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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 %% +%% %% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Basic book class. Required. %% +%% amsmath: Basic AMS math package. Required. %% +%% amssymb: Basic AMS symbols package. Required. %% +%% graphicx: Basic graphics package. Required. %% +%% makeidx: Basic index creation package. Required. %% +%% %% +%% %% +%% Producer's Comments: %% +%% %% +%% The page numbers in the static table of contents are gathered %% +%% with LaTeX page references, hence the file should be compiled %% +%% twice to get them right. %% +%% %% +%% latex (to generate dvi, and from this with appropriate tools %% +%% other formats) should work. The book contains 7 illustrations, %% +%% all in eps format. %% +%% %% +%% %% +%% Things to Check: %% +%% %% +%% Spellcheck: OK %% +%% LaCheck: OK %% +%% Lprep/gutcheck: OK %% +%% PDF pages, excl. Gutenberg boilerplate: 95 %% +%% PDF pages, incl. Gutenberg boilerplate: 104 %% +%% ToC page numbers: OK %% +%% Images: 7 image files (one is used twice in text) %% +%% Fonts: OK %% +%% %% +%% %% +%% Compile History: %% +%% %% +%% Feb 06: ss. Compiled with latex and dvipdf %% +%% Latex Version 3.14159 (te-latex 2.0.2-198.4) %% +%% latex numbersystem.tex %% +%% latex numbersystem.tex %% +%% dvipdf numbersystem.tex %% +%% %% +%% Mar 06: jt. Compiled with latex and dvipdfm. MiKTeX (XP) %% +%% latex 17920-t %% +%% latex 17920-t %% +%% latex 17920-t %% +%% dvipdfm 17920-t %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +\documentclass{book} +\listfiles +\usepackage{amsfonts, amsmath, amssymb, graphicx, makeidx} +\usepackage[latin1]{inputenc} +\pagestyle{plain} + + \renewcommand\chapter{\secdef\mychaptera\mychapterb} + +\newcommand\mychaptera[2][default]{% + \clearpage% + \newpage% + \refstepcounter{chapter}% + \label{chapter:\thechapter}% + \markboth{\S~\thechapter. #1}{\S~\thechapter. #1}% + \addcontentsline{toc}{chapter}{% + \protect\numberline{\thechapter.}\hspace{2em}#1}% + \vspace{2em}% + \begin{center}\begin{Large}% + \textbf{\thechapter. #2}% + \end{Large}\end{center}% + \vspace{1em}% +} + +\newcommand\mychapterb[1]{% + \clearpage% + \newpage% + \markboth{#1}{#1}% + \vspace{2em}% + \begin{center}\begin{Large}% + \textbf{#1}% + \end{Large}\end{center}% + \vspace{1em}% +} + +\renewcommand\thechapter{\arabic{chapter}} + +\begin{document} + +\thispagestyle{empty} +\small +\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 +\pagenumbering{gobble} +\begin{verbatim} + +End of Project Gutenberg's Number-System of Algebra, by Henry Fine + +*** 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 + + +*** This file should be named 17920-t.tex or 17920-t.zip *** +*** or 17920-pdf.pdf or 17920-pdf.pdf *** +This and all associated files of various formats will be found in: + https://www.gutenberg.org/1/7/9/2/17920/ + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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DOMAIN IN THE UNITED STATES. + +Procedures for determining public domain status are described in +the "Copyright How-To" at https://www.gutenberg.org. + +No investigation has been made concerning possible copyrights in +jurisdictions other than the United States. 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