 1854 a brick tablet was
found near Senkereh on the Euphrates, certainly older than 1600
\textsc{b.~c.}, on one face of which is impressed a table of the squares, on
the other, a table of the cubes of the numbers from 1 to 60. The
squares of $1$, $2$, $\ldots$ $7$ are written in the ordinary
decimal notation, but $8^2$, or $64$, the first number in the
table greater than $60$, is written $1$, $4$ ($1 \times 60 + 4$);
similarly $9^2$, and so on to $59^2$, which is written $58$, $1$
($58 \times 60 +1$); while $60^2$ is written $1$. The same
notation is followed in the table of cubes, and on other tablets
which have since been found. This is a positional system, and it
only lacks a symbol for $0$ of being a perfect positional system.

The inventors of the $0$-symbol and the modern complete decimal
positional system of notation were the Indians, a race of the
finest arithmetical gifts.

The earlier Indian notation is decimal but not positional. It has
characters for 10, 100, etc., as well as for $1$, $2$, $\ldots$
$9$, and, on the other hand, no $0$.

Most of the Indian characters have been traced back to an old
alphabet\footnote{Dr.~Isaac Taylor, in his book ``The Alphabet,''
names this alphabet the Indo-Bactrian. Its earliest and most
important monument is the version of the edicts of King Asoka at
Kapur-di-giri. In this inscription, it may be added, numerals are
denoted by strokes, as $|, ||, |||, ||||, |||||$.} in use in
Northern India 200 \textsc{b.~c.} The original of each numeral
symbol 4, 5, 6, 7, 8 (?), 9, is the initial letter in this
alphabet of the corresponding numeral word (see table on page
89,\footnote{Columns 1--5, 7, 8 of the table on page~89 are taken
from Taylor's Alphabet, II, p.~266; column~6, from Cantor's
Geschichte der Mathematik.} column~1). The characters first occur
as numeral signs in certain inscriptions which are assigned to the
1st and 2d centuries \textsc{a.~d.~} (column~2 of table). Later they
took the forms given in column~3 of the table.

When 0 was invented and the positional notation replaced the old
notation cannot be exactly determined. It was certainly later than
400 \textsc{a.~d.}, and there is no evidence that it was earlier than 500
\textsc{a.~d.} The earliest known instance of a date written in the new
notation is 738 \textsc{a.~d.} By the time that 0 came in, the other
characters had developed into the so-called Devanagari numerals
(table, column 4), the classical numerals of the Indians.

The perfected Indian system probably passed over to the Arabians
in 773 \textsc{a.~d.}, along with certain astronomical writings. However
that may be, it was expounded in the early part of the 9th century
by Alkhwarizm\^{i}, and from that time on spread gradually
throughout the Arabian world, the numerals taking different forms
in the East and in the West.

Europe in turn derived the system from the Arabians in the 12th
century, the ``Gobar'' \, numerals (table, column 5) of the
Arabians of Spain being the pattern forms of the European numerals
(table, column 7). The arithmetic founded on the new system was at
first called \textit{algorithm} (after Alkhwarizm\^{i}), to
distinguish it from the arithmetic of the abacus which it came to
replace.

A word must be said with reference to this arithmetic on the
abacus. In the primitive abacus, or reckoning table, unit counters
were used, and a number represented by the appropriate number of
these counters in the appropriate columns of the instrument;
\textit{e.~g.} 321 by 3 counters in the column of 100's, 2 in the
column of 10's, and 1 in the column of units. The Romans employed
such an abacus in all but the most elementary reckonings, it was
in use in Greece, and is in use to-day in China.

Before the introduction of \textit{algorithm}, however, reckoning
on the abacus had been improved by the use in its columns of
separate characters (called \textit{apices}) for each of the
numbers 1, 2, \ldots, 9, instead of the primitive unit counters.
This improved abacus reckoning was probably invented by Gerbert
(Pope Sylvester II.), and certainly used by him at Rheims about
970--980, and became generally known in the following century.

\begin{figure}[htbp]
\centering \includegraphics[scale=0.45]{images/fig5.eps}\\
\end{figure}

Now these apices are not Roman numerals, but symbols which do not
differ greatly from the Gobar numerals and are clearly, like them,
of Indian origin. In the absence of positive evidence a great
controversy has sprung up among historians of mathematics over the
immediate origin of the apices. The only earlier mention of them
occurs in a passage of the geometry of Boetius, which, if genuine,
was written about 500 \textsc{a.~d.} Basing his argument on this passage,
the historian Cantor urges that the earlier Indian numerals found
their way to Alexandria before her intercourse with the East was
broken off, that is, before the end of the 4th century, and were
transformed by Boetius into the apices. On the other hand, the
passage in Boetius is quite generally believed to be spurious, and
it is maintained that Gerbert got his apices directly or
indirectly from the Arabians of Spain, not taking the 0, either
because he did not learn of it, or because, being an abacist, he
did not appreciate its value.

At all events, it is certain that the Indo-Arabic numerals, 1, 2,
\ldots 9 (not 0), appeared in Christian Europe more than a century
before the complete positional system and \textit{algorithm}.

The Indians are the inventors not only of the positional decimal
system itself, but of most of the processes involved in elementary
reckoning with the system. Addition and subtraction they performed
quite as they are performed nowadays; multiplication they effected
in many ways, ours among them, but division cumbrously.

\chapter{THE FRACTION.}

\addcontentsline{toc}{section}{\numberline{}Primitive fractions}

\textbf{89. Primitive Fractions.} Of the artificial forms of
number---as we may call the fraction, the irrational, the
negative, and the imaginary in contradistinction to the positive
integer---all but the fraction are creations of the
mathematicians. They were devised to meet purely mathematical
rather than practical needs. The fraction, on the other hand, is
already present in the oldest numerical records---those of Egypt
and Babylonia---was reckoned with by the Romans, who were no
mathematicians, and by Greek merchants long before Greek
mathematicians would tolerate it in arithmetic.

The primitive fraction was a concrete thing, merely an aliquot
part of some larger thing. When a unit of measure was found too
large for certain uses, it was subdivided, and one of these
subdivisions, generally with a name of its own, made a new unit.
Thus there arose fractional units of measure, and in like manner
fractional coins.

In time the relation of the sub-unit to the corresponding
principal unit came to be abstracted with greater or less
completeness from the particular kind of things to which the units
belonged, and was recognized when existing between things of other
kinds. The relation was generalized, and a pure numerical
expression found for it.

\addcontentsline{toc}{section}{\numberline{}Roman fractions}

\textbf{90. Roman Fractions.} Sometimes, however, the relation was
never completely enough separated from the sub-units in which it
was first recognized to be generalized. The Romans, for instance,
never got beyond expressing all their fractions in terms of the
\textit{uncia}, \textit{sicilicus}, etc., names originally of
subdivisions of the old unit coin, the \textit{as}.

\addcontentsline{toc}{section}{\numberline{}Egyptian (the Book of
Ahmes)}

\textbf{91. Egyptian Fractions.} Races of better mathematical
endowments than the Romans, however, had sufficient appreciation
of the fractional relation to generalize it and give it an
arithmetical symbolism.

The ancient Egyptians had a very complete symbolism of this sort.
They represented any fraction whose numerator is 1 by the
denominator simply, written as an integer with a dot over it, and
resolved all other fractions into sums of such unit fractions. The
oldest mathematical treatise known,---a papyrus\footnote{The Rhind
papyrus of the British Museum; translated by A. Eisenlohr,
Leipzig, 1877.} roll entitled ``Directions for Attaining to the
Knowledge of All Dark Things,'' \, written by a scribe named Ahmes
in the reign of Ra-\"{a}-us (therefore before 1700 \textsc{b.~c.}), after
the model, as he says, of a more ancient work,---opens with a
table which expresses in this manner the quotient of 2 by each odd
number from 5 to 99. Thus the quotient of 2 by 5 is written
$\dot{3}$ $\dot{15}$, by which is meant $\displaystyle \frac{1}{3}
+ \frac{1}{15}$; and the  quotient of 2 by 13, $\dot{8}$ $
\dot{52}$ $\dot{104}$. Only  $\displaystyle \frac{2}{3}$, among
the  fractions having numerators which differ from 1, gets
recognition as a distinct fraction and receives a symbol of its
own.

\addcontentsline{toc}{section}{\numberline{}Babylonian or
sexagesimal}

\textbf{92. Babylonian or Sexagesimal Fractions.} The fractional
notation of the Babylonian astronomers is of great interest
intrinsically and historically. Like their notation of integers it
is a sexagesimal positional notation. The denominator is always 60
or some power of 60 indicated by the position of the numerator,
which alone is written. The fraction $\displaystyle \frac{3}{8}$,
for instance, which is equal to $\displaystyle \frac{22}{60} +
\frac{30}{60^2}$, would in this notation be written 22 30. Thus
the ability to represent fractions by a single integer or a
sequence of integers, which the Egyptians secured by the use of
fractions having a common numerator, 1, the Babylonians found in
fractions having common denominators and the principle of
position. The Egyptian system is superior in that it gives an
exact expression of every quotient, which the Babylonian can in
general do only approximately. As regards practical usefulness,
however, the Babylonian is beyond comparison the better system.
Supply the 0-symbol and substitute 10 for 60, and this notation
becomes that of the modern decimal fraction, in whose distinctive
merits it thus shares.

As in their origin, so also in their subsequent history, the
sexagesimal fractions are intimately associated with astronomy.
The astronomers of Greece, India, and Arabia all employ them in
reckonings of any complexity, in those involving the lengths of
lines as well as in those involving the measures of angles. So the
Greek astronomer, Ptolemy (150 \textsc{a.~d.}), in the \textit{Almagest}
($\mu\epsilon\gamma\Acute{\alpha}\lambda\eta$
$\sigma\Acute{\upsilon}\nu\tau\alpha\xi\iota\varsigma$) measures
chords as well as arcs in degrees, minutes, and seconds---the
degree of chord being the 60th part of the radius as the degree of
arc is the 60th part of the arc subtended by a chord equal to the
radius.

The sexagesimal fraction held its own as the fraction \textit{par
excellence} for scientific computation until the 16th century,
when it was displaced by the decimal fraction in all uses except
the measurement of angles.

\addcontentsline{toc}{section}{\numberline{}Greek}

\textbf{93. Greek Fractions.} Fractions occur in Greek
writings---both mathematical and non-mathematical---much earlier
than Ptolemy, but not in arithmetic.\footnote{The usual method of
expressing fractions was to write the numerator with an accent,
and after it the denominator twice with a double accent:
\textit{e.~g.} $\displaystyle \iota\zeta^\prime~\kappa\alpha^{\prime\prime}~\kappa\alpha^{\prime\prime}
=\frac{17}{21}$.    Before sexagesimal fractions came into vogue
actual reckonings with fractions were effected by unit fractions,
of which only the denominators (doubly accented) were written.}
The Greeks drew as sharp a distinction between pure arithmetic,
$\grave{\alpha}\rho\iota\theta\mu\eta\tau\iota\kappa\Acute{\eta}$,
and the art of reckoning, $\lambda o
\gamma\iota\sigma\tau\iota\kappa\Acute{\eta}$, as between pure and
metrical geometry. The fraction was relegated to $\lambda o
\gamma\iota\sigma\tau\iota\kappa\Acute{\eta}$. There is no place
in a pure science for artificial concepts, no place, therefore,
for the fraction in
$\Acute{\alpha}\rho\iota\theta\mu\eta\tau\iota\kappa\Acute{\eta}$;
such was the Greek position. Thus, while the metrical
geometers---as Archimedes (250 \textsc{b.~c.}), in his ``Measure of the
Circle'' \, ($\kappa \Acute{\upsilon}\kappa\lambda o \upsilon$
$\mu\Acute{\epsilon}\tau\rho\eta\sigma\iota\varsigma$), and Hero
(120 \textsc{b.~c.})---employ fractions, neither of the treatises on Greek
arithmetic before Diophantus (300 \textsc{a.~d.~}) which have come down to
us---the 7th, 8th, 9th books of Euclid's ``Elements''
(300~\textsc{b.~c.}), and the ``Introduction to Arithmetic''
([$\epsilon\iota\sigma\alpha\gamma\omega\gamma\acute{\eta}\
  \alpha\rho\iota\theta\mu\eta\tau\iota\kappa\grave{\eta}$])
of Nicomachus (100~\textsc{a.~d.})---recognizes the fraction. They
do, it is true, recognize the fractional relation. Euclid, for
instance, expressly declares that any number is either a multiple,
a part, or parts ([$\mu\grave{\epsilon}\rho\eta$]), \textit{i.~e.}\
multiple of a part, of every other number (Euc.~VII,~4), and he
demonstrates such theorems as these:

\emph{If $A$ be the same parts of $B$ that $C$ is of $D$, then the
sum or difference of $A$ and $C$ is the same parts of the sum or
difference of $B$ and $D$ that $A$ is of $B$} (VII,~6 and~8).

\emph{If $A$ be the same parts of $B$ that $C$ is of $D$, then,
alternately, $A$ is the same parts of $C$ that $B$ is of $D$}
(VII,~10).

But the relation is expressed by two integers, that which
indicates the part and that which indicates the multiple. It is a
ratio, and Euclid has no more thought of expressing it except by
\emph{two} numbers than he has of expressing the ratio of two
geometric magnitudes except by two magnitudes. There is no
conception of a single number, the fraction proper, the quotient
of one of these integers by the other.

In the $\alpha\rho\iota\theta\mu\eta\tau\iota\kappa\grave{\alpha}$
of Diophantus, on the other hand, the last and transcendently the
greatest achievement of the Greeks in the science of number, the
fraction is granted the position in elementary arithmetic which it
has held ever since.

\chapter{ORIGIN OF THE IRRATIONAL.}

\addcontentsline{toc}{section}{\numberline{}Discovery of
irrational lines. Pythagoras}

\textbf{94. The Discovery of Irrational Lines.} The Greeks
attributed the discovery of the Irrational to the mathematician
and philosopher Pythagoras\footnote{\label{Summary of the history
of Greek mathematics}This is the explicit declaration of the most
reliable document extant on the history of geometry before Euclid,
a chronicle of the ancient geometers which Proclus (\textsc{a.~d.~} 450) gives
in his commentary on Euclid, deriving it from a history written by
Eudemus about 330 \textsc{b.~c.} This chronicle credits the Egyptians with
the discovery of geometry and Thales (600 \textsc{b.~c.~}) with having first
introduced this study into Greece.

Thales and Pythagoras are the founders of the Greek mathematics.
But while Thales should doubtless be credited with the first
conception of an abstract deductive geometry in contradistinction
to the practical empirical geometry of Egypt, the glory of
realizing this conception belongs chiefly to Pythagoras and his
disciples in the Greek cities of Italy (Magna Gr\ae cia); for they
established the principal theorems respecting rectilineal figures.
To the Pythagoreans the discovery of many of the elementary
properties of numbers is due, as well as the geometric form which
characterized the Greek theory of numbers throughout its history.

In the middle of the fifth century before Christ Athens became the
principal centre of mathematical activity. There Hippocrates of
Chios (430 \textsc{b.~c.~}) made his contributions to the geometry of the
circle, Plato (380 \textsc{b.~c.}) to geometric method, The\ae tetus (380
\textsc{b.~c.}) to the doctrine of incommensurable magnitudes, and Eudoxus
(360 \textsc{b.~c.}) to the theory of proportion. There also was begun the
study of the conics.

About 300 \textsc{b.~c.~} the mathematical centre of the Greeks shifted to
Alexandria, where it remained.

The third century before Christ is the most brilliant period in
Greek mathematics. At its beginning---in Alexandria---Euclid lived
and taught and wrote his Elements, collecting, systematizing, and
perfecting the work of his predecessors. Later (about 250)
Archimedes of Syracuse flourished, the greatest mathematician of
antiquity and founder of the science of mechanics; and later still
(about 230) Apollonius of Perga, ``the great geometer,'' \, whose
Conics marks the culmination of Greek geometry.

Of the later Greek mathematicians, besides Hero and Diophantus, of
whom an account is given in the text, and the great summarizer of
the ancient mathematics, Pappus (300 \textsc{a.~d.}), only the famous
astronomers Hipparchus (130 \textsc{b.~c.}) and Ptolemy (150 \textsc{a.~d.~}) call for
mention here. To them belongs the invention of trigonometry and
the first trigonometric tables, tables of chords.

The dates in this summary are from Gow's Hist.\ of Greek Math.}
(525~\textsc{b.~c.}).

If, as is altogether probable,\footnote{Compare Cantor, Geschichte
der Mathematik, p.~153.} the most famous theorem of
Pythagoras---that \textit{the square on the hypothenuse of a right
triangle is equal to the sum of the squares on the other two
sides}---was suggested to him by the fact that $\displaystyle 3^2
+4^2 = 5^2$, in connection with the fact that the triangle whose
sides are 3, 4, 5, is right-angled,---for both almost certainly
fell within the knowledge of the Egyptians,---he would naturally
have sought, after he had succeeded in demonstrating the geometric
theorem generally, for number triplets corresponding to the sides
of any right triangle as do 3, 4, 5 to the sides of the particular
triangle.

The search of course proved fruitless, fruitless even in the case
which is geometrically the simplest, that of the isosceles right
triangle. To discover that it was \textit{necessarily} fruitless;
in the face of preconceived ideas and the apparent testimony of
the senses, to conceive that lines may exist which have no common
unit of measure, however small that unit be taken; to demonstrate
that the hypothenuse and side of the isosceles right triangle
actually are such a pair of lines, was the great achievement of
Pythagoras.\footnote{\label{Old Greek demonstration that the side
and diagonal of a square are incommensurable}His demonstration may
easily have been the following, which was old enough in
Aristotle's time (340 \textsc{b.~c.}) to be made the subject of a popular
reference, and which is to be found at the end of the 10th book in
all old editions of Euclid's Elements:

If there be any line which the side and diagonal of a square both
contain an exact number of times, let their lengths in terms of
this line be $a$ and $b$ respectively; then $b^2=2a^2$.

The numbers $a$ and $b$ may have a common factor, $\gamma$; so
that $a=\alpha\gamma$ and $b=\beta\gamma$, where $\alpha$ and
$\beta$ are prime to each other. The equation $b^2=2a^2$ then
reduces, on the removal of the factor $\gamma^2$ common to both
its members, to $\beta^2=2\alpha^2$.

From this equation it follows that $\beta^2$, and therefore
$\beta$, is an even number, and hence that $\alpha$ which is prime
to $\beta$ is odd.

But set $\beta=2\beta'$, where $\beta'$ is integral, in the
equation $\beta^2=2\alpha^2$; it becomes $4\beta'^2=2\alpha^2$, or
$2\beta'^2=\alpha^2$, whence $\alpha^2$, and therefore $\alpha$,
is even.

$\alpha$ has thus been proven to be both odd and even, and is
therefore not a number.}

\addcontentsline{toc}{section}{\numberline{}Consequences of this
discovery in Greek mathematics}

\textbf{95. Consequences of this Discovery in Greek Mathematics.}
One must know the antecedents and follow the consequences of this
discovery to realize its great significance. It was the first
recognition of the fundamental difference between the geometric
magnitudes and number, which Aristotle formulated brilliantly 200
years later in his famous distinction between the continuous and
the discrete, and as such was potent in bringing about that
complete banishment of numerical reckoning from geometry which is
so characteristic of this department of Greek mathematics in its
best, its creative period.

No one before Pythagoras had questioned the possibility of
expressing all size relations among lines and surfaces in terms of
number,---rational number of course. Indeed, except that it
recorded a few facts regarding congruence of figures gathered by
observation, the Egyptian geometry was nothing else than a meagre
collection of formulas for computing areas. The earliest geometry
was metrical.

But to the severely logical Greek no alternative seemed possible,
when once it was known that lines exist whose lengths---whatever
unit be chosen for measuring them---cannot both be integers, than
to have done with number and measurement in geometry altogether.
Congruence became not only the final but the sole test of
equality. For the study of size relations among unequal magnitudes
a pure geometric theory of proportion was created, in which
proportion, not ratio, was the primary idea, the method of
exhaustions making the theory available for figures bounded by
curved lines and surfaces.

The outcome was the system of geometry which Euclid expounds in
his Elements and of which Apollonius makes splendid use in his
Conics, a system absolutely free from extraneous concepts or
methods, yet, within its limits, of great power.

It need hardly be added that it never occurred to the Greeks to
meet the difficulty which Pythagoras' discovery had brought to
light by inventing an \textit{irrational number}, itself
incommensurable with rational numbers. For artificial concepts
such as that they had neither talent nor liking.

On the other hand, they did develop the theory of irrational
magnitudes as a department of their geometry, the irrational line,
surface, or solid being one incommensurable with some chosen
(rational) line, surface, solid. Such a theory forms the content
of the most elaborate book of Euclid's Elements, the 10th.

\addcontentsline{toc}{section}{\numberline{}Greek approximate
values of irrationals}

\textbf{96. Approximate Values of Irrationals.} In the practical
or metrical geometry which grew up after the pure geometry had
reached its culmination, and which attained in the works of Hero
the Surveyor almost the proportions of our modern elementary
mensuration,\footnote{The formula $\displaystyle
\sqrt{s(s-a)(s-b)(s-c)}$ for the area of a triangle in terms of
its sides is due to Hero.} \textit{approximate values} of
irrational numbers played a very important rôle. Nor do such
approximations appear for the first time in Hero. In Archimedes'
``Measure of the Circle'' \, a number of excellent approximations
occur, among them the famous approximation $\displaystyle
\frac{22}{7}$ for $\pi$, the ratio of the circumference of a
circle to its diameter. The approximation $\displaystyle
\frac{7}{5}$ for $\displaystyle \sqrt{2}$ is reputed to be as old
as Plato.

It is not certain how these approximations were
effected.\footnote{\label{Greek methods of approximation}Many
attempts have been made to discover the methods of approximation
used by Archimedes and Hero from an examination of their results,
but with little success. The formula $\displaystyle \sqrt{a^2\pm
b}=a\pm\frac{b}{2a}$ will account for some of the simpler
approximations, but no single method or set of methods have been
found which will account for the more difficult. See G\"{u}nther:
Die quadratischen Irrationalit\"{a}ten der Alten und deren
Entwicklungsmethoden. Leipzig, 1882. Also in Handbuch der
klassischen Altertums-Wissenschaft, 11ter. Halbband.} They involve
the use of some method for extracting square roots. The earliest
explicit statement of the method in common use to-day for
extracting square roots of numbers (whether exactly or
approximately) occurs in the commentary of Theon of Alexandria
(380 \textsc{a.~d.~}) on Ptolemy's \textit{Almagest}. Theon, who like Ptolemy
employs sexagesimal fractions, thus finds the length of the side
of a square containing $4500^\circ$ to be $67^\circ 1'
55^{\prime\prime}$.

\textbf{97. The Later History of the Irrational} is deferred to
the chapters which follow (\S\S~106, 108, 112, 121, 129).

It will be found that the Indians permitted the simplest forms of
irrational numbers, surds, in their algebra, and that they were
followed in this by the Arabians and the mathematicians of the
Renaissance, but that the general irrational did not make its way
into algebra until after Descartes.


\chapter{ORIGIN OF THE NEGATIVE AND THE IMAGINARY\@. THE EQUATION.}

\addcontentsline{toc}{section}{\numberline{}The equation in
Egyptian mathematics}

\textbf{98. The Equation in Egyptian Mathematics.} While the
irrational originated in geometry, the negative and the imaginary
are of purely algebraic origin. They sprang directly from the
algebraic equation.

The authentic history of the equation, like that of geometry and
arithmetic, begins in the book of the old Egyptian scribe Ahmes.
For Ahmes, quite after the present method, solves numerical
problems which admit of statement in an equation of the first
degree involving one unknown quantity.\footnote{His symbol for the
unknown quantity is the word \textit{hau}, meaning heap.}

\addcontentsline{toc}{section}{\numberline{}In the earlier Greek
mathematics}

\textbf{99. In the Earlier Greek Mathematics.} The equation was
slow in arousing the interest of Greek mathematicians. They were
absorbed in geometry, in a geometry whose methods were essentially
non-algebraic.

To be sure, there are occasional signs of a concealed algebra
under the closely drawn geometric cloak. Euclid solves three
geometric problems which, stated algebraically, are but the three
forms of the quadratic; $x^2+ax =b^2$, $x^2 = ax+b^2$, $x^2 + b^2
= ax$.\footnote{Elements, VI, 29, 28; Data, 84, 85.} And the
Conics of Apollonius, so astonishing if regarded as a product of
the pure geometric method used in its demonstrations, when stated
in the language of algebra, as recently it has been stated by
Zeuthen,\footnote{Die Lehre von den Kegelschnitten im Altertum.
Copenhagen, 1886.} almost convicts its author of the use of
algebra as his instrument of investigation.

\addcontentsline{toc}{section}{\numberline{}Hero of Alexandria}

\textbf{100. Hero.} But in the writings of Hero of Alexandria (120
\textsc{b.~c.}) the equation first comes clearly into the light
again. Hero was a man of practical genius whose aim was to make
the rich pure geometry of his predecessors available for the
surveyor. With him the rigor of the old geometric method is
relaxed; proportions, even equations, among the \textit{measures}
of magnitudes are permitted where the earlier geometers allow only
proportions among the magnitudes themselves; the theorems of
geometry are stated metrically, in formulas; and more than all
this, the equation becomes a recognized geometric instrument.

Hero gives for the diameter of a circle in terms of $s$, the sum
of diameter, circumference, and area, the formula:\footnote{See
Cantor; Geschichte der Mathematik, p.~341.}
\[
d=\frac{\sqrt{154s+841}-29}{11}
\]
He could have reached this formula only by \textit{solving a
quadratic equation}, and that not geometrically,---the nature of
the oddly constituted quantity $s$ precludes that
supposition,---but by a purely algebraic reckoning like the
following:

The area of a circle in terms of its diameter being $\displaystyle
\frac{\pi d^2}{4}$, the  length of its circumference $\pi d$, and
$\pi$ according to Archimedes' approximation $\displaystyle
\frac{22}{7}$, we have the equation:
\[
s = d+\frac{\pi d^2}{4}+\pi d, \; \text{ or } \;
\frac{11}{14}d^2+\frac{29}{7}d=s. \]

Clearing of fractions, multiplying by 11, and completing the
square,
\[
121 d^2 + 638d+ 841 = 154 s + 841,
\]
whence
\[
11 d + 29 =\sqrt{154 s + 841},
\]
or
\[d=\frac{\sqrt{154s+841}-29}{11}.
\]

Except that he lacked an algebraic symbolism, therefore, Hero was
an algebraist, an algebraist of power enough to solve an affected
quadratic equation.

\addcontentsline{toc}{section}{\numberline{}Diophantus of
Alexandria}

\textbf{101. Diophantus.} (300 \textsc{a.~d.}?). The last of the
Greek mathematicians, Diophantus of Alexandria, was a great
algebraist.

The period between him and Hero was not rich in creative
mathematicians, but it must have witnessed a gradual development
of algebraic ideas and of an algebraic symbolism.

At all events, in the
$\Grave{\alpha}\rho\iota\theta\mu\eta\tau\iota\kappa\Acute{\alpha}$
of Diophantus the algebraic equation has been supplied with a
symbol for the unknown quantity, its powers and the powers of its
reciprocal to the 6th, and a symbol for equality. Addition is
represented by mere juxtaposition, but there is a special symbol,
see Figure A, for subtraction. On the other hand, there are
no general symbols for known quantities,---symbols to serve the
purpose which the first letters of the alphabet are made to serve
in elementary algebra nowadays,---therefore no literal
coefficients and no general formulas.

\begin{figure}[htbp]
\centering \includegraphics[scale=0.5]{images/symbol.eps}\\
\textsc{Fig. A.}
\end{figure}

With the symbolism had grown up many of the formal rules of
algebraic reckoning also. Diophantus prefaces the
$\alpha\rho\iota\theta\mu\eta\tau\iota\kappa\grave{\alpha}$ with
rules for the addition, subtraction, and multiplication of
polynomials. He states expressly that the product of two
subtractive terms is additive.

The $\alpha\rho\iota\theta\mu\eta\tau\iota\kappa\grave{\alpha}$
itself is a collection of problems concerning numbers, some of
which are solved by determinate algebraic equations, some by
indeterminate.

Determinate equations are solved which have given positive
integers as coefficients, and are of any of the forms $ax^m =
bx^n$, $ax^2 + bx = c$, $ax^2 +c = bx$, $ax^2 = bx + c$; also a
single cubic equation, $ax^3 + x = 4x^2 + 4$. In reducing
equations to these forms, equal quantities in opposite members are
cancelled and subtractive terms in either member are rendered
additive by transposition to the other member.

The indeterminate equations are of the form $y^2 = ax^2 + bx + c$,
Diophantus regarding any pair of positive \textit{rational}
numbers (integers or fractions) as a solution which, substituted
for $y$ and $x$, satisfy the equation.\footnote{\label{Diophantine
equations}The designation ``Diophantine equations,'' commonly
applied to indeterminate equations of the first degree when
investigated for integral solutions, is a striking misnomer.
Diophantus nowhere considers such equations, and, on the other
hand, allows fractional solutions of indeterminate equations of
the second degree.} These equations are handled with marvellous
dexterity in the
$\alpha\rho\iota\theta\mu\eta\tau\iota\kappa\grave{\alpha}$. No
effort is made to develop general comprehensive methods, but each
exercise is solved by some clever device suggested by its
individual peculiarities. Moreover, the discussion is never
exhaustive, one solution sufficing when the possible number is
infinite. Yet until some trace of indeterminate equations earlier
than the
$\alpha\rho\iota\theta\mu\eta\tau\iota\kappa\grave{\alpha}$ is
discovered, Diophantus must rank as the originator of this
department of mathematics.

The determinate quadratic is solved by the method which we have
already seen used by Hero. The equation is first multiplied
throughout by a number which renders the coefficient of $x^2$ a
perfect square, the ``square is completed,'' the square root of
both members of the equation taken, and the value of $x$ reckoned
out from the result. Thus from $ax^2+c=bx$ is derived first the
equation
\begin{align*}
a^2x^2+ac&=abx,\\
\mbox{then} \quad a^2x^2 - abx +\left(\frac{b}{2}\right)^2 &= \left(\frac{b}{2}\right)^2 - ac,\\
\mbox{then} \quad ax-\frac{b}{2} &= \sqrt{\left(\frac{b}{2}\right)^2-ac},\\
\mbox{and finally,} \quad
x&=\frac{\frac{b}{2}+\sqrt{\left(\frac{b}{2}\right)^2-ac}}{a}.
\end{align*}

The solution is regarded as possible only when the number under
the radical is a perfect square (it must, of course, be positive),
and only one root---that belonging to the positive value of the
radical---is ever recognized.

Thus the number system of Diophantus contained only the positive
integer and fraction; the irrational is excluded; and as for the
negative, there is no evidence that a Greek mathematician ever
conceived of such a thing,---certainly not Diophantus with his
three classes and one root of affected quadratics. The position of
Diophantus is the more interesting in that in the
$\alpha\rho\iota\theta\mu\eta\tau\iota\kappa\grave{\alpha}$ the
Greek science of number culminates.

\addcontentsline{toc}{section}{\numberline{}The Indian
mathematics. Âryabha\d{t}\d{t}a, Brahmagupta, Bhâskara}

\textbf{102. The Indian Mathematics.} The pre-eminence in
mathematics passed from the Greeks to the Indians. Three
mathematicians of India stand out above the rest:
\emph{Âryabha\d{t}\d{t}a} (born 476 \textsc{a.~d.}), \emph{Brahmagupta}
(born 598 \textsc{a.~d.}), \emph{Bhâskara} (born 1114 \textsc{a.~d.}) While all are
in the first instance astronomers, their treatises also contain
full expositions of the mathematics auxiliary to astronomy, their
reckoning, algebra, geometry, and trigonometry.\footnote{The
mathematical chapters of Brahmagupta and Bhâskara have been
translated into English by Colebrooke: ``Algebra, Arithmetic, and
Mensuration, from the Sanscrit of Brahmagupta and Bhâskara,''
1817; those of Âryabha\d{t}\d{t}a into French by L. Rodet (Journal
Asiatique, 1879).}

An examination of the writings of these mathematicians and of the
remaining mathematical literature of India leaves little room for
doubt that the Indian geometry was taken bodily from Hero, and the
algebra---whatever there may have been of it before
\^Aryabha\d{t}\d{t}a---at least powerfully affected by Diophantus.
Nor is there occasion for surprise in this. \^Aryabha\d{t}\d{t}a
lived two centuries after Diophantus and six after Hero, and
during those centuries the East had frequent communication with
the West through various channels. In particular, from Trajan's
reign till later than 300~\textsc{a.~d.~} an active commerce was
kept up between India and the east coast of Egypt by way of the
Indian Ocean.

Greek geometry and Greek algebra met very different fates in
India. The Indians lacked the endowments of the geometer. So far
from enriching the science with new discoveries, they seem with
difficulty to have kept alive even a proper understanding of
Hero's metrical formulas. But algebra flourished among them
wonderfully. Here the fine talent for reckoning which could create
a perfect numeral notation, supported by a talent equally fine for
symbolical reasoning, found a great opportunity and made great
achievements. With Diophantus algebra is no more than an art by
which disconnected numerical problems are solved; in India it
rises to the dignity of a science, with general methods and
concepts of its own.

\addcontentsline{toc}{section}{\numberline{}Its algebraic
symbolism}

\textbf{103. Its Algebraic Symbolism.} First of all, the Indians
devised a complete, and in most respects adequate, symbolism.
Addition was represented, as by Diophantus, by mere juxtaposition;
subtraction, exactly as addition, except that a dot was written
over the coefficient of the subtrahend. The syllable \textit{bha}
written after the factors indicated a product; the divisor written
under the dividend, a quotient; a syllable, \textit{ka}, written
before a number, its (irrational) square root; one member of an
equation placed over the other, their equality. The equation was
also provided with symbols for any number of unknown quantities
and their powers.

\addcontentsline{toc}{section}{\numberline{}Its invention of the
negative}

\textbf{104. Its Invention of the Negative.} The most note-worthy
feature of this symbolism is its representation of subtraction. To
remove the subtractive symbol from between minuend and subtrahend
(where Diophantus had placed his symbol, see Figure A.) to attach it wholly to
the subtrahend
and then connect this modified subtrahend with the minuend
additively, is, formally considered, to transform the subtraction
of a positive quantity into the addition of the corresponding
negative. It suggests what other evidence makes certain, that
\textit{algebra owes to India the immensely useful concept of the
absolute negative.}

Thus one of these dotted numbers is allowed to stand by itself as
a member of an equation. Bh\^{a}skara recognizes the double sign
of the square root, as well as the impossibility of the square
root of a negative number (which is very interesting, as being the
first dictum regarding the imaginary), and no longer ignores
either root of the quadratic. More than this, recourse is had to
the same expedients for interpreting the negative, for attaching a
concrete physical idea to it, as are in common use to-day. The
primary meaning of the very name given the negative was
\textit{debt}, as that given the positive was \textit{means}. The
opposition between the two was also pictured by lines described in
opposite directions.

\addcontentsline{toc}{section}{\numberline{}Its use of zero}

\textbf{105. Its Use of Zero.} But the contributions of the
Indians to the fund of algebraic concepts did not stop with the
absolute negative.

They made a number of 0, and though some of their reckonings with
it are childish, Bh\^{a}skara, at least, had sufficient
understanding of the nature of the ``quotient'' \, $\displaystyle
\frac{a}{0}$ (infinity) to say ``it suffers no change, however
much it is increased or diminished.'' \, He associates it with
Deity.

\addcontentsline{toc}{section}{\numberline{}Its use of irrational
numbers}

\textbf{106. Its Use of Irrational Numbers.} Again, the Indians
were the first to reckon with irrational square roots as with
numbers; Bhâskara extracting square roots of binomial surds and
rationalizing irrational denominators of fractions even when these
are polynomial. Of course they were as little able rigorously to
justify such a procedure as the Greeks; less able, in fact, since
they had no equivalent of the method of exhaustions. But it
probably never occurred to them that justification was necessary;
they seem to have been unconscious of the gulf fixed between the
discrete and continuous. And here, as in the case of 0 and the
negative, with the confidence of apt and successful reckoners,
they were ready to pass immediately from numerical to purely
symbolical reasoning, ready to trust their processes even where
formal demonstration of the right to apply them ceased to be
attainable. Their skill was too great, their instinct too true, to
allow them to go far wrong.

\addcontentsline{toc}{section}{\numberline{}Its treatment of
determinate and indeterminate equations}

\textbf{107. Determinate and Indeterminate Equations in Indian
Algebra.} As regards equations---the only changes which the Indian
algebraists made in the treatment of determinate equations were
such as grew out of the use of the negative. This brought the
triple classification of the quadratic to an end and secured
recognition for both roots of the quadratic.

Brahmagupta solves the quadratic by the rule of Hero and
Diophantus, of which he gives an explicit and general statement.
Çrîdhara, a mathematician of some distinction belonging to the
period between Brahmagupta and Bhâskara, made the improvement of
this method which consists in first multiplying the equation
throughout by four times the coefficient of the square of the
unknown quantity and so preventing the occurrence of fractions
under the radical sign.\footnote{This method still goes under the
name ``Hindoo method.''}

Bhâskara also solves a few cubic and biquadratic equations by
special devices.

The theory of indeterminate equations, on the other hand, made
great progress in India. The achievements of the Indian
mathematicians in this beautiful but difficult department of the
science are as brilliant as those of the Greeks in geometry. They
created the doctrine of the indeterminate equation of the first
degree, $ax + by = c$, which they treated for integral solutions
by the method of continued fractions in use to-day. They worked
also with equations of the second degree of the forms $ax^2 + b =
cy^2$, $xy = ax + by + c$, originating general and comprehensive
methods where Diophantus had been content with clever devices.

\addcontentsline{toc}{section}{\numberline{}The Arabian
mathematics. Alkhwarizmî, Alkarchî, Alchayyâmî}

\textbf{108. The Arabian Mathematics.} The Arabians were the
instructors of modern Europe in the ancient mathematics. The
service which they rendered in the case of the numeral notation
and reckoning of India they rendered also in the case of the
geometry, algebra, and astronomy of the Greeks and Indians. Their
own contributions to mathematics are unimportant. Their
receptiveness for mathematical ideas was extraordinary, but they
had little originality.

The history of Arabian mathematics begins with the reign of
Alman\d{s}ûr (754--775),\footnote{It was Alman\d{s}ûr who
transferred the throne of the caliphs from Damascus to Bagdad
which immediately became not only the capital city of Islam, but
its commercial and intellectual centre.} the second of the Abbasid
caliphs.

It is related (by Ibn-al-Adamî, about 900) that in this reign, in
the year 773, an Indian brought to Bagdad certain astronomical
writings of his country, which contained a method called
``Sindhind,'' for computing the motions of the stars,---probably
portions of the Siddhânta of Brahmagupta,---and that Alfazârî was
commissioned by the caliph to translate them into
Arabic.\footnote{This translation remained the guide of the
Arabian astronomers until the reign of Almamûn (813--833), for whom
Alkhwarizmî prepared his famous astronomical tables (820). Even
these were based chiefly on the ``Sindhind,'' though some of the
determinations were made by methods of the Persians and Ptolemy.}
Inasmuch as the Indian astronomers put full expositions of their
reckoning, algebra, and geometry into their treatises, Alfazârî's
translation laid open to his countrymen a rich treasure of
mathematical ideas and methods.

It is impossible to set a date to the entrance of Greek ideas.
They must have made themselves felt at Damascus, the residence of
the later Omayyad caliphs, for that city had numerous inhabitants
of Greek origin and culture. But the first translations of Greek
mathematical writings were made in the reign of Hârûn Arraschîd
(786--809), when Euclid's Elements and Ptolemy's Almagest were put
into Arabic. Later on, translations were made of Archimedes,
Apollonius, Hero, and last of all, of Diophantus (by Abû'l Wafâ,
940--998).

The earliest mathematical author of the Arabians is Alkhwarizmî,
who flourished in the first quarter of the 9th century. Besides
astronomical tables, he wrote a treatise on algebra and one on
reckoning (elementary arithmetic). The latter has already been
mentioned. It is an exposition of the positional reckoning of
India, the reckoning which mediæval Europe named after him
\textit{Algorithm}.

The treatise on algebra bears a title in which the word
\textit{Algebra} appears for the first time: viz., \textit{Aldjebr
walmukâbala}. Aldjebr (\textit{i.~e.} reduction) signifies the
making of all terms of an equation positive by transferring
negative terms to the opposite member of the equation;
\textit{almukâbala} (\textit{i.~e.} opposition), the cancelling of
equal terms in opposite members of an equation.

Alkhwarizmî's classification of equations of the 1st and 2d
degrees is that to which these processes would naturally lead,
viz.:

\[
\begin{array}{lll}
ax^2 = bx, & bx^2 = c, & bx = c,\\

x^2 + bx = c, & x^2 + c = bx, & x^2 = bx + c.
\end{array}
\]

These equations he solves separately, following up the solution in
each case with a geometric demonstration of its correctness. He
recognizes both roots of the quadratic when they are positive. In
this respect he is Indian; in all others---the avoidance of
negatives, the use of geometric demonstration---he is Greek.

Besides Alkhwarizmî, the most famous algebraists of the Arabians
were \textit{Alkarchî} and \textit{Alchayyâmî}, both of whom lived
in the 11th century.

Alkarchî gave the solution of equations of the forms:

\[ax^{2p}+bx^p=c, ax^{2p}+c=bx^p, bx^p+c=ax^{2p}.\]

He also reckoned with irrationals, the equations

\[\sqrt{8}+\sqrt{18}=\sqrt{50}, \sqrt[3]{54}-\sqrt[3]{2}=\sqrt[3]{16},\]

being pretty just illustrations of his success in this field.

Alchayyâmî was the first mathematician to make a systematic
investigation of the cubic equation. He classified the various
forms which this equation takes when all its terms are positive,
and solved each form geometrically---by the intersections of
conics.\footnote{\label{Alchayyami method of solving cubics by the
intersections of conics}Thus suppose the equation $x^3+bx=a$,
given.

For $b$ substitute the quantity $p^2$, and for $a$, $p^2r$. Then
$x^3=p^3(r-x)$.

Now this equation is the result of eliminating $y$ from between
the two equations, $x^2=py$, $y^2=x(r-x)$; the first of which is
the equation of a parabola, the second, of a circle.

Let these two curves be constructed; they will intersect in one
real point distinct from the origin, and the abscissa of this
point is a root of $x^3+bx=a$. See Hankel, Geschichte der
Mathematik, p.~279.

This method is of greater interest in the history of geometry than
in that of algebra. It involves an anticipation of some of the
most important ideas of Descartes' \textit{Géométrie} (see p.~118).} A pure algebraic solution of the cubic he believed
impossible.

Like Alkhwarizmî, Alkarchî and Alchayyâmî were Eastern Arabians.
But early in the 8th century the Arabians conquered a great part
of Spain. An Arabian realm was established there which became
independent of the Bagdad caliphate in 747, and endured for 300
years. The intercourse of these Western Arabians with the East was
not frequent enough to exercise a controlling influence on their
æsthetic or scientific development. Their mathematical productions
are of a later date than those of the East and almost exclusively
arithmetico-algebraic. They constructed a formal algebraic
notation which went over into the Latin translations of their
writings and rendered the path of the Europeans to a knowledge of
the doctrine of equations easier than it would have been, had the
Arabians of the East been their only instructors. The best known
of their mathematicians are \textit{Ibn Aflah} (end of 11th
century), \textit{Ibn Albannâ} (end of 13th century),
\textit{Alkasâdî} (15th century).

\addcontentsline{toc}{section}{\numberline{}Arabian algebra Greek
rather than Indian}

\textbf{109. Arabian Algebra Greek rather than Indian.} Thus, of
the three greater departments of the Arabian mathematics, the
Indian influence gained the mastery in reckoning only.

The Arabian geometry is Greek through and through.

While the algebra contains both elements, the Greek predominates.
Indeed, except that both roots of the quadratic are recognized,
the doctrine of the determinate equation is altogether Greek. It
avoids the negative almost as carefully as Diophantus does; and in
its use of the geometric method of demonstration it is actuated by
a spirit less modern still---the spirit in which Euclid may have
conceived of algebra when he solved his geometric quadratics.

The theory of indeterminate equations seldom goes beyond
Diophantus; where it does, it is Indian.

The Arabian trigonometry is based on Ptolemy's, but is its
superior in two important particulars. It employs the sine where
Ptolemy employs the chord (being in this respect Indian), and has
an algebraic instead of a geometric form. Some of the methods of
approximation used in reckoning out trigonometric tables show
great cleverness. Indeed, the Arabians make some amends for their
ill-advised return to geometric algebra by this excellent
achievement in algebraic geometry.

The preference of the Arabians for Greek algebra was especially
unfortunate in respect to the negative, which was in consequence
forced to repeat in Europe the fight for recognition which it had
already won in India.

\addcontentsline{toc}{section}{\numberline{}Mathematics in Europe
before the twelfth century}

\textbf{110. Mathematics in Europe before the Twelfth Century.}
The Arabian mathematics found entrance to Christian Europe in the
12th century. During this century and the first half of the next a
good part of its literature was translated into Latin.

Till then the plight of mathematics in Europe had been miserable
enough. She had no better representatives than the Romans, the
most deficient in the sense for mathematics of all cultured
peoples, ancient or modern; no better literature than the
collection of writings on surveying known as the \textit{Codex
Arcerianus}, and the childish arithmetic and geometry of Boetius.

Prior to the 10th century, however, Northern Europe had not
sufficiently emerged from barbarism to call even this paltry
mathematics into requisition. What learning there was was confined
to the cloisters. Reckoning (\textit{computus}) was needed for the
Church calendar and was taught in the cloister schools established
by Alcuin (735--804) under the patronage of Charlemagne. Reckoning
was commonly done on the fingers. Not even was the multiplication
table generally learned. Reference would be made to a written copy
of it, as nowadays reference is made to a table of logarithms. The
Church did not need geometry, and geometry in any proper sense did
not exist.

\addcontentsline{toc}{section}{\numberline{}Gerbert}

\textbf{111. Gerbert.} But in the 10th century there lived a man
of true scientific interests and gifts, Gerbert,\footnote{See
§88.} Bishop of Rheims, Archbishop of Ravenna, and finally Pope
Sylvester II\@. In him are the first signs of a new life for
mathematics. His achievements, it is true, do not extend beyond
the revival of Roman mathematics, the authorship of a geometry
based on the \textit{Codex Arcerianus}, and a method for effecting
division on the abacus with apices. Yet these achievements are
enough to place him far above his contemporaries. His influence
gave a strong impulse to mathematical studies where interest in
them had long been dead. He is the forerunner of the intellectual
activity ushered in by the translations from the Arabic, for he
brought to life the feeling of the need for mathematics which
these translations were made to satisfy.

\addcontentsline{toc}{section}{\numberline{}Entrance of the
Arabian mathematics. Leonardo}

\textbf{112. Entrance of the Arabian Mathematics. Leonardo.} It
was the elementary branch of the Arabian mathematics which took
root quickest in Christendom---reckoning with nine digits and 0.

\textit{Leonardo} of Pisa---\textit{Fibonacci}, as he was also
called---did great service in the diffusion of the new learning
through his \textit{Liber Abaci} (1202 and 1228), a remarkable
presentation of the arithmetic and algebra of the Arabians, which
remained for centuries the fund from which reckoners and
algebraists drew and is indeed the foundation of the modern
science.

The four fundamental operations on integers and fractions are
taught after the Arabian method; the extraction of the square root
and the doctrine of irrationals are presented in their pure
algebraic form; quadratic equations are solved and applied to
quite complicated problems; \textit{negatives are accepted when
they admit of interpretation as debt}.

The last fact illustrates excellently the character of the
\textit{Liber Abaci}. It is not a mere translation, but an
independent and masterly treatise in one department of the new
mathematics.

Besides the \textit{Liber Abaci}, Leonardo wrote the
\textit{Practica Geometriae}, which contains much that is best of
Euclid, Archimedes, Hero, and the elements of trigonometry; also
the \textit{Liber Quadratorum}, a collection of original algebraic
problems most skilfully handled.

\addcontentsline{toc}{section}{\numberline{}Mathematics during the
age of Scholasticism}

\textbf{113. Mathematics during the Age of Scholasticism.}
Leonardo was a great mathematician,\footnote{\label{Jordanus
Nemorarius}Besides Leonardo there flourished in the first quarter
of the 13th century an able German mathematician, \textit{Jordanus
Nemorarius}. He was the author of a treatise entitled \textit{De
numeris datis}, in which known quantities are for the first time
represented by letters, and of one \textit{De trangulis} which is
a rich though rather systemless collection of theorems and
problems principally of Greek and Arabian origin. See Günther:
Geschichte des mathemathischen Unterrichts im deutschen
Mittelalter, p.~156.} but fine as his work was, it bore no fruit
until the end of the 15th century. In him there had been a
brilliant response to the Arabian impulse. But the awakening was
only momentary; it quickly yielded to the heavy lethargy of the
``dark'' ages.

The age of scholasticism, the age of devotion to the forms of
thought, logic and dialectics, is the age of greatest dulness and
confusion in mathematical thinking.\footnote{\label{The summa of
Luca Pacioli}Compare Hankel, Geschichte der Mathematik, pp.~349--352. To the unfruitfulness of these centuries the
\textit{Summa} of \textit{Luca Pacioli} bears witness. This book,
which has the distinction of being the earliest book on algebra
printed, appeared in 1494, and embodies the arithmetic, algebra,
and geometry of the time just preceding the Renaissance. It
contains not an idea or method not already presented by Leonardo.
Even in respect to algebraic symbolism it surpasses the
\textit{Liber Abaci} only to the extent of using abbreviations for
a few frequently recurring words, as p.\ for ``plus,'' and R.\ for
``res'' (the unknown quantity). And this is not to be regarded as
original with Pacioli for the Arabians of Leonardo's time made a
similar use of abbreviations. In a translation made by Gerhard of
Cremona (12th century) from an unknown Arabic original the letters
\textit{r} (radix), $c$ (census), $d$ (dragma) are used to
represent the unknown quantity, its square, and the absolute term
respectively.

The \textit{Summa} of Pacioli has great merits, notwithstanding
its lack of originality. It satisfied the mathematical needs of
the time. It is very comprehensive, containing full and excellent
instruction in the art of reckoning after the methods of Leonardo,
for the merchant-man, and a great variety of matter of a purely
theoretical interest also---representing the elementary theory of
numbers, algebra, geometry, and the application of algebra to
geometry. Compare Cantor, Geschichte der Mathematik, II, p.~308.

\label{Regiomontanus}It should be added that the 15th century
produced a mathematician who deserves a distinguished place in the
general history of mathematics on account of his contributions to
trigonometry, the astronomer \textit{Regiomontanus} (1436--1476).
Like Jordanus, he was a German.} Algebra owes the entire period
but a single contribution; the concept of the fractional power.
Its author was Nicole Oresme (died 1382), who also gave a symbol
for it and the rules by which reckoning with it is governed.

\addcontentsline{toc}{section}{\numberline{}The Renaissance.
Solution of the cubic and biquadratic equations}

\textbf{114. The Renaissance. Solution of the Cubic and
Biquadratic Equations.} The first achievement in algebra by the
mathematicians of the Renaissance was the algebraic solution of
the cubic equation: a fine beginning of a new era in the history
of the science.

The cubic $x^3 + mx = n$ was solved by \textit{Ferro} of Bologna
in 1505, and a second time and independently, in 1535, by Ferro's
countryman, \textit{Tartaglia}, who by help of a transformation
made his method apply to $x^3 \pm mx^2 = \pm n$ also. But
\textit{Cardan} of Milan was the first to publish the solution, in
his \textit{Ars Magna},\footnote{The proper title of this work is:
``Artis magnae sive de regulis Algebraicis liber unus.'' It has
stolen the title of Cardan's ``Ars magna Arithmeticae,'' published
at Basel, 1570.} 1545.

The \textit{Ars Magna} records another brilliant discovery: the
solution---after a general method---of the biquadratic $x^4 + 6x^2
+ 36 = 60x$ by \textit{Ferrari}, a pupil of Cardan.

Thus in Italy, within fifty years of the new birth of algebra,
after a pause of sixteen centuries at the quadratic, the limits of
possible attainment in the algebraic solution of equations were
reached; for the algebraic solution of the general equation of a
degree higher than 4 is impossible, as was first demonstrated by
Abel.\footnote{Mémoire sur les Equations Algébriques: Christiania,
1826. Also in Crelle's Journal, I, p.~65.}

The general solution of higher equations proving an obstinate
problem, nothing was left the searchers for the roots of equations
but to devise a method of working them out approximately. In this
the French mathematician \textit{Vieta} (1540--1603) was
successful, his method being essentially the same as that now
known as Newton's.

\addcontentsline{toc}{section}{\numberline{}The negative in the
algebra of this period. First appearance of the imaginary}

\textbf{115. The Negative in the Algebra of this Period. First
Appearance of the Imaginary.} But the general equation presented
other problems than the discovery of rules for obtaining its
roots; the nature of these roots and the relations between them
and the coefficients of the equation invited inquiry.

We witness another phase of the struggle of the negative for
recognition. The imaginary is now ready to make common cause with
it.

Already in the \textit{Ars Magna} Cardan distinguishes between
\textit{numeri veri}---the positive integer, fraction, and
irrational,---and \textit{numeri ficti}, or \textit{falsi}---the
negative and the square root of the negative. Like Leonardo, he
tolerates negative roots of equations when they admit of
interpretation as ``debitum,'' not otherwise. While he has no
thought of accepting imaginary roots, he shows that if $5 +
\sqrt{-15}$ be substituted for $x$ in $x(10 - x) = 40$, that
equation is satisfied; which, of course, is all that is meant
nowadays when $5 + \sqrt{-15}$ is called a root. His declaration
that $5 \pm \sqrt{-15}$ are ``vere sophistica'' does not detract
from the significance of this, the earliest recorded instance of
reckoning with the imaginary. It ought perhaps to be added that
Cardan is not always so successful in these reckonings; for in
another place he sets

\[
\frac{1}{4}(-\sqrt{-\frac{1}{4}}) = \sqrt{\frac{1}{64}} =
\frac{1}{8}
\]

Following Cardan, \textit{Bombelli}\footnote{L'Algebra, 1579. He
also formally states rules for reckoning with $\pm \sqrt{-1}$ and
$a + b \sqrt{-1}$.} reckoned with imaginaries to good purpose,
explaining by their aid the irreducible case in Cardan's solution
of the cubic.

On the other hand, neither Vieta nor his distinguished follower,
the Englishman \textit{Harriot} (1560--1621), accept even negative
roots; though Harriot does not hesitate to perform algebraic
reckonings on negatives, and even allows a negative to constitute
one member of an equation.

\addcontentsline{toc}{section}{\numberline{}Algebraic symbolism.
Vieta and Harriot}

\textbf{116. Algebraic Symbolism. Vieta and Harriot.} Vieta and
Harriot, however, did distinguished service in perfecting the
symbolism of algebra; Vieta, by the systematic use of letters to
represent known quantities,---algebra first became ``literal'' or
``universal arithmetic'' in his hands,\footnote{\label{Algebraic
symbolism}There are isolated instances of this use of letters much
earlier than Vieta in the \textit{De numeris datis} of Jordanus
Nemorarius, and in the \textit{Algorithmus demonstratus} of the
same author. But the credit of making it the general practice of
algebraists belongs to Vieta.}---Harriot, by ridding algebraic
statements of every non-symbolic element, of everything but the
letters which represent quantities known as well as unknown,
symbols of operation, and symbols of relation. Harriot's
\textit{Artis Analyticae Praxis} (1631) has quite the appearance
of a modern algebra.\footnote{One has only to reflect how much of
the power of algebra is due to its admirable symbolism to
appreciate the importance of the \textit{Artis Analyticae Praxis},
in which this symbolism is finally established. But one addition
of consequence has since been made to it, integral and fractional
exponents introduced by Descartes (1637) and Wallis (1659).

Harriot substituted small letters for the capitals used by Vieta,
but followed Vieta in representing known quantities by consonants
and unknown by vowels. The present convention of representing
known quantities by the earlier letters of the alphabet, unknown
by the later, is due to Descartes.

Vieta's notation is unwieldy and ill adapted to purposes of
algebraic reckoning. Instead of restricting itself, as Harriot's
does, to the use of brief and easily apprehended conventional
symbols, it also employs words subject to the rules of syntax.
Thus for $A^3 - 3B^2A = Z$ (or $aaa - 3bba = z$, as Harriot would
have written it), Vieta writes \textit{A cubus - B quad 3 in A
aequatur Z solido}. In this respect Vieta is inferior not only to
Harriot, but to several of his predecessors and notably to his
contemporary, the Dutch mathematician Stevinus (1548--1620), who
would, for instance, have written $x^2 + 3x - 8$ as $1* + 3* -
8*$. The geometric affiliations of Vieta's notation are obvious.
It suggests the Greek arithmetic.

It is surprising that algebraic symbolism should owe so little to
the great Italian algebraists of the 16th century. Like Pacioli
(see note, p.~113) they were content with a few abbreviations for
words, a ``syncopated'' notation, as it has been called, and an
incomplete one at that.

The current symbols of operation and relation are chiefly of
English and German origin, having been invented or adopted as
follows: viz. $=$, by \textit{Recorde} in 1556; $\sqrt{}$, by
\textit{Rudolf} in 1525; the \textit{vinculum}, by \textit{Vieta}
in 1591; \textit{brackets}, by \textit{Bombelli}, 1572; $\div$, by
\textit{Rahn} in 1659; $\times ,
>, <,$ by \textit{Harriot} in 1631. The signs $+$ and $-$ occur in a
15th century manuscript discovered by Gerhardt at Vienna. The
notations $a - b$ and $\frac{a}{b}$ for the fraction were adopted
from the Arabians.}

\addcontentsline{toc}{section}{\numberline{}The fundamental
theorem of algebra. Harriot and Girard}

\textbf{117. Fundamental Theorem of Algebra. Harriot and Girard.}
Harriot has been credited with the discovery of the ``fundamental
theorem'' of algebra---the theorem that the number of roots of an
algebraic equation is the same as its degree. The \textit{Artis
Analyticae Praxis} contains no mention of this theorem---indeed,
by ignoring negative and imaginary roots, leaves no place for it;
yet Harriot develops systematically a method which, if carried far
enough, leads to the discovery of this theorem as well as to the
relations holding between the roots of an equation and its
coefficients.

By multiplying together binomial factors which involve the unknown
quantity, and setting their product equal to 0, he builds
``canonical'' equations, and shows that the roots of these
equations---the only roots, he says---are the positive values of
the unknown quantity which render these binomial factors 0. Thus
he builds $aa - ba - ca = -bc$, in which $a$ is the unknown
quantity, out of the factors $a - b, a + c$, and proves that $b$
is a root of this equation and the only root, the negative root
$c$ being totally ignored.

While no attempt is made to show that if the terms of a ``common''
equation be collected in one member, this can be separated into
binomial factors, the case of canonical equations raised a strong
presumption for the soundness of this view of the structure of an
equation.

The first statement of the fundamental theorem and of the
relations between coefficients and roots occurs in a remarkably
clever and modern little book, the \textit{Invention Nouvelle en
l'Algebre}, of \textit{Albert Girard}, published in Amsterdam in
1629, two years earlier, therefore, than the \textit{Artis
Anatyticae Praxis}. Girard stands in no fear of imaginary roots,
but rather insists on the wisdom of recognizing them. They never
occur, he says, except when real roots are lacking, and then in
number just sufficient to fill out the entire number of roots to
equality with the degree of the equation.

Girard also anticipated Descartes in the geometrical
interpretation of negatives. But the \textit{Invention Nouvelle}
does not seem to have attracted much notice, and the genius and
authority of Descartes were needed to give the interpretation
general currency.



\chapter{ACCEPTANCE OF THE NEGATIVE, THE GENERAL IRRATIONAL, AND THE
IMAGINARY AS NUMBERS.}

\addcontentsline{toc}{section}{\numberline{}Descartes'
\textit{Géométrie} and the negative}

\textbf{118. Descartes' Géométrie and the Negative.} The
\textit{Géométrie} of Descartes appeared in 1637. This famous
little treatise enriched geometry with a general and at the same
time simple and natural method of investigation: the method of
representing a geometric curve by an equation, which, as Descartes
puts it, expresses generally the relation of its points to those
of some chosen line of reference.\footnote{See Géométrie, Livre
II\@. In Cousin's edition of Descartes' works, Vol. V, p.~337.} To
form such equations Descartes represents line segments by
letters,---the known by $a, b, c,$ etc., the unknown by $x$ and
$y$. He supposes a perpendicular, $y$, to be dropped from any
point of the curve to the line of reference, and then the equation
to be found from the known properties of the curve which connects
$y$ with $x$, the distance of $y$ from a fixed point of the line
of reference. This is the equation of the curve in that it is
satisfied by the $x$ and $y$ of each and every
curve-point.\footnote{Descartes fails to recognize a number of the
conventions of our modern Cartesian geometry. He makes no formal
choice of two axes of reference, calls abscissas $y$ and ordinates
$x$, and as frequently regards as positive ordinates below the
axis of abscissas as ordinates above it.} To meet the difficulty
that the mere length of the perpendicular ($y$) from a curve-point
will not indicate to which side of the line of reference the point
lies, Descartes makes the convention that perpendiculars on
opposite sides of this line (and similarly intercepts ($x$) on
opposite sides of the point of reference) shall have opposite
algebraic signs.

This convention gave the negative a new position in mathematics.
Not only was a ``real'' interpretation here found for it, the lack
of which had made its position so difficult hitherto, but it was
made indispensable, placed on a footing of equality with the
positive. The acceptance of the negative in algebra kept pace with
the spread of Descartes' analytical method in geometry.

\addcontentsline{toc}{section}{\numberline{}Descartes' geometric
algebra}

\textbf{119. Descartes' Geometric Algebra.} But the
\textit{Géométrie} has another and perhaps more important claim on
the attention of the historian of algebra. The entire method of
the book rests on the assumption---made only tacitly, to be sure,
and without knowledge of its significance---that two algebras are
formally identical whose fundamental operations are formally the
same; \textit{i.~e.} subject to the same laws of combination.

For the algebra of the \textit{Géométrie} is not, as is commonly
said, mere numerical algebra, but what may for want of a better
name be called the algebra of line segments. Its symbolism is the
same as that of numerical algebra; but symbols which there
represent numbers here represent line segments. Not only is this
the case with the letters $a, b, x, y,$ etc., which are mere names
(\textit{noms}) of line segments, not their numerical measures,
but with the algebraic combinations of these letters. $a + b$ and
$a - b$ are respectively the sum and difference of the line
segments $a$ and $b$; $ab$, the fourth proportional to an assumed
unit line, $a$, and $b$; $\frac{a}{b}$, the fourth proportional to
$b, a,$ and the unit line; and $\sqrt{a}, \sqrt[3]{a}$, etc., the
first, second, etc., mean proportionals to the unit line and
$a$.\footnote{Géométrie, Livre I.\ Ibid.\ pp.~313--314.}

Descartes' justification of this use of the symbols of numerical
algebra is that the geometric constructions of which he makes $a +
b, a - b,$ etc., represent the results are ``the same'' as
numerical addition, subtraction, multiplication, division, and
evolution, respectively. Moreover, since all geometric
constructions which determine line segments may be resolved into
combinations of these constructions as the operations of numerical
algebra into the fundamental operations, the correspondence which
holds between these fundamental constructions and operations holds
equally between the more complex constructions and operations. The
entire system of the geometric constructions under consideration
may therefore be regarded as formally identical with the system of
algebraic operations, and be represented by the same symbolism.

In what sense his fundamental constructions are ``the same'' as
the fundamental operations of arithmetic, Descartes does not
explain. The true reason of their formal identity is that both are
controlled by the commutative, associative, and distributive laws.
Thus in the case of the former as of the latter, $ab = ba$, and
$a(bc) = abc$; for the fourth proportional to the unit line, $a$,
and $b$ is the same as the fourth proportional to the unit line,
$b$, and $a$; and the fourth proportional to the unit line, $a$,
and $bc$ is the same as the fourth proportional to the unit line,
$ab$, and $c$. But this reason was not within the reach of
Descartes, in whose day the fundamental laws of numerical algebra
had not yet been discovered.

\addcontentsline{toc}{section}{\numberline{}The continuous
variable. Newton. Euler}

\textbf{120. The Continuous Variable. Newton. Euler.} It is
customary to credit the \textit{Géométrie} with having introduced
the \textit{continuous variable} into mathematics, but without
sufficient reason. Descartes prepared the way for this concept,
but he makes no use of it in the \textit{Géométrie}. The $x$ and
$y$ which enter in the equation of a curve he regards not as
variables but as indeterminate quantities, a pair of whose values
correspond to each curve-point.\footnote{Géométrie, Livre II.
Ibid.\ pp.~337--338.} The real author of this concept is Newton
(1642--1727), of whose great invention, the method of fluxions,
continuous variation, ``flow,'' is the fundamental idea.

But Newton's calculus, like Descartes' algebra, is geometric
rather than purely numerical, and his followers in England, as
also, to a less extent, the followers of his great rival,
Leibnitz, on the continent, in employing the calculus, for the
most part conceive of variables as lines, not numbers. The
geometric form again threatened to become paramount in
mathematics, and geometry to enchain the new ``analysis'' as it
had formerly enchained the Greek arithmetic. It is the great
service of \textit{Euler} (1707--1783) to have broken these fetters
once for all, to have accepted the \textit{continuously variable
number} in its purity, and therewith to have created the pure
analysis. For the relations of continuously variable numbers
constitute the field of the pure analysis; its central concept,
the \textit{function}, being but a device for representing their
interdependence.

\addcontentsline{toc}{section}{\numberline{}The general
irrational}

\textbf{121. The General Irrational.} While its concern with
variables puts analysis in a certain opposition to elementary
algebra, concerned as this is with constants, its establishment of
the continuously variable number in mathematics brought about a
rich addition to the number-system of algebra---the
\textit{general irrational}. Hitherto the only irrational numbers
had been ``surds,'' impossible roots of rational numbers;
henceforth their domain is as wide as that of all possible lines
incommensurable with any assumed unit line.

\addcontentsline{toc}{section}{\numberline{}The imaginary, a
recognized analytical instrument}

\textbf{122. The Imaginary, a Recognized Analytical Instrument.}
Out of the excellent results of the use of the negative grew a
spirit of toleration for the imaginary. Increased attention was
paid to its properties. Leibnitz noticed the real sum of conjugate
imaginaries (1676--7); Demoivre discovered (1730) the famous
theorem
\[
(\cos \theta + i \sin \theta)^{n} = \cos n\theta + i \sin n\theta;
\]
and Euler (1748) the equation
\[
\cos \theta + i \sin \theta = e^{i\theta},
\]
which plays so great a rôle in the modern theory of functions.

Euler also, practising the method of expressing complex numbers in
terms of modulus and angle, formed their products, quotients,
powers, roots, and logarithms, and by many brilliant discoveries
multiplied proofs of the power of the imaginary as an analytical
instrument.

\addcontentsline{toc}{section}{\numberline{}Argand's geometric
representation of the imaginary}

\textbf{123. Argand's Geometric Representation of the Imaginary.}
But the imaginary was never regarded as anything better than an
algebraic fiction---to be avoided, where possible, by the
mathematician who prized purity of method---until a method was
discovered for representing it geometrically. A Norwegian,
\textit{Wessel},\footnote{See W.~W.~Beman in Proceedings of the
American Association for the Advancement of Science, 1897.}
published such a method in 1797, and a Frenchman, \textit{Argand},
the same method independently in 1806.

As +1 and -1 may be represented by unit lines drawn in opposite
directions from any point, $O$, and as $i$ (\textit{i.~e.}
$\sqrt{-1}$) is a mean proportional to +1 and -1, it occurred to
Argand to represent this symbol by the line whose direction with
respect to the line +1 is the same as the direction of the line -1
with respect to it; viz., the unit perpendicular through $O$ to
the 1-line. Let only the \textit{direction} of the 1-line be
fixed, the position of the point $O$ in the plane is altogether
indifferent.

Between the segments of a given line, whether taken in the same or
opposite directions, the equation holds:
\[AB+BC=AC.\]
It means nothing more, however, when the directions of $AB$ and
$BC$ are opposite, than that the result of carrying a moving point
from $A$ first to $B$, and thence back to $C$, is the same as
carrying it from $A$ direct to $C$. But in this sense the equation
holds equally when $A, B, C$ are not in the same right line.

Given, therefore, a complex number, $a+ib$; choose any point $A$
in the plane; from it draw a line $AB$, of length $a$, in the
direction of the 1-line, and from $B$ a line $BC$, of length $b$,
in the direction of the $i$-line. The line $AC$, thus fixed in
length and direction, but situated anywhere in the plane, is
Argand's picture of $a+ib$.

Argand's skill in the use of his new device was equal to the
discovery of the demonstration given in §54, that every algebraic
equation has a root.

\addcontentsline{toc}{section}{\numberline{}Gauss. The complex
number}

\textbf{124. Gauss. The Complex Number.} The method of
representing complex numbers in common use to-day, that described
in §42, is due to Gauss. He was already in possession of it in
1811, though he published no account of it until 1831.

To Gauss belongs the conception of $i$ as an independent unit
co-ordinate with 1, and of $a+ib$ as a \textit{complex} number, a
sum of multiples of the units 1 and $i$; his also is the name
``complex number'' and the concept of complex numbers in general,
whereby $a + ib$ secures a footing in the theory of numbers as
well as in algebra.

He too, and not Argand, must be credited with really breaking down
the opposition of mathematicians to the imaginary. Argand's
\textit{Essai} was little noticed when it appeared, and soon
forgotten; but there was no withstanding the great authority of
Gauss, and his precise and masterly presentation of this
doctrine.\footnote{See Gauss, Complete Works, II, p.~174.}




\chapter{RECOGNITION OF THE PURELY SYMBOLIC CHARACTER OF ALGEBRA\@.
QUATERNIONS\@. AUSDEHNUNGSLEHRE.}

\addcontentsline{toc}{section}{\numberline{}The principle of
permanence. Peacock }

\textbf{125. The Principle of Permanence.} Thus, one after
another, the fraction, irrational, negative, and imaginary, gained
entrance to the number-system of algebra. Not one of them was
accepted until its correspondence to some actually existing thing
had been shown, the fraction and irrational, which originated in
relations among actually existing things, naturally making good
their position earlier than the negative and imaginary, which grew
immediately out of the equation, and for which a ``real''
interpretation had to be sought.

Inasmuch as this correspondence of the artificial numbers to
things extra-arithmetical, though most interesting and the reason
of the practical usefulness of these numbers, has not the least
bearing on the nature of their position in \textit{pure}
arithmetic or algebra; after all of them had been accepted as
numbers, the necessity remained of justifying this acceptance by
purely algebraic considerations. This was first accomplished,
though incompletely, by the English mathematician,
\textit{Peacock}.\footnote{Arithmetical and Symbolical Algebra,
1830 and 1845; especially the later edition. Also British
Association Reports, 1833.}

Peacock begins with a valuable distinction between
\textit{arithmetical} and \textit{symbolical} algebra. Letters are
employed in the former, but only to represent positive integers
and fractions, subtraction being limited, as in ordinary
arithmetic, to the case where subtrahend is less than minuend. In
the latter, on the other hand, the symbols are left altogether
general, untrammelled at the outset with any particular meanings
whatsoever.

It is then \textit{assumed} that the rules of operation applying
to the symbols of arithmetical algebra apply without alteration in
symbolical algebra; \textit{the meanings of the operations
themselves and their results being derived from these rules of
operation.}

This assumption Peacock names the \textit{Principle of Permanence
of Equivalent Forms}, and illustrates its use as
follows:\footnote{Algebra, edition of 1845, §§~631, 569, 639.}

In arithmetical algebra, when $a > b, c > d$, it may readily be
demonstrated that
\[
(a - b)(c - d) = ac - ad - bc + bd.
\]

By the principle of permanence, it follows that
\[
(0 - b)(0 - d) = 0 \times 0 - 0 \times d - b \times 0 + bd, \\
\textrm{or} (-b)(-d) = bd. \]

Or again. In arithmetical algebra $a^m a^n = a^{m+n}$, when $m$
and $n$ are positive integers. Applying the principle of
permanence,
\begin{align*}
(a^{\frac{p}{q}})^q & =  a^{\frac{p}{q}} \cdot a^{\frac{p}{q}} \cdots \textrm{to} q \textrm{factors}\\
 & = a^{\frac{p}{q} + \frac{p}{q} + \cdots \textrm{to} q \textrm{terms}} \\
 & =  a^p, \\
\textrm{whence} \quad a^{\frac{p}{q}} & =  \sqrt[q]{a^p}.
\end{align*}

Here the meanings of the product $(-b)(-d)$ and of the symbol
$a^{\frac{p}{q}}$ are both derived from certain rules of operation
in arithmetical algebra.

Peacock notices that the symbol = also has a wider meaning in
symbolical than in arithmetical algebra; for in the former = means
that ``the expression which exists on one side of it is the result
of an operation which is indicated on the other side of it and not
performed.''\footnote{Algebra, Appendix, §631.}

He also points out that the terms ``real'' and ``imaginary'' or
``impossible'' are relative, depending solely on the meanings
attaching to the symbols in any particular application of algebra.
For a quantity is real when it can be shown to correspond to any
real or possible existence; otherwise it is
imaginary.\footnote{Ibid.\ §557.} The solution of the problem: to
divide a group of 5 men into 3 equal groups, is imaginary though a
positive fraction, while in Argand's geometry the so-called
imaginary is real.

The principle of permanence is a fine statement of the assumption
on which the reckoning with artificial numbers depends, and the
statement of the nature of this dependence is excellent. Regarded
as an attempt at a complete presentation of the doctrine of
artificial numbers, however, Peacock's Algebra is at fault in
classing the positive fraction with the positive integer and not
with the negative and imaginary, where it belongs, in ignoring the
most difficult of all artificial numbers, the irrational, in not
defining artificial numbers as symbolic results of operations, but
principally in not subjecting the operations themselves to a final
analysis.

\addcontentsline{toc}{section}{\numberline{}The fundamental laws
of algebra. ``Symbolical algebras.'' Gregory }

\textbf{126. The Fundamental laws of Algebra. ``Symbolical
Algebras.''} Of the fundamental laws to which this analysis leads,
two, the commutative and distributive, had been noticed years
before Peacock by the inventors of symbolic methods in the
differential and integral calculus as being common to number and
the operation of differentiation. In fact, one of these
mathematicians, \textit{Servois},\footnote{Gergonne's Annales,
1813. One must go back to Euclid for the earliest known
recognition of any of these laws. Euclid demonstrated, of integers
(Elements, VII, 16), that $ab = ba$.} introduced the names
\textit{commutative} and \textit{distributive}.

Moreover, Peacock's contemporary, \textit{Gregory}, in a paper
``On the Real Nature of Symbolical Algebra,'' which appeared in
the interim between the two editions of Peacock's
Algebra,\footnote{In 1838. See The Mathematical Writings of D.~F.~Gregory,
p.~2. Among other writings of this period, which promoted
a correct understanding of the artificial numbers, should be
mentioned Gregory's interesting paper, ``On a Difficulty in the
Theory of Algebra,'' Writings, p.~235, and De Morgan's papers ``On
the Foundation of Algebra'' (1839, 1841; Cambridge Philosophical
Transactions, VII).} had restated these two laws, and had made
their significance very clear.

To Gregory the formal identity of complex operations with the
differential operator and the operations of numerical algebra
suggested the comprehensive notion of algebra embodied in his fine
definition: ``symbolical algebra is the science which treats of
the combination of operations defined not by their nature, that
is, by what they are or what they do, but by the laws of
combination to which they are subject.''

This definition recognizes the possibility of an entire class of
algebras, each characterized primarily not by its subject-matter,
but by \textit{its operations and the formal laws to which they
are subject}; and in which the algebra of the complex number $a +
ib$ and the system of operations with the differential operator
are included, the two (so far as their laws are identical) as one
and the same particular case.

So long, however, as no ``algebras'' existed whose laws differed
from those of the algebra of number, this definition had only a
speculative value, and the general acceptance of the dictum that
the laws regulating its operations constituted the essential
character of algebra might have been long delayed had not
Gregory's paper been quickly followed by the discovery of two
``algebras,'' the \textit{quaternions} of \textit{Hamilton} and
the \textit{Ausdehnungslehre} of \textit{Grassmann}, in which one
of the laws of the algebra of number, the commutative law for
multiplication, had lost its validity.

\addcontentsline{toc}{section}{\numberline{}Hamilton's
quaternions}

\textbf{127. Quaternions.} According to his own account of the
discovery,\footnote{Philosophical Magazine, II, Vol. 25, 1844.}
Hamilton came upon \textit{quaternions} in a search for a second
imaginary unit to correspond to the perpendicular which may be
drawn in space to the lines 1 and $i$.

In pursuance of this idea he formed the expressions, $a + ib + jc,
x + iy + jz$, in which $a$, $b$, $c$, $x$, $y$, $z$ were supposed
to be real numbers, and $j$ the new imaginary unit sought, and set
their product

\[
(a + ib + jc)(x + iy + jz) = ax - by - cz + i(ay + bx) \
                                    + j(az + cx) + ij(bz + cy).
\]

The question then was, what interpretation to give $ij$. It would
not do to set it equal to $a' + ib' + jc'$, for then the theorem
that the modulus of a product is equal to the product of the
moduli of its factors, which it seemed indispensable to maintain,
would lose its validity; unless, indeed, $a' = b' = c' = 0$, and
therefore $ij = 0$, a very unnatural supposition, inasmuch as $1i$
is different from 0.

No course was left for destroying the $ij$ term, therefore, but to
make its coefficient, $bz + cy$, vanish, which was tantamount to
supposing, since $b, c, y, z$ are perfectly general, that $ji =
-ij$.

Accepting this hypothesis, \textit{denial of the commutative law}
as it was, Hamilton was driven to the conclusion that the system
upon which he had fallen contained at least three imaginary units,
the third being the product $ij$. He called this $k$, took as
general complex numbers of the system, $a + ib + jc + kd, x + iy +
jz + kw,$ \textit{quaternions}, built their products, and assuming

\begin{align*}
i^2 &= j^2 = k^2 = -1 \\
ij &= -ji = k         \\
jk &= -kj = i         \\
ki &= -ik = j,
\end{align*} found that the modulus law was
fulfilled.

A geometrical interpretation was found for the ``\textit{imaginary
triplet}'' $ib + jc + kd$, by making its coefficients, $b, c, d$,
the rectangular co-ordinates of a point in space; the line drawn
to this point from the origin picturing the triplet by its length
and direction. Such directed lines Hamilton named
\textit{vectors}.

To interpret geometrically the multiplication of $i$ into $j$, it
was then only necessary to conceive of the $j$ axis as rigidly
connected with the $i$ axis, and \textit{turned by it} through a
right angle in the $jk$ plane, into coincidence with the $k$ axis.
The geometrical meanings of other operations followed readily.

In a second paper, published in the same volume of the
Philosophical Magazine, Hamilton compares in detail the laws of
operation in \textit{quaternions} and the algebra of number, for
the first time explicitly stating and naming the
\textit{associative} law.

\addcontentsline{toc}{section}{\numberline{}Grassmann's
Ausdehnungslehre }

\textbf{128. Grassmann's Ausdehnungslehre.} In the
\textit{Ausdehnungslehre}, as Grassmann first presented it, the
elementary magnitudes are vectors.

The fact that the equation $AB + BC = AC$ always holds among the
segments of a line, when account is taken of their directions as
well as their lengths, suggested the probable usefulness of
directed lengths in general, and led Grassmann, like Argand, to
make trial of this definition of addition for the general case of
three points, $A$, $B$, $C$, not in the same right line.

But the outcome was not great until he added to this his
definition of the product of two vectors. He took as the product
$ab$, of two vectors, $a$ and $b$, the parallelogram generated by
$a$ when its initial point is carried along $b$ from initial to
final extremity.

This definition makes a product vanish not only when one of the
vector factors vanishes, but also when the two are parallel. It
clearly conforms to the distributive law. On the other hand, since

\begin{align*}
(a+b)(a+b) & = aa+ab+ba+bb, \\
\text{and} \qquad (a+b)(a+b) & = aa =bb=0, \\
ab+ba & = 0, \; \text{or} \; ba=-ab,
\end{align*}
the commutative law for multiplication has lost its validity, and,
as in quaternions, an interchange of factors brings about a change
in the sign of the product.

The opening chapter of Grassmann's first treatise on the
\textit{Ausdehnungslehre} (1844) presents with admirable clearness
and from the general standpoint of what he calls ``Formenlehre''
\, (the doctrine of forms), the fundamental laws to which
operations are subject as well in the \textit{Ausdehnungslehre} as
in common algebra.

\addcontentsline{toc}{section}{\numberline{}The fully developed
doctrine of the artificial forms of number. Hankel. Weierstrass.
G. Cantor}

\textbf{129. The Doctrine of the Artificial Numbers fully
Developed.} The discovery of quaternions and the
\textit{Ausdehnungslehre} made the algebra of number in reality
what Gregory's definition had made it in theory, no longer the
sole algebra, but merely one of a class of algebras. A higher
standpoint was created, from which the laws of this algebra could
be seen in proper perspective. Which of these laws were
distinctive, and what was the significance of each, came out
clearly enough when numerical algebra could be compared with other
algebras whose characteristic laws were not the same as its
characteristic laws.

The doctrine of the artificial numbers regarded from this point of
view---as symbolic results of the operations which the fundamental
laws of algebra define---was fully presented for the negative,
fraction, and imaginary, by \textit{Hankel}, in his
\textit{Complexe Zahlensystemen} (1867). Hankel re-announced
Peacock's principle of permanence. The doctrine of the irrational
now accepted by mathematicians is due to \textit{Weierstrass} and
\textit{G. Cantor} and \textit{Dedekind}.\footnote{See Cantor in
Mathematische Annalen, V, p.~123, XXI, p.~567. The first paper was
written in 1871. In the second, Cantor compares his theory with
that of Weierstrass, and also with the theory proposed by Dedekind
in his \textit{Stetigkeit und irrationals Zahlen} (1872).

The theory of the irrational, set forth in Chapter IV of the first
part of this book, is Cantor's.}

\addcontentsline{toc}{section}{\numberline{}Recent literature }

A number of interesting contributions to the literature of the
subject have been made recently; among them a
paper\footnote{Journal für die reine und angewandte Mathematik,
Vol. 101, p.~337.} by Kronecker in which methods are proposed for
avoiding the artificial numbers by the use of congruences and
``indeterminates,'' and papers\footnote{Göttinger Nachrichten for
1884, p.~395; 1885, p.~141; 1889, p.~34, p.~237. Leipziger
Berichte for 1889, p.~177, p.~290, p.~400. Mathemathische Annalen,
XXXIII, p.~49.} by Weierstrass, Dedekind, Hölder, Study,
Scheffer, and Schur, all relating to the theory of general complex
numbers built from $n$ fundamental units (see page 40).


\textsc{SUPPLEMENTARY NOTE, 1902}. An elaborate and profound
analysis of the number-concept from the ordinal point of view is
made by Dedekind in his \textit{Was sind und was sollen die
Zahlen?} (1887). This essay, together with that on irrational
numbers cited above, has been translated by W.~W.~Beman, and
published by the Open Court Company, Chicago, 1901.

The same point of view is taken by Kronecker in the memoir above
mentioned, and by Helmholtz in his \textit{Zählen und Messen}
(Zeller-Jubeläum, 1887).

G. Cantor discusses the general notion of cardinal number, and
extends it to infinite groups and assemblages in his now famous
Memoirs on the theory of infinite assemblages. See particularly
Mathematische Annalen, XLVI, p.~489.

Very recently much attention has been given to the question: What
is the \textit{simplest} system of consistent and independent
laws---or ``axioms,'' as they are called---by which the
fundamental operations of the ordinary algebra may be defined? A
very complete \textit{résumé} of the literature may be found in a
paper by O. Hölder in Leipziger Berichte, 1901. See also E.~V.~Huntington
in Transactions of the American Mathematical Society,
Vol. III, p.~264. 
\newpage
\newpage

\small
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