% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% The Project Gutenberg EBook of A Short Account of the History of        %
% Mathematics, by W. W. Rouse Ball                                        %
%                                                                         %
% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
% re-use it under the terms of the Project Gutenberg License included     %
% with this eBook or online at www.gutenberg.org                          %
%                                                                         %
%                                                                         %
% Title: A Short Account of the History of Mathematics                    %
%                                                                         %
% Author: W. W. Rouse Ball                                                %
%                                                                         %
% Release Date: May 28, 2010 [EBook #31246]                               %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: ISO-8859-1                                      %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICS ***               %
%                                                                         %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %

\def\ebook{31246}
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%%                                                                  %%
%% Packages and substitutions:                                      %%
%%                                                                  %%
%% book:        Document class. Required.                           %%
%% geometry:    Enhanced page layout package. Required.             %%
%% babel:       For Greek words. Required.                          %%
%% type1ec:     Provides smooth (Type 1) Greek font. Recommended.   %%
%% amsmath:     AMS mathematics enhancements. Required.             %%
%% amssymb:     AMS symbols e.g. \therefore. Required.              %%
%% inputenc:    Standard DP encoding. Required.                     %%
%% babel:       Provide accented Greek. Required.                   %%
%% makeidx:     Allows index creation.  Required.                   %%
%% array:       Enhancement to arrays.  Required.                   %%
%% graphicx:    Allows inclusion of images. Required.               %%
%% perpage:     Resets footnote markers every page. Required.       %%
%% ifthen:      Logical conditionals. Required.                     %%
%% wrapfig:     Wrap text around images. Required.                  %%
%% indentfirst: Indent after headings. Required.                    %%
%% fancyhdr:    Enhanced running headers and footers. Required.     %%
%% verbatim:    For PG license text. Required.                      %%
%% multicol:    Automatically balance index columns. Required.      %%
%% hhline:      Enhanced table borders. Required.                   %%
%% hyperref:    Hypertext embellishments for pdf output. Required.  %%
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%% Things to Check:                                                 %%
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%% Smoothreading pool: Yes                                          %%
%% lacheck: OK                                                      %%
%%   False positives:                                               %%
%%     Complaints from preamble                   (before line 600) %%
%%     Complaints from lprep config and log file (after line 20500) %%
%%     Possible unwanted space at "{"                (19 instances) %%
%%     \ldots should be \cdots                       (16 instances) %%
%%     Use ' to end quotation, not `                  (7 instances) %%
%%     punctuation mark "!" should be placed after end of math mode %%
%%                                                    (2 instances) %%
%%                                                                  %%
%% Lprep: OK, no warnings                                           %%
%% Gutcheck: OK                                                     %%
%% PDF pages: 466                                                   %%
%% PDF page size: 5.25" x 8.65"                                     %%
%% PDF bookmarks: point to periods & chapters, preface, note,       %%
%%                table of contents, index, PG licensing            %%
%% PDF document info: filled in                                     %%
%% PDF Reader displays document title in window title bar           %%
%% ToC page numbers: OK                                             %%
%% Images: 43 PDFs, with sources for some images included in the    %%
%%         images/source subfolder                                  %%
%% Tables/Figures: Positioned correctly, i.e. no large whitespace   %%
%%         at bottom of previous page. Check following pages:       %%
%%         18, 20, 29, 32, 33, 52, 56, 63, 65, 68, 74, 82, 90, 103  %%
%%         104, 131, 153, 157, 159, 168, 204, 231, 234, 256, 265    %%
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%% pdflatex                                                         %%
%% makeindex                                                        %%
%% pdflatex x3                                                      %%
%%                                                                  %%
%% Compile History:                                                 %%
%% Nov 2008: v l pyner                                              %%
%%           MiKTeX 2.7, Windows Vista                              %%
%% Jan 2010: windymilla (Nigel Blower)                              %%
%%           MiKTeX 2.7, TeXnicCenter, Windows XP Pro               %%
%%           Compiled (includes makeindex) four times               %%
%%                                                                  %%
%%                                                                  %%
%% May 2010: pglatex.                                               %%
%%   Compile this project with:                                     %%
%%   pdflatex 31246-t.tex                                           %%
%%   makeindex 31246-t.idx                                          %%
%%   pdflatex 31246-t.tex ..... THREE times                         %%
%%                                                                  %%
%%   pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6)                 %%
%%                                                                  %%
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\listfiles

\makeindex

\makeatletter

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            marginratio=1:1]{geometry}[2008/11/13]
\usepackage[10pt]{type1ec}[2002/09/07]
\usepackage[latin1]{inputenc}[2006/05/05]
\usepackage[polutonikogreek, english]{babel}[2008/07/06]
\usepackage{amsmath}[2000/07/18]
\usepackage{amssymb}[2002/01/22]
\usepackage{makeidx}[2000/03/29]
\usepackage{array}[2005/08/23]
\usepackage{graphicx}[1999/02/16]
\usepackage{perpage}[2006/07/15]
\usepackage{ifthen}[2001/05/26]
\usepackage{wrapfig}[2003/01/31]
\usepackage{indentfirst}[1995/11/23]
\usepackage{fancyhdr}% no date stamp
\usepackage{verbatim}[2003/08/22]
\usepackage{multicol}[2006/05/18]
\usepackage{hhline}[1994/05/23]

% PDF attributes
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\usepackage[pdftex,
  hyperref,
  hyperfootnotes=false,
  pdftitle={The Project Gutenberg eBook \#\ebook: A Short Account of the History of Mathematics},
  pdfsubject={History of Mathematics},
  pdfauthor={W. W. Rouse Ball},
  pdfkeywords={Greg Lindahl, Viv, Juliet Sutherland, Nigel Blower, Project Gutenberg Online Distributed Proofreading Team},
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% customise footnotes (restart numbering on each page; no rule, but increase gap a little)
\MakePerPage{footnote}
\renewcommand\footnoterule{}
\addtolength{\skip\footins}{4pt}

% (re-)define operators
\let\arccos\undefined
\let\arcsin\undefined
\DeclareMathOperator\arccos{arc~cos}
\DeclareMathOperator\arcsin{arc~sin}
\DeclareMathOperator\cosec{cosec}
\DeclareMathOperator\vers{vers}

% define use of centered dot
\DeclareInputText{183}{\ifmmode\mathord{\cdot}\else\textperiodcentered\fi}
% Enable ° to be used in math without warnings
\DeclareInputText{176}{\ifmmode{{}^\circ}\else\textdegree\fi}

% used for old-style continued fractions
\newcommand{\cdrop}[1]{\mathbin{\raisebox{-1.5ex}{#1}}}
\newcommand{\cpad}{\phantom{^2}}

% slanted Greek text looks more like original
\newcommand{\mytextgreek}[1]{\textgreek{\itshape#1}} % italic Greek looks more like original
\newcommand{\omicron}{\mathord{o}}
\newcommand{\Chi}{\mathord{\text{X}}}
\newcommand{\Eta}{\mathord{\text{H}}}
\renewcommand{\sum}{\Sigma} % matches original more closely

% each part represents a 'period' in math history
\newcommand\nbpart[3]{%
	\clearpage%
  \pdfbookmark[1]{#1---#2}{#1---#2}%
  \vspace*{100\p@}%
  {\parindent \z@ \centering
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    \vskip 15\p@
    \large\includegraphics[height=\baselineskip]{./images/#3.pdf}
    \vskip 15\p@
  }%
  \thispagestyle{first}}

% customise chapter heading
\def\@makeschapterhead#1{%
  \vspace*{100\p@}%
  {\parindent \z@ \centering
    \normalfont
    \interlinepenalty\@M
    #1\par\nobreak
    \vskip 15\p@
  }}

% \nbChapter{chapternum}{chapter title}{page header}{subtitle}{footnote}
\newcommand\nbChapter[5]{
  \chapter*{\pdfbookmark[2]{#1. #2}{#1. #2}\nblabel{chapter:#1}CHAPTER #1.}%
  {%
    \begin{center}%
      \textsc{\MakeLowercase{#2.}}%
      \ifthenelse{\equal{#4}{}}%
        {\ifthenelse{\equal{#5}{}}%
          {}{\footnote{#5}}}%
        {\ifthenelse{\equal{#5}{}}%
          {}{\footnote{#5}}\\\textsc{\MakeLowercase{#4}}}%
    \end{center}%
    % put chapter title in header, unless different header supplied
    \fancyhead[L]{{\footnotesize CH. #1}]}%
    \ifthenelse{\equal{#3}{}}%
      {\fancyhead[C]{\small\MakeUppercase{#2}}}%
      {\fancyhead[C]{\small\MakeUppercase{#3}}}%
  }%
  \thispagestyle{first}}

% customise section heading
\renewcommand\section{\@startsection {section}{1}{\z@}%
                                     {-3.5ex \@plus -1ex \@minus -.2ex}%
                                     {1.5ex \@plus.2ex}%
                                     {\centering\normalsize\itshape}}

% format a chapter entry in the ToC
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{0pt}{-1.5\baselineskip}{\baselineskip}%
{\centering\normalsize\scshape\interlinepenalty\@M}}

% Command to provide ToC period headings using images
% optional first argument multiplier to override default image height of \baselineskip
\newcommand\tocperiod[2][1]{\@startsection{section}{1}%
{0pt}{-2.5\baselineskip}{\baselineskip}%
{\centering\large\includegraphics[height=#1\baselineskip]{./images/#2.pdf}}*{}}

% phantom section before label so hyperref links to correct place
\newcommand\nblabel[1]{\phantomsection\label{#1}}

% Commands to label and hyperref to figures
\newcommand{\labelincludegraphics}[2][]{\phantomsection\label{fig:#2}\includegraphics[#1]{./images/#2.pdf}}
\newcommand{\figureref}[2]{\hyperref[fig:#1]{#2}}

% for small graphics images (e.g. math symbols) used inline in text
\newcommand{\inlinegraphics}[3]{\raisebox{#2}{\includegraphics[height=#1]{./images/#3.pdf}}}

% this definition can be replaced with just the \hfill if no leader dots preferred
\newcommand{\contentsfill}{\leaders\hbox{\kern 1em.\kern 1em}\hfill}

% binary dot operator used in math
\newcommand{\ctrdot}{\mathbin{.}}

\newlength\TmpPadLen % for PadTo
\newcommand{\PadTo}[3][c]{%
  \settowidth{\TmpPadLen}{$#2$}%
  \makebox[\TmpPadLen][#1]{$#3$}%
}

% Set up fancy pagestyle:
%   page number at outer edge of headers
\pagestyle{fancy}
\fancyhf{}
\fancyhead[R]{\thepage}%
\setlength\headheight{14.5pt}
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% Set up pagestyle for first page of each chapter:
%   page number centered in header
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\fancyhead[C]{\thepage}}

% customise index - using multicols to balance columns
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  \small%
  \setlength\columnseprule{0.5pt}%
  \setlength\columnsep{1.5em}%
  \begin{multicols}{2}[\chapter*{\Large INDEX.}]%
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  \setlength\parskip{0pt plus 0.1pt}%
  \fancyhead[C]{\small INDEX}%
  \fancyhead[L]{}%
  \thispagestyle{first}%
  \nbstretchyspace
  \renewcommand{\@idxitem}{\par\hangindent 20\p@}
  \renewcommand{\subitem}{\par\hangindent 20\p@ \hspace*{10\p@}}
  \let\item\@idxitem}%
{\end{multicols}}

% miscellaneous unusual symbols - may need minor adjustment for change of font size, etc.
\newlength\TmpLen % for general use
\newcommand\nblongplusline{%
  \settowidth{\TmpLen}{$-$}%
  \makebox[0.75\TmpLen]{$-$}\makebox[0.75\TmpLen]{$-$}\makebox[0.75\TmpLen]{$-$}\makebox[0.75\TmpLen]{$-$}}
\newcommand\nblongplus{%
  \mathbin{\PadTo{\nblongplusline}{\nblongplusline}%
     \llap{\PadTo{\nblongplusline}{|}}}}
\newcommand\nbshortplus{\PadTo{-}{-}\llap{\PadTo{-}{|}}}
\newcommand\nbdiffm{%
  \settowidth{\TmpLen}{$-$}%
  \makebox[0.75\TmpLen]{$-$}\makebox[0.75\TmpLen]{$-$}\mathord{:}}
\newcommand\nbdiffc\sim
\newcommand\nbbarrowdiv{%
  \settowidth{\TmpLen}{$÷$}%
  \makebox[0.5\TmpLen]{$÷$}\makebox[0.5\TmpLen]{$÷$}}
\newcommand\nbinftyleft{\raisebox{-0.3ex}{\rotatebox[origin=c]{90}{8}}}
\newcommand\nbinftyright{\raisebox{-0.3ex}{\rotatebox[origin=c]{-90}{8}}}
\newcommand\nbproportion{\mathbin{:\::}}
\newcommand\nboughtredgreater{%
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  \raisebox{1pt}{\begin{tabular}{@{}c@{}c@{}|}\hline\hspace*{1em}&\hspace*{0.7em}\\\cline{2-2}\end{tabular}}}
\newcommand\nboughtredless{%
  \renewcommand\arraystretch{0.6}%
  \raisebox{1pt}{\begin{tabular}{@{}c@{}c@{}|}\cline{2-2}\hspace*{1em}&\hspace*{0.7em}\\\hline\end{tabular}}}
\newcommand\nbne {\rlap{\PadTo{\mathord{=}}{\raisebox{1pt}{$\scriptstyle|$}}}\mathord{=}}
\newcommand\nbngt{\rlap{\PadTo{\mathord{>}}{|\mspace{2mu}}}\mathord{>}}
\newcommand\nbnlt{\rlap{\PadTo{\mathord{<}}{\mspace{2mu}|}}\mathord{<}}

% digit contained in a circle
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  \setlength\unitlength{1pt}%
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  \thinlines%
  \put(0,0){\circle{10}}%
  \put(-2,-2){\tiny #1}%
  \end{picture}%
}

% bombelli's symbol
\newcommand{\bombelli}[1]{%
  \setlength\unitlength{1pt}%
  \begin{picture}(12,12)(-6,-4)%
  \thinlines%
  \put(0,0){\oval(12,12)[b]}%
  \put(-2,-4){\tiny #1}%
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}

% inline table used for its border lines
\newcommand{\inlinetabular}[2]
  {\raisebox{-0.67pt}{\renewcommand\arraystretch{0.7}\begin{tabular}{#1}#2\end{tabular}}}

% quotes, etc using smaller font
\newenvironment{nbQuote}{\medskip\small}{\medskip}

% names of mathematicians are mainly bold, and followed by a space when they begin a paragraph
\newcommand\nbnamespace{\hspace{0.7em plus 0.4em minus 0.3em}}
\newcommand\nbname[1]{\textbf{#1}}
\newcommand\nbparagraph[1]{\nbname{#1}\nbnamespace}
% some topics are introduced with italic text followed by space
\newcommand\nbtopictitle[1]{\emph{#1}\nbnamespace}

% Commands to include long-topped sigma
\newsavebox{\Glyph}
\savebox{\Glyph}[8pt]{% Dimensions may require adjustment
  \raisebox{-0.6ex}{%
    \hbox{\includegraphics[width=8pt]{./images/sigma.pdf}}
    \kern -6pt%
  }
}
\newcommand{\Mysigma}{\usebox{\Glyph}}

% format mod as in original text
\renewcommand\pmod[1]{\;(\text{mod.\ } #1)}

% if first word of chapter is required to be small caps
\newcommand{\First}[1]{\textsc{#1}}

% standard skip for thought break
\newcommand{\tbskip}{\bigskip}
% hyphenate some awkward words
\hyphenation{Pyth-ago-ras Pyth-ago-rean hexa-gons quad-riv-ium quad-ra-trix quad-rics
Metro-dorus Leo-nardo Archi-medes Tar-ta-glia Des-cartes phys-ics Mer-senne Brah-ma-gup-ta
equi-noxes De-moivre La-place For-schung Sal-mon Alem-bert Syl-vester Thi-baut Helm-holtz
Inns-pruck Dede-kind tri-go-no-metry Hal-phen}

% for sensible insertion of boilerplate/licence,
% overlong lines will wrap and be indented 0.25in
\def\@xobeysp{~\hfil\discretionary{}{\kern\z@}{}\hfilneg}
\renewcommand\verbatim@processline{\leavevmode
  \null\kern-0.25in\the\verbatim@line\par}
\addto@hook\every@verbatim{\@totalleftmargin0.25in\scriptsize}

% reduce chance of widows/clubs
\widowpenalty=1000
\clubpenalty=1000

% allow some slack to avoid under/overfull boxes
\newcommand\nbstretchyspace{\spaceskip0.5em plus 0.3em minus 0.25em}
% allow tiny protrusions into the margin without warning
\setlength{\hfuzz}{0.5pt}

% Simple attempt to replace reference with Ibid if on same page
% Usage:
%    \footnoteibid{}{laba}{Full text}{Additional text a}
% ...\footnoteibid{laba}{labb}{Full text}{Additional text b}
\newcommand{\footnoteibid}[4]{%
	\label{#2}% define label here, and check against it on the next call
  \expandafter\ifx\csname r@#1\endcsname\relax%
    % label #1 undefined: output full reference
    \footnote{#3 #4}%
  \else%
    \@tempcnta\expandafter\expandafter\expandafter%
      \@secondoffive\csname r@#1\endcsname\relax%
    \@tempcntb\expandafter\expandafter\expandafter%
      \@secondoffive\csname r@#2\endcsname\relax%
    \ifnum\@tempcnta=\@tempcntb%
      % same page: output Ibid.
     \footnote{\textit{Ibid.} #4}%
    \else%
      % pagebreak intervenes: output full reference
      \footnote{#3 #4}%
    \fi%
  \fi%
}

% Sometimes the author has multiple references on a page to the same footnote
% so we have to go to some lengths to discover if there's an intervening
% pagebreak or not since our pagination is unlikely to match the original.
% If there's a pagebreak we need a second copy of the footnote,
% but if not we need to duplicate the footnotemark.
%
% For first in the sequence:
% \multifootnote{nameof sequence}{footnote text}
% for subsequent references to the same footnote there's no need to replicate the text:
% \multifootnote{nameof sequence}
\newcommand{\multifootnote}[1]{%
  \expandafter\ifx\csname mfn:#1:seq\endcsname\relax % first in sequence
    \let\next\storemultifootnote
    \expandafter\gdef\csname mfn:#1:seq\endcsname{1}%
  \else
  \expandafter\expandafter\expandafter\@tempcnta=\csname mfn:#1:seq\endcsname
  \xdef\mfn@previous{r@mfn:#1:\the\@tempcnta}%
  \advance\@tempcnta1\relax
  \expandafter\xdef\csname mfn:#1:seq\endcsname{\the\@tempcnta}% update sequence counter
  \label{mfn:#1:\the\@tempcnta}% use labels to test for being on the same page
  \xdef\mfn@current{r@mfn:#1:\the\@tempcnta}%
  \expandafter\expandafter\expandafter\ifx\expandafter\csname\mfn@current\endcsname\relax%
    % labels undefined: don't do anything fancy this time
    \expandafter\footnote{\csname mfn:#1:txt\endcsname}%
  \else
    \@tempcnta\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter
      \mfn@getpagenum\expandafter\csname\mfn@previous\endcsname\relax
    \@tempcntb\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter\expandafter
      \mfn@getpagenum\expandafter\csname\mfn@current\endcsname\relax
    \ifnum\@tempcnta=\@tempcntb % footnotes on same page: just issue the series footnotemark
      \expandafter\expandafter\expandafter
        \@xfootnotemark\expandafter\expandafter\expandafter[\csname mfn:#1:num\endcsname]%
    \else % new footnote, and reset the series footnotemark
      \expandafter\footnote{\csname mfn:#1:txt\endcsname}%
      \expandafter\xdef\csname mfn:#1:num\endcsname{\the\c@footnote}%
    \fi
  \fi
  \let\next\@gobble
  \fi
  \next{#1}}
\newcommand\storemultifootnote[2]{% save footnote text and series marker
  \label{mfn:#1:1}\footnote{#2}%
  \expandafter\xdef\csname mfn:#1:num\endcsname{\the\c@footnote}%
  \expandafter\long\expandafter\def\csname mfn:#1:txt\endcsname{#2}}
\AtBeginDocument{\@ifpackageloaded{hyperref}{%
  \let\mfn@getpagenum\@secondoffive
  }{\let\mfn@getpagenum\@secondoftwo}}

% For magic squares we use a picture environment rather than tabular/array
% We first redefine the horizontal/vertical lines for the picture environment
% to give them squarecap ends a la PostScript: this makes the corners of
% frames join up neatly
\def\@hline{\advance\@linelen\@wholewidth
  \ifnum \@xarg <\z@ \hskip -\@linelen \hskip\@halfwidth
    \else\hskip-\@halfwidth\fi
  \vrule \@height \@halfwidth \@depth \@halfwidth \@width \@linelen
  \ifnum \@xarg <\z@ \hskip -\@linelen \hskip\@halfwidth
    \else\hskip-\@halfwidth\fi}
\def\@upline{%
  \hb@xt@\z@{\advance\@linelen\@halfwidth\hskip -\@halfwidth
   \vrule \@width \@wholewidth \@height \@linelen \@depth \@halfwidth\hss}}
\def\@downline{%
  \hb@xt@\z@{\advance\@linelen\@halfwidth\hskip -\@halfwidth
   \vrule \@width \@wholewidth \@height \@halfwidth \@depth \@linelen \hss}}
% Now we set up a MagicSquare environment, which is a bit like a tabular
% except that the entries must be enclosed in braces if they have > 1 digit
% \begin{MagicSquare}{horiz order}[optional vertical order, defaults to square]
%   {entry} & {entry} & ... & {entry}\\
% \end{MagicSquare}
%
% \begin{NapierRods}{horiz order}[optional vertical order, defaults to square]
%   {entry} & {entry} & ... & {entry}\\
%   {entry/entry} & {entry/entry} & ... & {entry/entry}\\
% \end{NapierRods}
%
% switch to allow table entries to be set in sans serif
\newif\ifusesans
\newif\ifNapier
\def\Sq@r#1{\vbox to\SqHt{\vss\hbox to\SqHt{\MagicFontSpec\hss$\vphantom
  {\SqHtDefault}\ifusesans\textsf{#1}\else#1\fi$\hss}\vss}}
\def\SqHtDefault{1}
\def\Cell(#1,#2;#3){\put(#1,#2){\framebox{\Sq@r{#3}}}}
\def\MagicSquare#1{\catcode`\&=\active
  \@ifnextchar[{\M@gicSqu@re#1}{\M@gicSqu@re#1[#1]}}
\long\def\M@gicSqu@re#1[#2]#3\end{%
  \ifNapier\let\\\MSqN@xtC@llSw\else\let\\\MSqN@xtC@ll\fi
  \begin{picture}(#1,#2)
   \@tempcnta=#2
   \MSqN@xtC@ll#3
   \put(0,0){\line(0,1){#2}}
   \put(0,0){\line(1,0){#1}}
   \put(#1,0){\line(0,1){#2}}
   \put(0,#2){\line(1,0){#1}}
  \end{picture}\end}
\let\endMagicSquare\empty
\let\MSqHorizAdvance\@ne
\let\MSqVertAdvance\m@ne
\long\def\MSqN@xtCell#1{\advance\@tempcntb\MSqHorizAdvance
  \Cell(\@tempcntb,\@tempcnta;#1)}
\long\def\MSqN@xtC@ll{\ifnum\@tempcnta=\z@\let\next\relax\else
  \advance\@tempcnta\MSqVertAdvance\@tempcntb=-\MSqHorizAdvance
  \let\next\MSqN@xtCell\fi\next}
\def\MSqN@xtC@llSw{\let\Cell\Napier\let\\\MSqN@xtC@ll\MSqN@xtC@ll}
{\catcode`\&=\active
\global\let&=\MSqN@xtCell}
\unitlength=1.5em
\linethickness{0.14em}
\fboxsep\z@
\def\SqHt{1.5em}
\def\NHt{0.75em} % half the \SqHt
\let\MagicFontSpec\footnotesize
% The Napier's rods diagram is similar to a magic square, but from the
% second row onwards there are two numbers separated by a diagonal dashed line
\def\NapierRods{\Napiertrue\MagicSquare}
\def\Napier(#1,#2;#3){\put(#1,#2){\framebox{\N@pier#3;}}}
\def\N@pier#1/#2;{\raise\NHt\vbox to\NHt{\vss\hbox to\NHt{\MagicFontSpec\hss$\vphantom
  {\SqHtDefault}\ifusesans\textsf{#1}\else#1\fi$\hss}}\hfil
  \vbox to\SqHt{\vss\hbox to0pt{\hss\rotatebox[origin=c]{45}{\tiny-\;-\;-\;-\;-}\hss}\vss}\hfil
  \vbox to\NHt{\hbox to\NHt{\MagicFontSpec\hss$\vphantom
  {\SqHtDefault}\ifusesans\textsf{#2}\else#2\fi$\hss}\vss}}
\let\endNapierRods\empty

% DPalign and DPgather definitions
\providecommand\shortintertext\intertext
\newcount\DP@lign@no
\newtoks\DP@lignb@dy
\newif\ifDP@cr
\newif\ifbr@ce
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%[**TN: spellings "show" and "shew" are both used in the original text - retained]

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\begin{verbatim}
The Project Gutenberg EBook of A Short Account of the History of
Mathematics, by W. W. Rouse Ball

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


Title: A Short Account of the History of Mathematics

Author: W. W. Rouse Ball

Release Date: May 28, 2010 [EBook #31246]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICS ***
\end{verbatim}

\frontmatter

\pdfbookmark[0]{A SHORT ACCOUNT OF THE HISTORY OF MATHEMATICS}{THE HISTORY OF MATHEMATICS}

\thispagestyle{empty}

\begin{center}
\vspace*{\stretch{1}}
{\Large A SHORT ACCOUNT}

\vspace{10mm}

{\small OF THE}

\vspace{10mm}

{\Huge HISTORY OF MATHEMATICS}

\vspace{30mm}

{\small BY}

\vspace{10mm}

W.~W.~ROUSE BALL

{\scriptsize FELLOW OF TRINITY COLLEGE, CAMBRIDGE}

\vspace*{\stretch{2}}
DOVER PUBLICATIONS, INC.

NEW YORK
\end{center}

\clearpage
\thispagestyle{empty}
\vspace*{\stretch{1}}
\begin{center}
This new Dover edition, first published in
1960, is an unabridged and unaltered republication
of the author's last revision---the fourth
edition which appeared in 1908.

\vspace*{\stretch{2}}

\emph{International Standard Book Number: 0-486-20630-0\\
Library of Congress Catalog Card Number: 60-3187}

\vspace{30mm}

Manufactured in the United States of America

Dover Publications, Inc.

180 Varick Street

New York, N. Y. 10014
\end{center}

\clearpage
\thispagestyle{empty}
\begin{center}
\small
Produced by Greg Lindahl, Viv, Juliet Sutherland, Nigel
Blower and the Online Distributed Proofreading Team at
http://www.pgdp.net
\end{center}

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  \subsection*{{\normalsize\centering\itshape Transcriber's Notes}}
  \small

  A small number of minor typographical errors and inconsistencies
  have been corrected. References to figures such as ``on the next page''
  have been replaced with text such as ``below'' which is more suited to an eBook.

  Such changes are documented in the \LaTeX\ source:
  {\ttfamily\footnotesize\%[**TN: text of note]}
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\chapter*{%
\pdfbookmark[1]{INTRODUCTION}{INTRODUCTION}%
\pdfbookmark[2]{Preface}{Preface}%
\Large PREFACE.}
\nblabel{png003}
\fancyhead[C]{\small PREFACE}
\thispagestyle{empty}

\First{The} subject-matter of this book is a historical summary
of the development of mathematics, illustrated by the
lives and discoveries of those to whom the progress of the
science is mainly due. It may serve as an introduction
to more elaborate works on the subject, but primarily it
is intended to give a short and popular account of those
leading facts in the history of mathematics which many
who are unwilling, or have not the time, to study it
systematically may yet desire to know.

The first edition was substantially a transcript of
some lectures which I delivered in the year 1888 with
the object of giving a sketch of the history, previous to
the nineteenth century, that should be intelligible to any
one acquainted with the elements of mathematics. In
the second edition, issued in 1893, I rearranged parts
of it, and introduced a good deal of additional matter.

The scheme of arrangement will be gathered from the
table of contents at the end of this preface. Shortly it
is as follows. The \hyperref[chapter:I]{first chapter} contains a brief statement
of what is known concerning the mathematics of
the Egyptians and Phoenicians; this is introductory to
the history of mathematics under Greek influence. The
subsequent history is divided into three periods: first,
that under Greek influence, chapters \hyperref[chapter:II]{\textsc{ii}} to \hyperref[chapter:VII]{\textsc{vii}}; second,
that of the middle ages and renaissance, chapters \hyperref[chapter:VIII]{\textsc{viii}} to
\hyperref[chapter:XIII]{\textsc{xiii}}; and lastly that of modern times, chapters \hyperref[chapter:XIV]{\textsc{xiv}} to
\hyperref[chapter:XIX]{\textsc{xix}}.

In discussing the mathematics of these periods I
have confined myself to giving the leading events in the
history, and frequently have passed in silence over men
or works whose influence was comparatively unimportant.
Doubtless an exaggerated view of the discoveries of those
mathematicians who are mentioned may be caused by
the non-allusion to minor writers who preceded and
prepared the way for them, but in all historical sketches
this is to some extent inevitable, and I have done my
best to guard against it by interpolating remarks on the
progress of the science at different times. \index{Astronomy.!Descriptive astronomy outside range of work}Perhaps also
I should here state that generally I have not referred
to the results obtained by practical astronomers and
physicists unless there was some mathematical interest
in them. In quoting results I have commonly made
use of modern notation; the reader must therefore
recollect that, while the matter is the same as that
of any writer to whom allusion is made, his proof is
sometimes translated into a more convenient and familiar
language.

The greater part of my account is a compilation from
existing histories or memoirs, as indeed must be necessarily
the case where the works discussed are so numerous
and cover so much ground. When authorities disagree I
have generally stated only that view which seems to me
to be the most probable; but if the question be one of
importance, I believe that I have always indicated that
there is a difference of opinion about it.

I think that it is undesirable to overload a popular
account with a mass of detailed references or the
authority for every particular fact mentioned. \index{Cantor, M., ref.\ to}For the
history previous to 1758, I need only refer, once for all,
to the closely printed pages of M.~Cantor's monumental
\emph{Vorlesungen über die Geschichte der Mathematik} (hereafter
alluded to as Cantor), which may be regarded
as the standard treatise on the subject, but usually
I have given references to the other leading
authorities on which I have relied or with which
I am acquainted. My account for the period subsequent
to 1758 is generally based on the memoirs or
monographs referred to in the footnotes, but the main
facts to 1799 have been also enumerated in a supplementary
volume issued by Prof.~Cantor last year. I
hope that my footnotes will supply the means of studying
in detail the history of mathematics at any specified
period should the reader desire to do so.

My thanks are due to various friends and correspondents
who have called my attention to points in the
previous editions. I shall be grateful for notices of
additions or corrections which may occur to any of my
readers.

\begin{flushright}
{\large W.~W.~ROUSE BALL.} \qquad
\end{flushright}

{\scriptsize TRINITY COLLEGE, CAMBRIDGE.}

\chapter*{\pdfbookmark[2]{Note}{Note}\Large NOTE.}
\nblabel{png006}
\thispagestyle{plain}

\First{The} fourth edition was stereotyped in 1908, but no
material changes have been made since the issue of the
second edition in 1893, other duties having, for a few
years, rendered it impossible for me to find time for any
extensive revision. Such revision and incorporation of
recent researches on the subject have now to be postponed
till the cost of printing has fallen, though advantage has
been taken of reprints to make trivial corrections and
additions.

\begin{flushright}
{\large W.~W.~R.~B. \qquad}
\end{flushright}

{\scriptsize TRINITY COLLEGE, CAMBRIDGE.}

\chapter*{\pdfbookmark[2]{Table of Contents}{Table of Contents}%
\large\bfseries TABLE OF CONTENTS.}

\fancyhead[C]{\small TABLE OF CONTENTS}

\thispagestyle{first}

\nblabel{png007a}\begin{center}
\footnotesize
\hfill \textsc{page}\\*
\textsc{Preface}\contentsfill  \pageref{png003}\\*
\textsc{Table of Contents}\contentsfill  \pageref{png007a}

\tocchapter*{Chapter I\@. Egyptian and Phoenician Mathematics.}

The history of mathematics begins with that of the Ionian Greeks\contentsfill \pageref{png023b}\\*
Greek indebtedness to Egyptians and Phoenicians\contentsfill \pageref{png024a}\\
Knowledge of the science of numbers possessed by the Phoenicians\contentsfill \pageref{png024b}\\
Knowledge of the science of numbers possessed by the Egyptians\contentsfill \pageref{png025a}\\
Knowledge of the science of geometry possessed by the Egyptians\contentsfill \pageref{png027a}\\*
Note on ignorance of mathematics shewn by the Chinese\contentsfill \pageref{png031a}

\tocperiod{firstperiod}

\emph{This period begins with the teaching of Thales, circ.~\emph{600~\textsc{b.c.},} and ends
with the capture of Alexandria by the Mohammedans in or about \emph{641~\textsc{a.d.}}
The characteristic feature of this period is the development of geometry.}

\tocchapter*{Chapter II\@. The Ionian and Pythagorean Schools.\\*
Circ.~600~\textsc{b.c.}--400~\textsc{b.c.}}

Authorities\contentsfill \pageref{png034b}\\*
\emph{The Ionian School}\contentsfill \pageref{png035a}\\
\textsc{Thales}, 640--550~\textsc{b.c.}\contentsfill \pageref{png035b}\\
\qquad His geometrical discoveries\contentsfill \pageref{png036a}\\
\qquad His astronomical teaching\contentsfill \pageref{png038a}\\
Anaximander. Anaximenes. Mamercus. Mandryatus\contentsfill \pageref{png039a}\\
\emph{The Pythagorean School}\contentsfill \pageref{png040a}\\
\textsc{Pythagoras}, 569--500~\textsc{b.c.}\contentsfill \pageref{png040b}\\
\qquad The Pythagorean teaching\contentsfill \pageref{png041a}\\
\qquad The Pythagorean geometry\contentsfill \pageref{png043a}\\
\qquad The Pythagorean theory of numbers\contentsfill \pageref{png045a}\\
Epicharmus. Hippasus. Philolaus. Archippus. Lysis\contentsfill \pageref{png049a}\\
\textsc{Archytas}, circ.~400~\textsc{b.c.}\contentsfill \pageref{png049b}\\
\qquad His solution of the duplication of a cube\contentsfill \pageref{png050a}\\
Theodorus. Timaeus. Bryso\contentsfill \pageref{png051a}\\
\emph{Other Greek Mathematical Schools in the Fifth Century}~\textsc{b.c.}\contentsfill  \pageref{png051b}\\
Oenopides of Chios\contentsfill \pageref{png051c}\\*
Zeno of Elea. Democritus of Abdera\contentsfill \pageref{png052a}

\tocchapter*{Chapter III\@. The Schools of Athens and Cyzicus. \\*
Circ.~420--300~\textsc{b.c.}}

Authorities\contentsfill \pageref{png054b}\\*
Mathematical teachers at Athens prior to 420~\textsc{b.c.}\contentsfill \pageref{png055a}\\
\qquad Anaxagoras. The Sophists. Hippias (The quadratrix).\contentsfill \pageref{png055b}\\
\qquad Antipho\contentsfill \pageref{png057a}\\
Three problems in which these schools were specially interested\contentsfill \pageref{png058a}\\
\textsc{Hippocrates} of Chios, circ.~420~\textsc{b.c.}\contentsfill \pageref{png058b}\\
\qquad Letters used to describe geometrical diagrams\contentsfill \pageref{png059a}\\
\qquad Introduction in geometry of the method of reduction\contentsfill \pageref{png060a}\\
\qquad The quadrature of certain lunes\contentsfill \pageref{png060b}\\
\qquad The problem of the duplication of the cube\contentsfill \pageref{png062a}\\
Plato, 429--348~\textsc{b.c.}\contentsfill \pageref{png063a}\\
\qquad Introduction in geometry of the method of analysis\contentsfill \pageref{png064a}\\
\qquad Theorem on the duplication of the cube\contentsfill \pageref{png065a}\\
\textsc{Eudoxus}, 408--355~\textsc{b.c.}\contentsfill \pageref{png065b}\\
\qquad Theorems on the golden section\contentsfill \pageref{png065c}\\
\qquad Introduction of the method of exhaustions\contentsfill \pageref{png066a}\\
Pupils of Plato and Eudoxus\contentsfill \pageref{png067a}\\
\textsc{Menaechmus}, circ.~340~\textsc{b.c.}\contentsfill \pageref{png067b}\\
\qquad Discussion of the conic sections\contentsfill \pageref{png068a}\\
\qquad His two solutions of the duplication of the cube\contentsfill \pageref{png068b}\\
Aristaeus. Theaetetus\contentsfill \pageref{png069a}\\
Aristotle, 384--322~\textsc{b.c.}\contentsfill \pageref{png069b}\\*
Questions on mechanics. Letters used to indicate magnitudes\contentsfill \pageref{png069c}

\tocchapter*{Chapter IV\@. The First Alexandrian School. \\*
Circ.~300--30~\textsc{b.c.}}

Authorities\contentsfill \pageref{png071b}\\*
Foundation of Alexandria\contentsfill \pageref{png072a}\\
\emph{The Third Century before Christ}\contentsfill \pageref{png073a}\\
\textsc{Euclid}, circ.~330--275~\textsc{b.c.}\contentsfill \pageref{png073b}\\
\qquad Euclid's \emph{Elements}\contentsfill \pageref{png074a}\\
\qquad The \emph{Elements} as a text-book of geometry\contentsfill \pageref{png075a}\\
\qquad The \emph{Elements} as a text-book of the theory of numbers\contentsfill \pageref{png078a}\\
\qquad Euclid's other works\contentsfill \pageref{png081a}\\
Aristarchus, circ.~310--250~\textsc{b.c.}\contentsfill \pageref{png083a}\\
\qquad Method of determining the distance of the sun\contentsfill \pageref{png083b}\\
Conon. Dositheus. Zeuxippus. Nicoteles\contentsfill \pageref{png085a}\\
\textsc{Archimedes}, 287--212~\textsc{b.c.}\contentsfill \pageref{png085b}\\
\qquad His works on plane geometry\contentsfill \pageref{png088a}\\
\qquad His works on geometry of three dimensions\contentsfill \pageref{png091a}\\
\qquad His two papers on arithmetic, and the ``cattle problem''\contentsfill \pageref{png092a}\\
\qquad His works on the statics of solids and fluids\contentsfill \pageref{png094a}\\
\qquad His astronomy\contentsfill \pageref{png097a}\\
\qquad The principles of geometry assumed by Archimedes\contentsfill \pageref{png097b}\\
\textsc{Apollonius}, circ.~260--200~\textsc{b.c.}\contentsfill \pageref{png098a}\\
\qquad His conic sections\contentsfill \pageref{png098b}\\
\qquad His other works\contentsfill \pageref{png101a}\\
\qquad His solution of the duplication of the cube\contentsfill \pageref{png102a}\\
\qquad Contrast between his geometry and that of Archimedes\contentsfill \pageref{png103a}\\
Eratosthenes, 275--194~\textsc{b.c.}\contentsfill \pageref{png104a}\\
\qquad The Sieve of Eratosthenes\contentsfill \pageref{png104b}\\
\emph{The Second Century before Christ}\contentsfill \pageref{png105a}\\
Hypsicles (Euclid, book xiv). Nicomedes. Diocles\contentsfill \pageref{png106a}\\
Perseus. Zenodorus\contentsfill \pageref{png107a}\\%[**TN: Indented in original text]
\textsc{Hipparchus}, circ.~130~\textsc{b.c.}\contentsfill \pageref{png107b}\\
\qquad Foundation of scientific astronomy\contentsfill \pageref{png108a}\\
\qquad Foundation of trigonometry\contentsfill \pageref{png109a}\\
\textsc{Hero} of Alexandria, circ.~125~\textsc{b.c.}\contentsfill \pageref{png109b}\\
\qquad Foundation of scientific engineering and of land-surveying\contentsfill \pageref{png109c}\\
\qquad Area of a triangle determined in terms of its sides\contentsfill \pageref{png110a}\\
\qquad Features of Hero's works\contentsfill \pageref{png112a}\\
\emph{The First Century before Christ}\contentsfill \pageref{png112b}\\
Theodosius\contentsfill \pageref{png112c}\\
Dionysodorus\contentsfill \pageref{png113a}\\
\emph{End of the First Alexandrian School}\contentsfill \pageref{png113b}\\*
Egypt constituted a Roman province\contentsfill \pageref{png113c}

\tocchapter*{Chapter V\@. The Second Alexandrian School. \\*
30~\textsc{b.c.}--641~\textsc{a.d.}}

Authorities\contentsfill \pageref{png114b}\\*
\emph{The First Century after Christ}\contentsfill \pageref{png115a}\\
Serenus. Menelaus\contentsfill \pageref{png115b}\\
Nicomachus\contentsfill \pageref{png115c}\\
\qquad Introduction of the arithmetic current in medieval Europe\contentsfill \pageref{png116a}\\
\emph{The Second Century after Christ}\contentsfill \pageref{png116b}\\
Theon of Smyrna. Thymaridas\contentsfill \pageref{png116c}\\
\textsc{Ptolemy}, died in 168\contentsfill \pageref{png117a}\\
\qquad The \emph{Almagest}\contentsfill \pageref{png117b}\\
\qquad Ptolemy's astronomy\contentsfill \pageref{png117c}\\
\qquad Ptolemy's geometry\contentsfill \pageref{png119a}\\
\emph{The Third Century after Christ}\contentsfill \pageref{png120a}\\
Pappus, circ.~280\contentsfill \pageref{png120b}\\
\qquad The \mytextgreek{Sunagwg'h}, a synopsis of Greek mathematics\contentsfill \pageref{png120c}\\
\emph{The Fourth Century after Christ}\contentsfill \pageref{png122a}\\
Metrodorus. Elementary problems in arithmetic and algebra\contentsfill \pageref{png123a}\\
Three stages in the development of algebra\contentsfill \pageref{png124a}\\
\textsc{Diophantus}, circ.~320 (?)\contentsfill \pageref{png124b}\\
\qquad Introduction of syncopated algebra in his \emph{Arithmetic}\contentsfill \pageref{png125a}\\
\qquad The notation, methods, and subject-matter of the work\contentsfill \pageref{png125b}\\
\qquad His \emph{Porisms}\contentsfill \pageref{png131a}\\
\qquad Subsequent neglect of his discoveries\contentsfill \pageref{png131b}\\
Iamblichus\contentsfill \pageref{png131c}\\
Theon of Alexandria. Hypatia\contentsfill \pageref{png132a}\\
Hostility of the Eastern Church to Greek science\contentsfill \pageref{png132b}\\
\emph{The Athenian School (in the Fifth Century)}\contentsfill \pageref{png132c}\\
Proclus, 412--485. Damascius. Eutocius\contentsfill \pageref{png133a}\\
\emph{Roman Mathematics}\contentsfill \pageref{png134a}\\
Nature and extent of the mathematics read at Rome\contentsfill \pageref{png134b}\\
Contrast between the conditions at Rome and at Alexandria\contentsfill \pageref{png135a}\\
\emph{End of the Second Alexandrian School}\contentsfill \pageref{png136a}\\*
The capture of Alexandria, and end of the Alexandrian Schools\contentsfill \pageref{png136b}

\tocchapter*{Chapter VI\@. The Byzantine School. 641--1453.}

Preservation of works of the great Greek Mathematicians\contentsfill \pageref{png137b}\\*
Hero of Constantinople. Psellus. Planudes. Barlaam. Argyrus\contentsfill \pageref{png138a}\\
Nicholas Rhabdas, Pachymeres. Moschopulus (Magic Squares)\contentsfill \pageref{png139a}\\*
Capture of Constantinople, and dispersal of Greek Mathematicians\contentsfill \pageref{png141a}

\tocchapter*{Chapter VII\@. Systems of Numeration and Primitive Arithmetic.}

Authorities\contentsfill \pageref{png142b}\\*
Methods of counting and indicating numbers among primitive races\contentsfill \pageref{png142c}\\
Use of the abacus or swan-pan for practical calculation\contentsfill \pageref{png144a}\\
Methods of representing numbers in writing\contentsfill \pageref{png147a}\\
The Roman and Attic symbols for numbers\contentsfill \pageref{png147b}\\
The Alexandrian (or later Greek) symbols for numbers\contentsfill \pageref{png148a}\\
Greek arithmetic\contentsfill \pageref{png148b}\\*
Adoption of the Arabic system of notation among civilized races\contentsfill \pageref{png149a}

\tocperiod[1.83]{secondperiod}

\emph{This period begins about the sixth century, and may be said to end
with the invention of analytical geometry and of the infinitesimal calculus.
The characteristic feature of this period is the creation or development of
modern arithmetic, algebra, and trigonometry.}

\tocchapter*{Chapter VIII\@. The Rise Of Learning In Western Europe. \\*
Circ.~600--1200.}

Authorities\contentsfill \pageref{png151c}\\*
\emph{Education in the Sixth, Seventh, and Eighth Centuries}\contentsfill \pageref{png151b}\\
The Monastic Schools\contentsfill \pageref{png151d}\\
Boethius, circ.~475--526\contentsfill \pageref{png152a}\\
\qquad Medieval text-books in geometry and arithmetic\contentsfill \pageref{png153a}\\
Cassiodorus, 490--566. Isidorus of Seville, 570--636\contentsfill \pageref{png153b}\\
\emph{The Cathedral and Conventual Schools}\contentsfill \pageref{png154a}\\
The Schools of Charles the Great\contentsfill \pageref{png154b}\\
Alcuin, 735--804\contentsfill \pageref{png154c}\\
Education in the Ninth and Tenth Centuries\contentsfill \pageref{png156a}\\
Gerbert (Sylvester II.), died in 1003\contentsfill \pageref{png156b}\\
Bernelinus\contentsfill \pageref{png159a}\\
\emph{The Early Medieval Universities}\contentsfill \pageref{png159b}\\
Rise during the twelfth century of the earliest universities\contentsfill \pageref{png159c}\\
Development of the medieval universities\contentsfill \pageref{png160a}\\*
Outline of the course of studies in a medieval university\contentsfill \pageref{png161a}

\tocchapter*{Chapter IX\@. The Mathematics Of The Arabs.}

Authorities\contentsfill \pageref{png164b}\\*
\emph{Extent of Mathematics obtained from Greek Sources}\contentsfill \pageref{png164c}\\
The College of Scribes\contentsfill \pageref{png165a}\\
\emph{Extent of Mathematics obtained from the (Aryan) Hindoos}\contentsfill \pageref{png166a}\\
\textsc{Arya-Bhata}, circ.~530\contentsfill \pageref{png167a}\\
\qquad His algebra and trigonometry (in his \emph{Aryabhathiya})\contentsfill \pageref{png167b}\\
\textsc{Brahmagupta}, circ.~640\contentsfill \pageref{png167c}\\
\qquad His algebra and geometry (in his \emph{Siddhanta})\contentsfill \pageref{png168a}\\
\textsc{Bhaskara}, circ.~1140\contentsfill \pageref{png170a}\\
\qquad The \emph{Lilavati} or arithmetic; decimal numeration used\contentsfill \pageref{png170b}\\
\qquad The \emph{Bija Ganita} or algebra\contentsfill \pageref{png173a}\\
\emph{Development of Mathematics in Arabia}\contentsfill \pageref{png175a}\\
\textsc{Alkarismi} or \textsc{Al-Khw\={a}rizm\={i}}, circ.~830\contentsfill \pageref{png175b}\\
\qquad His \emph{Al-gebr we' l mukabala}\contentsfill \pageref{png176a}\\
\qquad His solution of a quadratic equation\contentsfill \pageref{png177a}\\
\qquad Introduction of Arabic or Indian system of numeration\contentsfill \pageref{png178a}\\
\textsc{Tabit ibn Korra}, 836--901; solution of a cubic equation\contentsfill \pageref{png178b}\\
Alkayami. Alkarki. Development of algebra\contentsfill \pageref{png179a}\\
Albategni. Albuzjani. Development of trigonometry\contentsfill \pageref{png181a}\\
Alhazen. Abd-al-gehl. Development of geometry\contentsfill \pageref{png181b}\\*
Characteristics of the Arabian School\contentsfill \pageref{png182a}

\tocchapter*{Chapter X\@. Introduction of Arabian Works into Europe.\\*
Circ.~1150--1450.}

\emph{The Eleventh Century}\contentsfill \pageref{png185a}\\*
Moorish Teachers. Geber ibn Aphla. Arzachel\contentsfill \pageref{png185b}\\
\emph{The Twelfth Century}\contentsfill \pageref{png185c}\\
Adelhard of Bath\contentsfill \pageref{png185d}\\
Ben-Ezra. Gerard. John Hispalensis\contentsfill \pageref{png186a}\\
\emph{The Thirteenth Century}\contentsfill \pageref{png187a}\\
\textsc{Leonardo of Pisa}, circ.~1175--1230\contentsfill \pageref{png187b}\\
\qquad The \emph{Liber Abaci}, 1202\contentsfill \pageref{png187c}\\
\qquad The introduction of the Arabic numerals into commerce\contentsfill \pageref{png188a}\\
\qquad The introduction of the Arabic numerals into science\contentsfill \pageref{png188b}\\
\qquad The mathematical tournament\contentsfill \pageref{png189a}\\
Frederick II., 1194--1250\contentsfill \pageref{png190a}\\
\textsc{Jordanus}, circ.~1220\contentsfill \pageref{png191a}\\
\qquad His \emph{De Numeris Datis}; syncopated algebra\contentsfill \pageref{png192a}\\
Holywood\contentsfill \pageref{png194a}\\
\textsc{Roger Bacon}, 1214--1294\contentsfill \pageref{png194b}\\
Campanus\contentsfill \pageref{png197a}\\
\emph{The Fourteenth Century}\contentsfill \pageref{png197b}\\
Bradwardine\contentsfill \pageref{png197c}\\
Oresmus\contentsfill \pageref{png198a}\\
The reform of the university curriculum\contentsfill \pageref{png199a}\\
\emph{The Fifteenth Century}\contentsfill \pageref{png200a}\\*
Beldomandi\contentsfill \pageref{png200b}

\tocchapter*{Chapter XI\@. The Development Of Arithmetic. \\*
Circ.~1300--1637.}

Authorities\contentsfill \pageref{png202b}\\*
The Boethian arithmetic\contentsfill \pageref{png202c}\\
Algorism or modern arithmetic\contentsfill \pageref{png203a}\\
The Arabic (or Indian) symbols: history of\contentsfill \pageref{png204a}\\
Introduction into Europe by science, commerce, and calendars\contentsfill \pageref{png206a}\\
Improvements introduced in algoristic arithmetic\contentsfill \pageref{png208a}\\
\qquad   (i)~Simplification of the fundamental processes\contentsfill \pageref{png208b}\\
\qquad  (ii)~Introduction of signs for addition and subtraction\contentsfill \pageref{png214a}\\
\qquad (iii)~Invention of logarithms, 1614\contentsfill \pageref{png215a}\\*
\qquad  (iv)~Use of decimals, 1619\contentsfill \pageref{png217a}

\tocchapter*{Chapter XII\@. The Mathematics of the Renaissance.\\*
Circ.~1450--1637.}

Authorities\contentsfill \pageref{png219b}\\*
Effect of invention of printing. The renaissance\contentsfill \pageref{png219c}\\
\emph{Development of Syncopated Algebra and Trigonometry}\contentsfill \pageref{png221a}\\
\textsc{Regiomontanus}, 1436--1476\contentsfill \pageref{png221b}\\
\qquad His \emph{De Triangulis} (printed in 1496)\contentsfill \pageref{png222a}\\
Purbach, 1423--1461. Cusa, 1401--1464. Chuquet, circ.~1484\contentsfill \pageref{png225a}\\
Introduction and origin of symbols $+$ and $-$\contentsfill \pageref{png226a}\\
Pacioli or Lucas di Burgo, circ.~1500\contentsfill \pageref{png228a}\\
\qquad His arithmetic and geometry, 1494\contentsfill \pageref{png229a}\\
Leonardo da Vinci, 1452--1519\contentsfill \pageref{png232a}\\
Dürer, 1471--1528. Copernicus, 1473--1543\contentsfill \pageref{png233a}\\
Record, 1510--1558; introduction of symbol for equality\contentsfill \pageref{png234a}\\%[**TN: "1510-1588" in original text]
Rudolff, circ.~1525. Riese, 1489--1559\contentsfill \pageref{png235a}\\
\textsc{Stifel}, 1486--1567\contentsfill \pageref{png235b}\\
\qquad His \emph{Arithmetica Integra}, 1544\contentsfill \pageref{png236a}\\
\textsc{Tartaglia}, 1500--1557\contentsfill \pageref{png237a}\\%[**TN: "1500-1559" in original text]
\qquad His solution of a cubic equation, 1535\contentsfill \pageref{png238a}\\
\qquad His arithmetic, 1556--1560\contentsfill \pageref{png239a}\\
\textsc{Cardan}, 1501--1576\contentsfill \pageref{png241a}\\
\qquad His \emph{Ars Magna}, 1545; the third work printed on algebra.\contentsfill \pageref{png242a}\\
\qquad His solution of a cubic equation\contentsfill \pageref{png244a}\\
Ferrari, 1522--1565; solution of a biquadratic equation\contentsfill \pageref{png245a}\\
Rheticus, 1514--1576. Maurolycus. Borrel. Xylander\contentsfill \pageref{png246a}\\
Commandino. Peletier. Romanus. Pitiscus. Ramus. 1515--1572\contentsfill \pageref{png247a}\\
Bombelli, circ.~1570\contentsfill \pageref{png248a}\\
\emph{Development of Symbolic Algebra}\contentsfill \pageref{png249a}\\
\textsc{Vieta}, 1540--1603\contentsfill \pageref{png249b}\\
\qquad The \emph{In Artem}; introduction of symbolic algebra, 1591\contentsfill \pageref{png251a}\\
\qquad Vieta's other works\contentsfill \pageref{png252a}\\
Girard, 1595--1632; development of trigonometry and algebra\contentsfill \pageref{png254a}\\%[**TN: "1590--1633" in original text]
\textsc{Napier}, 1550--1617; introduction of logarithms, 1614\contentsfill \pageref{png255a}\\
Briggs, 1561--1631; calculations of tables of logarithms\contentsfill \pageref{png256a}\\%[**TN: "1556--1631" in original text]
\textsc{Harriot}, 1560--1621; development of analysis in algebra\contentsfill \pageref{png257a}\\
Oughtred, 1574--1660\contentsfill \pageref{png258a}\\*
\emph{The Origin of the more Common Symbols in Algebra}\contentsfill \pageref{png259a}

\tocchapter*{Chapter XIII\@. The Close of the Renaissance. \\*
Circ.~1586--1637.}

Authorities\contentsfill \pageref{png264b}\\*
\emph{Development of Mechanics and Experimental Methods}\contentsfill \pageref{png264c}\\
\textsc{Stevinus}, 1548--1620\contentsfill \pageref{png264d}\\
\qquad Commencement of the modern treatment of statics, 1586\contentsfill \pageref{png265a}\\
\textsc{Galileo}, 1564--1642\contentsfill \pageref{png267a}\\
\qquad Commencement of the science of dynamics\contentsfill \pageref{png268a}\\
\qquad Galileo's astronomy\contentsfill \pageref{png269a}\\
Francis Bacon, 1561--1626. Guldinus, 1577--1643\contentsfill \pageref{png272a}\\
Wright, 1560--1615; construction of maps\contentsfill \pageref{png273a}\\
Snell, 1591--1626\contentsfill \pageref{png274a}\\
\emph{Revival of Interest in Pure Geometry}\contentsfill \pageref{png274b}\\
\textsc{Kepler}, 1571--1630\contentsfill \pageref{png274c}\\
\qquad His \emph{Paralipomena}, 1604; principle of continuity\contentsfill \pageref{png276a}\\
\qquad His \emph{Stereometria}, 1615; use of infinitesimals\contentsfill \pageref{png276b}\\
\qquad Kepler's laws of planetary motion, 1609 and 1619\contentsfill \pageref{png276c}\\
Desargues, 1593--1662\contentsfill \pageref{png277a}\\
\qquad His \emph{Brouillon project}; use of projective geometry\contentsfill \pageref{png277b}\\*
\emph{Mathematical Knowledge at the Close of the Renaissance}\contentsfill \pageref{png278a}

\tocperiod{thirdperiod}

\emph{This period begins with the invention of analytical geometry and the
infinitesimal calculus. The mathematics is far more complex than that
produced in either of the preceding periods: but it may be generally described
as characterized by the development of analysis, and its application
to the phenomena of nature.}

\tocchapter*{Chapter XIV\@. The History of Modern Mathematics.}

Treatment of the subject\contentsfill \pageref{png281b}\\*
Invention of analytical geometry and the method of indivisibles\contentsfill \pageref{png282a}\\
Invention of the calculus\contentsfill \pageref{png283a}\\
Development of mechanics\contentsfill \pageref{png283b}\\
Application of mathematics to physics\contentsfill \pageref{png284a}\\*
Recent development of pure mathematics\contentsfill \pageref{png285a}

\tocchapter*{Chapter XV\@. History of Mathematics from Descartes to Huygens.\quad
Circ.~1635--1675.}

Authorities\contentsfill \pageref{png286b}\\*
\textsc{Descartes}, 1596--1650\contentsfill \pageref{png286c}\\
\qquad His views on philosophy\contentsfill \pageref{png289a}\\
\qquad His invention of analytical geometry, 1637\contentsfill \pageref{png290a}\\
\qquad His algebra, optics, and theory of vortices\contentsfill \pageref{png294a}\\
\textsc{Cavalieri}, 1598--1647\contentsfill \pageref{png296a}\\
\qquad The method of indivisibles\contentsfill \pageref{png297a}\\
\textsc{Pascal}, 1623--1662\contentsfill \pageref{png299a}\\
\qquad His geometrical conics\contentsfill \pageref{png302a}\\
\qquad The arithmetical triangle\contentsfill \pageref{png302b}\\
\qquad Foundation of the theory of probabilities, 1654\contentsfill \pageref{png303a}\\
\qquad His discussion of the cycloid\contentsfill \pageref{png305a}\\
\textsc{Wallis}, 1616--1703\contentsfill \pageref{png306a}\\
\qquad The \emph{Arithmetica Infinitorum}, 1656\contentsfill \pageref{png307a}\\
\qquad Law of indices in algebra\contentsfill \pageref{png307b}\\
\qquad Use of series in quadratures\contentsfill \pageref{png308a}\\
\qquad Earliest rectification of curves, 1657\contentsfill \pageref{png309a}\\
\qquad Wallis's algebra\contentsfill \pageref{png310a}\\
\textsc{Fermat}, 1601--1665\contentsfill \pageref{png311a}\\
\qquad His investigations on the theory of numbers\contentsfill \pageref{png312a}\\
\qquad His use in geometry of analysis and of infinitesimals\contentsfill \pageref{png316a}\\
\qquad Foundation of the theory of probabilities, 1654\contentsfill \pageref{png318a}\\
\textsc{Huygens}, 1629--1695\contentsfill \pageref{png319a}\\
\qquad The \emph{Horologium Oscillatorium}, 1673\contentsfill \pageref{png320a}\\
\qquad The undulatory theory of light\contentsfill \pageref{png321a}\\
\emph{Other Mathematicians of this Time}\contentsfill \pageref{png323a}\\
Bachet\contentsfill \pageref{png323b}\\
Mersenne; theorem on primes and perfect numbers\contentsfill \pageref{png324a}\\%[**TN: "Marsenne" in original text]
Roberval. Van Schooten. Saint-Vincent\contentsfill \pageref{png325a}\\
Torricelli. Hudde. Frénicle\contentsfill \pageref{png326a}\\
De Laloubère. Mercator. Barrow; the differential triangle\contentsfill \pageref{png327a}\\
Brouncker; continued fractions\contentsfill \pageref{png330a}\\
James Gregory; distinction between convergent and divergent series\contentsfill \pageref{png331a}\\
Sir Christopher Wren\contentsfill \pageref{png332a}\\
Hooke. Collins\contentsfill \pageref{png333a}\\
Pell. Sluze. Viviani\contentsfill \pageref{png334a}\\*
Tschirnhausen. De la Hire. Roemer. Rolle\contentsfill \pageref{png335a}

\tocchapter*{Chapter XVI\@. The Life and Works of Newton.}

Authorities\contentsfill \pageref{png337b}\\*
Newton's school and undergraduate life\contentsfill \pageref{png338a}\\
Investigations in 1665--1666 on fluxions, optics, and gravitation\contentsfill \pageref{png339a}\\
\qquad His views on gravitation, 1666\contentsfill \pageref{png339b}\\
Researches in 1667--1669\contentsfill \pageref{png341a}\\
Elected Lucasian professor, 1669\contentsfill \pageref{png342a}\\
Optical lectures and discoveries, 1669--1671\contentsfill \pageref{png342b}\\
Emission theory of light, 1675\contentsfill \pageref{png344a}\\
The Leibnitz Letters, 1676\contentsfill \pageref{png345a}\\
Discoveries on gravitation, 1679\contentsfill \pageref{png348a}\\
Discoveries and lectures on algebra, 1673--1683\contentsfill \pageref{png348b}\\
Discoveries and lectures on gravitation, 1684\contentsfill \pageref{png351a}\\
The \emph{Principia}, 1685--1686\contentsfill \pageref{png352a}\\
\qquad The subject-matter of the \emph{Principia}\contentsfill \pageref{png353a}\\
\qquad Publication of the \emph{Principia}\contentsfill \pageref{png355a}\\
Investigations and work from 1686 to 1696\contentsfill \pageref{png356a}\\
Appointment at the Mint, and removal to London, 1696\contentsfill \pageref{png357a}\\
Publication of the \emph{Optics}, 1704\contentsfill \pageref{png357b}\\
\qquad Appendix on classification of cubic curves\contentsfill \pageref{png358a}\\
\qquad Appendix on quadrature by means of infinite series\contentsfill \pageref{png359a}\\
\qquad Appendix on method of fluxions\contentsfill \pageref{png361a}\\
The invention of fluxions and the infinitesimal calculus\contentsfill \pageref{png365a}\\
Newton's death, 1727\contentsfill \pageref{png366a}\\
List of his works\contentsfill \pageref{png366b}\\
Newton's character\contentsfill \pageref{png367a}\\*
Newton's discoveries\contentsfill \pageref{png369a}

\tocchapter*{Chapter XVII\@. Leibnitz and the Mathematicians
of the First Half of the Eighteenth Century.}

Authorities\contentsfill \pageref{png371b}\\*
\emph{Leibnitz and the Bernoullis}\contentsfill \pageref{png371c}\\
\textsc{Leibnitz}, 1646--1716\contentsfill \pageref{png371d}\\
\qquad His system of philosophy, and services to literature\contentsfill \pageref{png373a}\\
\qquad The controversy as to the origin of the calculus\contentsfill \pageref{png374a}\\
\qquad His memoirs on the infinitesimal calculus\contentsfill \pageref{png380a}\\
\qquad His papers on various mechanical problems\contentsfill \pageref{png381a}\\
\qquad Characteristics of his work\contentsfill \pageref{png383a}\\
\textsc{James Bernoulli}, 1654--1705\contentsfill \pageref{png384a}\\
\textsc{John Bernoulli}, 1667--1748\contentsfill \pageref{png385a}\\
The younger Bernouillis\contentsfill \pageref{png386a}\\
\emph{Development of Analysis on the Continent}\contentsfill \pageref{png387a}\\
L'Hospital, 1661--1704\contentsfill \pageref{png387b}\\
Varignon, 1654--1722. De Montmort. Nicole\contentsfill \pageref{png388a}\\
Parent. Saurin. De Gua. Cramer, 1704--1752\contentsfill \pageref{png389a}\\
Riccati, 1676--1754. Fagnano, 1682--1766\contentsfill \pageref{png390a}\\
\textsc{Clairaut}, 1713--1765\contentsfill \pageref{png391a}\\
\textsc{D'Alembert}, 1717--1783\contentsfill \pageref{png392a}\\
\qquad Solution of a partial differential equation of the second order\contentsfill \pageref{png394a}\\
Daniel Bernoulli, 1700--1782\contentsfill \pageref{png395a}\\
\emph{English Mathematicians of the Eighteenth Century}\contentsfill \pageref{png396a}\\
David Gregory, 1661--1708. Halley, 1656--1742\contentsfill \pageref{png397a}\\
Ditton, 1675--1715\contentsfill \pageref{png398a}\\
\textsc{Brook Taylor}, 1685--1731\contentsfill \pageref{png398b}\\
\qquad Taylor's theorem\contentsfill \pageref{png399a}\\
\qquad Taylor's physical researches\contentsfill \pageref{png400a}\\
Cotes, 1682--1716\contentsfill \pageref{png400b}\\
Demoivre, 1667--1754; development of trigonometry\contentsfill \pageref{png401a}\\
\textsc{Maclaurin}, 1698--1746\contentsfill \pageref{png402a}\\
\qquad His geometrical discoveries\contentsfill \pageref{png403a}\\
\qquad The \emph{Treatise of Fluxions}\contentsfill \pageref{png404a}\\
\qquad His propositions on attractions\contentsfill \pageref{png405a}\\*
Stewart, 1717--1785. Thomas Simpson, 1710--1761\contentsfill \pageref{png406a}

\tocchapter*{Chapter XVIII\@. Lagrange, Laplace, and their Contemporaries.\quad
Circ.~1740--1830.}

Characteristics of the mathematics of the period\contentsfill \pageref{png409b}\\*
\emph{Development of Analysis and Mechanics}\contentsfill \pageref{png411a}\\
\textsc{Euler}, 1707--1783\contentsfill \pageref{png411b}\\
\qquad The \emph{Introductio in Analysin Infinitorum}, 1748\contentsfill \pageref{png412a}\\
\qquad The \emph{Institutiones Calculi Differentialis}, 1755\contentsfill \pageref{png414a}\\
\qquad The \emph{Institutiones Calculi Integralis}, 1768--1770\contentsfill \pageref{png414b}\\
\qquad The \emph{Anleitung zur Algebra}, 1770\contentsfill \pageref{png415a}\\
\qquad Euler's works on mechanics and astronomy\contentsfill \pageref{png416a}\\
Lambert, 1728--1777\contentsfill \pageref{png418a}\\
Bézout, 1730--1783. Trembley, 1749--1811. Arbogast, 1759--1803\contentsfill \pageref{png419a}\\
\textsc{Lagrange}, 1736--1813\contentsfill \pageref{png419b}\\
\qquad Memoirs on various subjects\contentsfill \pageref{png421a}\\
\qquad The \emph{Mécanique analytique}, 1788\contentsfill \pageref{png424a}\\
\qquad The \emph{Théorie} and \emph{Calcul des fonctions}, 1797, 1804\contentsfill \pageref{png428a}\\
\qquad The \emph{Résolution des équations numériques}, 1798.\contentsfill \pageref{png428b}\\
\qquad Characteristics of Lagrange's work\contentsfill \pageref{png429a}\\
\textsc{Laplace}, 1749--1827\contentsfill \pageref{png430a}\\
\qquad Memoirs on astronomy and attractions, 1773--1784\contentsfill \pageref{png431a}\\
\qquad Use of spherical harmonics and the potential\contentsfill \pageref{png431b}\\
\qquad Memoirs on problems in astronomy, 1784--1786\contentsfill \pageref{png432a}\\
\qquad The \emph{Mécanique céleste} and \emph{Exposition du système du monde}\contentsfill \pageref{png433a}\\
\qquad The Nebular Hypothesis\contentsfill \pageref{png433b}\\
\qquad The Meteoric Hypothesis\contentsfill \pageref{png433c}\\
\qquad The \emph{Théorie analytique des probabilités}, 1812\contentsfill \pageref{png436a}\\
\qquad The Method of Least Squares\contentsfill \pageref{png436b}\\
\qquad Other researches in pure mathematics and in physics\contentsfill \pageref{png437a}\\
\qquad Characteristics of Laplace's work\contentsfill \pageref{png438a}\\
\qquad Character of Laplace\contentsfill \pageref{png439a}\\
\textsc{Legendre}, 1752--1833\contentsfill \pageref{png439b}\\
\qquad His memoirs on attractions\contentsfill \pageref{png440a}\\
\qquad The \emph{Théorie des nombres}, 1798\contentsfill \pageref{png441a}\\
\qquad Law of quadratic reciprocity\contentsfill \pageref{png441b}\\
\qquad The \emph{Calcul intégral} and the \emph{Fonctions elliptiques}\contentsfill \pageref{png442a}\\
Pfaff, 1765--1825\contentsfill \pageref{png443a}\\
\emph{Creation of Modern Geometry}\contentsfill \pageref{png443b}\\
Monge, 1746--1818\contentsfill \pageref{png444a}\\%[**TN: "1748--1818" in original text]
Lazare Carnot, 1753--1823. Poncelet, 1788--1867\contentsfill \pageref{png446a}\\
\emph{Development of Mathematical Physics}\contentsfill \pageref{png447a}\\
Cavendish, 1731--1810\contentsfill \pageref{png447b}\\
Rumford, 1753--1815. Young, 1773--1829\contentsfill \pageref{png448a}\\
Dalton, 1766--1844\contentsfill \pageref{png449a}\\
\textsc{Fourier}, 1768--1830\contentsfill \pageref{png450a}\\
Sadi Carnot; foundation of thermodynamics\contentsfill \pageref{png451a}\\
\textsc{Poisson}, 1781--1840\contentsfill \pageref{png451b}\\
Ampère, 1775--1836. Fresnel, 1788--1827. Biot, 1774--1862\contentsfill \pageref{png454a}\\
Arago, 1786--1853\contentsfill \pageref{png455a}\\
\emph{Introduction of Analysis into England}\contentsfill \pageref{png456a}\\
Ivory, 1765--1842\contentsfill \pageref{png457a}\\
The Cambridge Analytical School\contentsfill \pageref{png457b}\\
Woodhouse, 1773--1827\contentsfill \pageref{png458a}\\*
Peacock, 1791--1858. Babbage, 1792--1871. John Herschel, 1792--1871\contentsfill \pageref{png459a}

\tocchapter*{Chapter XIX\@. Mathematics of the Nineteenth Century.}

Creation of new branches of mathematics\contentsfill \pageref{png462b}\\*
Difficulty in discussing the mathematics of this century\contentsfill \pageref{png463a}\\
Account of contemporary work not intended to be exhaustive\contentsfill \pageref{png463b}\\
Authorities\contentsfill \pageref{png463c}\\
\textsc{Gauss}, 1777--1855\contentsfill \pageref{png465a}\\
\qquad Investigations in astronomy\contentsfill \pageref{png466a}\\
\qquad Investigations in electricity\contentsfill \pageref{png467a}\\
\qquad The \emph{Disquisitiones Arithmeticae}, 1801\contentsfill \pageref{png470a}\\
\qquad His other discoveries\contentsfill \pageref{png471a}\\
\qquad Comparison of Lagrange, Laplace, and Gauss\contentsfill \pageref{png472a}\\
Dirichlet, 1805--1859\contentsfill \pageref{png472b}\\
\emph{Development of the Theory of Numbers}\contentsfill \pageref{png473a}\\
Eisenstein, 1823--1852\contentsfill \pageref{png473b}\\
Henry Smith, 1826--1883\contentsfill \pageref{png474a}\\
Kummer, 1810--1893\contentsfill \pageref{png476a}\\
Notes on other writers on the Theory of Numbers\contentsfill \pageref{png477a}\\
\emph{Development of the Theory of Functions of Multiple Periodicity}\contentsfill \pageref{png479a}\\
\textsc{Abel}, 1802--1829. Abel's Theorem\contentsfill \pageref{png479b}\\
\textsc{Jacobi}, 1804--1851\contentsfill \pageref{png480a}\\
\textsc{Riemann}, 1826--1866\contentsfill \pageref{png482a}\\
Notes on other writers on Elliptic and Abelian Functions\contentsfill \pageref{png483a}\\
\textsc{Weierstrass}, 1815--1897\contentsfill \pageref{png484a}\\
Notes on recent writers on Elliptic and Abelian Functions\contentsfill \pageref{png485a}\\
\emph{The Theory of Functions}\contentsfill \pageref{png485b}\\
\emph{Development of Higher Algebra}\contentsfill \pageref{png486a}\\
\textsc{Cauchy}, 1789--1857\contentsfill \pageref{png487a}\\%[**TN: "1759--1857" in original text]
Argand, 1768--1822; geometrical interpretation of complex numbers\contentsfill \pageref{png489a}\\
\textsc{Sir William Hamilton}, 1805--1865; introduction of quaternions\contentsfill \pageref{png490a}\\
\textsc{Grassmann}, 1809--1877; his non-commutative algebra, 1844\contentsfill \pageref{png491a}\\%[**TN: "algebra, 1884" in original text]
Boole, 1815--1864. De~Morgan, 1806--1871\contentsfill \pageref{png492a}\\
\textsc{Galois}, 1811--1832; theory of discontinuous substitution groups\contentsfill \pageref{png493a}\\
\textsc{Cayley}, 1821--1895\contentsfill \pageref{png493b}\\
\textsc{Sylvester}, 1814--1897\contentsfill \pageref{png494a}\\
\textsc{Lie}, 1842--1889; theory of continuous substitution groups\contentsfill \pageref{png495a}\\
\textsc{Hermite}, 1822--1901\contentsfill \pageref{png496a}\\
Notes on other writers on Higher Algebra\contentsfill \pageref{png497a}\\
\emph{Development of Analytical Geometry}\contentsfill \pageref{png498a}\\
Notes on some recent writers on Analytical Geometry\contentsfill \pageref{png499a}\\
Line Geometry\contentsfill \pageref{png500a}\\
\emph{Analysis.} Names of some recent writers on Analysis\contentsfill \pageref{png500b}\\
\emph{Development of Synthetic Geometry}\contentsfill \pageref{png501a}\\
Steiner, 1796--1863\contentsfill \pageref{png501b}\\
Von Staudt, 1798--1867\contentsfill \pageref{png502a}\\
Other writers on modern Synthetic Geometry\contentsfill \pageref{png502b}\\
\emph{Development of Non-Euclidean Geometry}\contentsfill \pageref{png503a}\\
\qquad Euclid's Postulate on Parallel Lines\contentsfill \pageref{png504a}\\
\qquad Hyperbolic Geometry. Elliptic Geometry\contentsfill \pageref{png504b}\\
\qquad Congruent Figures\contentsfill \pageref{png506a}\\
\emph{Foundations of Mathematics. Assumptions made in the subject}\contentsfill \pageref{png507a}\\
\emph{Kinematics}\contentsfill \pageref{png507b}\\
\emph{Development of the Theory of Mechanics, treated Graphically}\contentsfill \pageref{png507c}\\
\emph{Development of Theoretical Mechanics, treated Analytically}\contentsfill \pageref{png509a}\\
Notes on recent writers on Mechanics\contentsfill \pageref{png510a}\\
\emph{Development of Theoretical Astronomy}\contentsfill \pageref{png511a}\\
Bessel, 1784--1846\contentsfill \pageref{png511b}\\
Leverrier, 1811--1877. Adams, 1819--1892\contentsfill \pageref{png512a}\\
Notes on other writers on Theoretical Astronomy\contentsfill \pageref{png513a}\\
Recent Developments\contentsfill \pageref{png515a}\\*
\emph{Development of Mathematical Physics}\contentsfill \pageref{png516a}

\begin{center}\rule{2cm}{1pt}\end{center}

\textsc{Index}\contentsfill \pageref{pngend}
\end{center}

\mainmatter

\pagestyle{fancy}

\nbChapter{I}{Egyptian and Phoenician Mathematics}{}{}{}
\index{Egyptian mathematics, chap. \textsc{i}}\nblabel{png023a}

\First{The} history of mathematics cannot with certainty be traced
back to any school or period before that of the Ionian Greeks.
The subsequent history may be divided into three periods, the
distinctions between which are tolerably well marked. The first
period is that of the history of mathematics under Greek influence,
this is discussed in chapters \hyperref[chapter:II]{\textsc{ii}} to \hyperref[chapter:VII]{\textsc{vii}}; the second is that
of the mathematics of the middle ages and the renaissance,
this is discussed in chapters \hyperref[chapter:VIII]{\textsc{viii}} to \hyperref[chapter:XIII]{\textsc{xiii}}; the third is that of
modern mathematics, and this is discussed in chapters \hyperref[chapter:XIV]{\textsc{xiv}} to
\hyperref[chapter:XIX]{\textsc{xix}}.

\index{Ionian School, the}\index{Phoenician mathematics|(}\nblabel{png023b}Although the history of mathematics commences with that
of the Ionian schools, there is no doubt that those Greeks who
first paid attention to the subject were largely indebted to
the previous investigations of the Egyptians and Phoenicians.
Our knowledge of the mathematical attainments of those races
is imperfect and partly conjectural, but, such as it is, it is here
briefly summarised. The definite history begins with the \hyperref[chapter:II]{next
chapter}.

On the subject of prehistoric mathematics, we may observe
in the first place that, though all early races which have left
records behind them knew something of numeration and
mechanics, and though the majority were also acquainted with
the elements of land-surveying, yet the rules which they
possessed were in general founded only on the results of observation
and experiment, and were neither deduced from nor did
they form part of any science. The fact then that various
nations in the vicinity of Greece had reached a high state of
civilisation does not justify us in assuming that they had studied
mathematics.

\nblabel{png024a}The only races with whom the Greeks of Asia Minor
(amongst whom our history begins) were likely to have come
into frequent contact were those inhabiting the eastern littoral
of the Mediterranean; and Greek tradition uniformly assigned
the special development of geometry to the Egyptians, and
that of the science of numbers either to the Egyptians or to the
Phoenicians. I discuss these subjects separately.

\index{Arithmetic.!z2@Pre-Hellenic|(}\nblabel{png024b}First, as to the science of \emph{numbers}. So far as the acquirements
of the Phoenicians on this subject are concerned it is
impossible to speak with certainty. The magnitude of the
commercial transactions of Tyre and Sidon necessitated a considerable
development of arithmetic, to which it is probable
the name of science might be properly applied. \index{Squares, table of}A Babylonian
table of the numerical value of the squares of a series of consecutive
integers has been found, and this would seem to indicate
that properties of numbers were studied. \index{Strabo, ref.\ to}\index{Bockh@Böckh on Babylonian measures}According to Strabo
the Tyrians paid particular attention to the sciences of numbers,
navigation, and astronomy; they had, we know, considerable
commerce with their neighbours and kinsmen the Chaldaeans;
and Böckh says that they regularly supplied the weights and
measures used in Babylon. \index{Chaldean mathematics}Now the Chaldaeans had certainly
paid some attention to arithmetic and geometry, as is shown
by their astronomical calculations; and, whatever was the
extent of their attainments in arithmetic, it is almost certain
that the Phoenicians were equally proficient, while it is likely
that the knowledge of the latter, such as it was, was communicated
to the Greeks. On the whole it seems probable that the
early Greeks were largely indebted to the Phoenicians for their
knowledge of practical arithmetic or the art of calculation, and
perhaps also learnt from them a few properties of numbers. It
may be worthy of note that \index{Pythagoras!ref.\ to}Pythagoras was a Phoenician; and
according to \index{Herodotus, ref.\ to}Herodotus, but this is more doubtful, \index{Thales!ref.\ to}Thales was
also of that race.

I may mention that the almost universal use of the \index{Abacus,!ref.\ to}abacus
or swan-pan rendered it easy for the ancients to add and
subtract without any knowledge of theoretical arithmetic.
These instruments will be described later in chapter \hyperref[chapter:VII]{\textsc{vii}}; it
will be sufficient here to say that they afford a concrete way
of representing a number in the decimal scale, and enable the
results of addition and subtraction to be obtained by a merely
mechanical process. This, coupled with a means of representing
the result in writing, was all that was required for practical
purposes.

\index{Rhind papyrus, the|(}\nblabel{png025a}We are able to speak with more certainty on the arithmetic
of the Egyptians. About forty years ago a hieratic papyrus,\footnote
  {See \emph{Ein mathematisches Handbuch der alten Aegypter}, by \index{Eisenlohr, ref.\ to}A.~Eisenlohr,
  second edition, Leipzig, 1891; see also \index{Cantor, M., ref.\ to}Cantor, chap.~i; and \emph{A Short
  History of Greek Mathematics}, by \index{Gow, ref.\ to}J.~Gow, Cambridge, 1884, arts.~12--14.
  Besides these authorities the papyrus has been discussed in memoirs by
  \index{Rodet, ref.\ to}L.~Rodet, \index{Favaro, A., ref.\ to}A.~Favaro, V.~Bobynin, and \index{Weyr, ref.\ to}E.~Weyr.}
forming part of the Rhind collection in the British Museum,
was deciphered, which has thrown considerable light on their
mathematical attainments. \index{Bobynin on Ahmes}The manuscript was written by a
scribe named \index{Ahmes|(}Ahmes at a date, according to Egyptologists,
considerably more than a thousand years before Christ, and it
is believed to be itself a copy, with emendations, of a treatise
more than a thousand years older. The work is called ``directions
for knowing all dark things,'' and consists of a collection of
problems in arithmetic and geometry; the answers are given, but
in general not the processes by which they are obtained. It appears
to be a summary of rules and questions familiar to the priests.

\index{Fractions, continued!treatment of}The first part deals with the reduction of fractions of the
form $2/(2n+1)$ to a sum of fractions each of whose numerators
is unity: for example, Ahmes states that $\frac{2}{29}$ is the sum of
$\frac{1}{24}$, $\frac{1}{58}$, $\frac{1}{174}$, and $\frac{1}{232}$; and $\frac{2}{97}$ is the sum of $\frac{1}{56}$, $\frac{1}{679}$, and $\rule{0ex}{2.1ex}\frac{1}{776}$. %[**TN: adjust line spacing with \rule or fractions clash]
In all the examples $n$ is less than 50. Probably he had no
rule for forming the component fractions, and the answers
given represent the accumulated experiences of previous writers:
\index{Fractions, continued!treatment of}in one solitary case, however, he has indicated his method, for,
after having asserted that $\frac{2}{3}$ is the sum of $\frac{1}{2}$ and $\frac{1}{6}$, he adds that
therefore two-thirds of one-fifth is equal to the sum of a half of
a fifth and a sixth of a fifth, that is, to $\frac{1}{10} + \frac{1}{30}$.

That so much attention was paid to fractions is explained by
the fact that in early times their treatment was found difficult.
The Egyptians and Greeks simplified the problem by reducing
a fraction to the sum of several fractions, in each of which the
numerator was unity, the sole exception to this rule being the
fraction $\frac{2}{3}$. This remained the Greek practice until the sixth
century of our era. The Romans, on the other hand, generally
kept the denominator constant and equal to twelve, expressing
the fraction (approximately) as so many twelfths. \index{Angle,!sexagesimal division}The Babylonians
did the same in astronomy, except that they used sixty
as the constant denominator; and from them through the Greeks
the modern division of a \index{Degrees, angular}\index{Sexagesimal angles}degree into sixty equal parts is derived.
Thus in one way or the other the difficulty of having to consider
changes in both numerator and denominator was evaded. To-day
when using decimals we often keep a fixed denominator, thus
reverting to the Roman practice.

After considering fractions Ahmes proceeds to some examples
of the fundamental processes of arithmetic. \index{Multiplication, processes of}In multiplication
he seems to have relied on repeated additions. Thus in one
numerical example, where he requires to multiply a certain
number, say $a$, by 13, he first multiplies by 2 and gets $2a$, then
he doubles the results and gets $4a$, then he again doubles the
result and gets $8a$, and lastly he adds together $a$, $4a$, and $8a$.
Probably division was also performed by repeated subtractions, but,
as he rarely explains the process by which he arrived at a result,
this is not certain. After these examples Ahmes goes on to the
solution of some simple numerical equations. For example, he
says ``heap, its seventh, its whole, it makes nineteen,'' by which
he means that the object is to find a number such that the sum
of it and one-seventh of it shall be together equal to 19; and he
gives as the answer $16 + \frac{1}{2} + \frac{1}{8}$, which is correct.

\nblabel{xref:005}\index{Addition, processes of!symbols for}The arithmetical part of the papyrus indicates that he had
some idea of algebraic symbols. \index{Unknown quantity, word or symbol for}\index{Heap for unknown number}The unknown quantity is
always represented by the symbol which means a heap; \index{Plus!symbols for}addition
is sometimes represented by a pair of legs walking forwards,
\index{Minus!z1@symbols for}\index{Subtraction,!symbols for}\index{Negative sign}subtraction by a pair of legs walking backwards or by a flight
of arrows; \index{Equality, symbols for}and equality by the sign
% Superimpose a rotated less-than symbol and a minus sign
\rlap{\rotatebox[origin=c]{-17.9}{\resizebox{\width}{0.7\height}{$<$}}}% less-than is scaled smaller vertically, then rotated.
\hspace{0.053em}%                                                        small space for correct horizontal positioning of minus sign.
\raisebox{3pt}{\resizebox{1.022\width}{0.7\height}{$-$}}.%               minus sign is raised and stretched to match less-than.

The latter part of the book contains various geometrical
problems to which I allude later. He concludes the work with
some arithmetico-algebraical questions, two of which deal with
arithmetical progressions and seem to indicate that he knew
how to sum such series.\index{Arithmetic.!z2@Pre-Hellenic|)}

\index{Geometry.!z1@Egyptian geometry|(}\nblabel{png027a}Second, as to the science of \emph{geometry}. \index{Geometryz@Geometry, origin of|(}Geometry is supposed
to have had its origin in land-surveying; but while it is difficult
to say when the study of numbers and calculation---some knowledge
of which is essential in any civilised state---became a
science, it is comparatively easy to distinguish between the
abstract reasonings of geometry and the practical rules of the
land-surveyor. Some methods of land-surveying must have
been practised from very early times, but the universal tradition
of antiquity asserted that the origin of geometry was to be
sought in Egypt. That it was not indigenous to Greece, and
that it arose from the necessity of surveying, is rendered the
more probable by the derivation of the word from \mytextgreek{g\~h}, the earth, %[**TN: accent is inverted breve in original]
and \mytextgreek{metr'ew}, I measure. Now the Greek geometricians, as far as
we can judge by their extant works, always dealt with the
science as an abstract one: they sought for theorems which
should be absolutely true, and, at any rate in historical times,
would have argued that to measure quantities in terms of a
unit which might have been incommensurable with some of the
magnitudes considered would have made their results mere
approximations to the truth. The name does not therefore
refer to their practice. It is not, however, unlikely that it
indicates the use which was made of geometry among the
Egyptians from whom the Greeks learned it. This also agrees
with the Greek traditions, which in themselves appear probable;
for \index{Herodotus, ref.\ to}Herodotus states that the periodical inundations of the Nile
(which swept away the landmarks in the valley of the river,
and by altering its course increased or decreased the taxable
value of the adjoining lands) rendered a tolerably accurate
system of surveying indispensable, and thus led to a systematic
study of the subject by the priests.

We have no reason to think that any special attention was
paid to geometry by the Phoenicians, or other neighbours of the
Egyptians. \index{Jews, science of}A small piece of evidence which tends to show that
the Jews had not paid much attention to it is to be found in
the mistake made in their sacred books,\footnote
  {I.~Kings, chap.~vii, verse~23, and II.~Chronicles, chap.~iv, verse~2.}
where it is stated that
the circumference of a circle is three times its diameter: \index{Babylonians, mathematics of}the
Babylonians\footnote
  {See \index{Oppert, ref.\ to}J.~Oppert, \emph{Journal Asiatique}, August 1872, and October 1874.}
 also reckoned that \index{P1@$\pi$,!z1@value of}$\pi$ was equal to 3.\index{Geometryz@Geometry, origin of|)}

Assuming, then, that a knowledge of geometry was first
derived by the Greeks from Egypt, we must next discuss the
range and nature of Egyptian geometry.\footnote
  {See \index{Eisenlohr, ref.\ to}Eisenlohr; \index{Cantor, M., ref.\ to}Cantor, chap.~ii; \index{Gow, ref.\ to}Gow, arts.~75, 76; and \emph{Die
  Geometrie der alten Aegypter}, by \index{Weyr, ref.\ to}E.~Weyr, Vienna, 1884.}
That some geometrical
results were known at a date anterior to Ahmes's work
seems clear if we admit, as we have reason to do, that, centuries
before it was written, the following method of obtaining a right
angle was used in laying out the ground-plan of certain buildings.
\index{Orientation of Egyptian temples}The Egyptians were very particular about the exact
orientation of their temples; and they had therefore to obtain
with accuracy a north and south line, as also an east and west
line. By observing the points on the horizon where a star rose
and set, and taking a plane midway between them, they could
obtain a north and south line. To get an east and west line,
which had to be drawn at right angles to this, certain professional
\index{Rope-fasteners, Egyptian}``rope-fasteners'' were employed. These men used a
rope $ABCD$ divided by knots or marks at $B$ and $C$, so that the
lengths $AB$, $BC$, $CD$ were in the ratio $3:4:5$. The length $BC$
was placed along the north and south line, and pegs $P$ and $Q$
inserted at the knots $B$ and $C$. The piece $BA$ (keeping it
stretched all the time) was then rotated round the peg $P$, and
similarly the piece $CD$ was rotated round the peg $Q$, until the
ends $A$ and $D$ coincided; the point thus indicated was marked
by a peg $R$. The result was to form a triangle $PQR$ whose
sides $RP$, $PQ$, $QR$ were in the ratio $3:4:5$. The angle of the
triangle at $P$ would then be a right angle, and the line $PR$
would give an east and west line. \index{Inductive geometry|(}A similar method is constantly
used at the present time by practical engineers for
measuring a right angle. The property employed can be
deduced as a particular case of \index{Eucn@Euc.!a48@\textsc{i}, 48. ref.\ to}Euc.~\textsc{i}, 48; and there is reason
to think that the Egyptians were acquainted with the results of
this proposition and of \index{Eucn@Euc.!a47@\textsc{i}, 47. ref.\ to}Euc.~\textsc{i}, 47, for triangles whose sides are
in the ratio mentioned above. They must also, there is little
doubt, have known that the latter proposition was true for an
isosceles right-angled triangle, as this is obvious if a floor be
paved with tiles of that shape. But though these are interesting
facts in the history of the Egyptian arts we must not press
them too far as showing that geometry was then studied as a
science. Our real knowledge of the nature of Egyptian geometry
depends mainly on the Rhind papyrus.

Ahmes commences that part of his papyrus which deals with
geometry by giving some numerical instances of the contents of
barns. Unluckily we do not know what was the usual shape
of an Egyptian barn, but where it is defined by three linear
measurements, say $a$, $b$, and $c$, the answer is always given as
if he had formed the expression $a × b × (c+\frac{1}{2} c)$. He next
proceeds to find the areas of certain rectilineal figures; if the
text be correctly interpreted, some of these results are wrong.
He then goes on to find the area of a circular field of diameter
12---no unit of length being mentioned---and gives the result
as $(d-\frac{1}{9}d)^{2}$, where $d$ is the diameter of the circle: this is
equivalent to taking $3.1604$ as the value of \index{P1@$\pi$,!z1@value of}$\pi$, the actual value
being very approximately $3.1416$. Lastly, Ahmes gives some
problems on pyramids. \index{Trigonometry.!z1@Ideas of, in Rhind papyrus|(}These long proved incapable of interpretation,
but \index{Cantor, M., ref.\ to}Cantor and \index{Eisenlohr, ref.\ to}Eisenlohr have shown that Ahmes
was attempting to find, by means of data obtained from the
measurement of the external dimensions of a building, the
ratio of certain other dimensions which could not be directly
measured: his process is equivalent to determining the trigonometrical
ratios of certain angles. The data and the results
given agree closely with the dimensions of some of the existing
pyramids. Perhaps all Ahmes's geometrical results were intended
only as approximations correct enough for practical purposes.\index{Trigonometry.!z1@Ideas of, in Rhind papyrus|)}

It is noticeable that all the specimens of Egyptian geometry
which we possess deal only with particular numerical problems
and not with general theorems; and even if a result be stated
as universally true, it was probably proved to be so only by a
wide induction. We shall see later that Greek geometry was
from its commencement deductive. There are reasons for thinking
that Egyptian geometry and arithmetic made little or no
progress subsequent to the date of Ahmes's work; and though
for nearly two hundred years after the time of Thales Egypt
was recognised by the Greeks as an important school of mathematics,
it would seem that, almost from the foundation of the
Ionian school, the Greeks outstripped their former teachers.\index{Geometry.!z1@Egyptian geometry|)}\index{Inductive geometry|)}

It may be added that Ahmes's\index{Ahmes|)} book gives us much that idea
of Egyptian mathematics which we should have gathered from
statements about it by various Greek and Latin authors, who
lived centuries later. Previous to its translation it was commonly
thought that these statements exaggerated the acquirements of
the Egyptians, and its discovery must increase the weight to be
attached to the testimony of these authorities.\index{Rhind papyrus, the|)}

We know nothing of the applied mathematics (if there were
any) of the Egyptians or Phoenicians.\index{Phoenician mathematics|)} The astronomical attainments
of the Egyptians and \index{Chaldean mathematics}Chaldaeans were no doubt considerable,
though they were chiefly the results of observation: the
Phoenicians are said to have confined themselves to studying
what was required for navigation. Astronomy, however, lies
outside the range of this book.

\index{Chinese, early mathematics|(}I do not like to conclude the chapter without a brief mention
of the Chinese, since at one time it was asserted that they were
familiar with the sciences of arithmetic, geometry, mechanics,
optics, navigation, and astronomy nearly three thousand years
ago, and a few writers were inclined to suspect (for no evidence
was forthcoming) that some knowledge of this learning had
filtered across Asia to the West. It is true that at a very early
period the Chinese were acquainted with several geometrical or
rather architectural implements, such as the rule, square, compasses,
and level; with a few mechanical machines, such as the
wheel and axle; that they knew of the characteristic property
of the magnetic needle; and were aware that astronomical events
occurred in cycles. But the careful investigations of \index{Sedillot@Sédillot, ref.\ to}L.~A.~Sédillot\footnote
  {See \index{Boncompagni, ref.\ to}Boncompagni's \emph{Bulletino di bibliografia e di storia delle scienze
  matematiche e fisiche} for May, 1868, vol.~i, pp.~161--166. On Chinese
  mathematics, mostly of a later date, see \index{Cantor, M., ref.\ to}Cantor, chap.~xxxi.}
have shown that the Chinese made no serious attempt
to classify or extend the few rules of arithmetic or geometry
with which they were acquainted, or to explain the causes of
the phenomena which they observed.

\nblabel{png031a}The idea that the Chinese had made considerable progress
in theoretical mathematics seems to have been due to a misapprehension
of the Jesuit missionaries who went to China
in the sixteenth century. \index{Arabs!q2@introduced into China}In the first place, they failed to
distinguish between the original science of the Chinese and
the views which they found prevalent on their arrival---the
latter being founded on the work and teaching of Arab or
Hindoo missionaries who had come to China in the course of
the thirteenth century or later, and while there introduced a
knowledge of spherical trigonometry. In the second place,
finding that one of the most important government departments
was known as the Board of Mathematics, they supposed
that its function was to promote and superintend mathematical
studies in the empire. Its duties were really confined to the
annual preparation of an almanack, the dates and predictions
in which regulated many affairs both in public and domestic
life. All extant specimens of these almanacks are defective
and, in many respects, inaccurate.

\index{Rhind papyrus, the!ref.\ to}The only geometrical theorem with which we can be certain
that the ancient Chinese were acquainted is that in certain cases
(namely, when the ratio of the sides is $3:4:5$, or $1:1:\surd 2$)
the area of the square described on the hypotenuse of a right-angled
triangle is equal to the sum of the areas of the squares
described on the sides. It is barely possible that a few
geometrical theorems which can be demonstrated in the quasi-experimental
way of superposition were also known to them.
Their arithmetic was decimal in notation, but their knowledge
seems to have been confined to the art of calculation by means
of the swan-pan, and the power of expressing the results in
writing. Our acquaintance with the early attainments of the
Chinese, slight though it is, is more complete than in the case
of most of their contemporaries. It is thus specially instructive,
and serves to illustrate the fact that a nation may possess considerable
skill in the applied arts while they are ignorant of the
sciences on which those arts are founded.\index{Chinese, early mathematics|)}

From the foregoing summary it will be seen that our knowledge
of the mathematical attainments of those who preceded
the Greeks is very limited; but we may reasonably infer that
from one source or another the early Greeks learned the use of
the abacus\index{Abacus,!ref.\ to} for practical calculations, symbols for recording the
results, and as much mathematics as is contained or implied in
the \index{Rhind papyrus, the!ref.\ to}Rhind papyrus. It is probable that this sums up their
indebtedness to other races. In the next six chapters I shall
trace the development of mathematics under Greek influence.

\nbpart{FIRST PERIOD}{Mathematics Under Greek Influence}{greek}
\nblabel{png033a}

\emph{This period begins with the teaching of Thales, circ.~\emph{600~\textsc{b.c.},}
and ends with the capture of Alexandria by the Mohammedans
in or about \emph{641~\textsc{a.d.}} The characteristic feature of this period
is the development of Geometry.}
\medskip

It will be remembered that I commenced the \hyperref[chapter:I]{last chapter} by
saying that the history of mathematics might be divided into
three periods, namely, that of mathematics under Greek influence,
that of the mathematics of the middle ages and of the renaissance,
and lastly that of modern mathematics. The next four chapters
(chapters~\hyperref[chapter:II]{\textsc{ii}}, \hyperref[chapter:III]{\textsc{iii}}, \hyperref[chapter:IV]{\textsc{iv}} and \hyperref[chapter:V]{\textsc{v}}) deal with the history of mathematics
under Greek influence: to these it will be convenient to
add one (chapter~\hyperref[chapter:VI]{\textsc{vi}}) on the Byzantine school, since through it
the results of Greek mathematics were transmitted to western
Europe; and another (chapter~\hyperref[chapter:VII]{\textsc{vii}}) on the systems of numeration
which were ultimately displaced by the system introduced by the
Arabs. I should add that many of the dates mentioned in these
chapters are not known with certainty, and must be regarded as
only approximately correct.

There appeared in December 1921, just before this reprint
was struck off, \index{Heath, Sir T. L.}Sir~T.~L.~Heath's work in 2 volumes on the
History of Greek Mathematics. This may now be taken as the
standard authority for this period.

\nbChapter{II}{The Ionian and Pythagorean Schools}{Ionian and Pythagorean Schools}{Circ. 600~B.C.--400~B.C.}
  {The history of these schools has been discussed by G.~Loria in his \emph{Le Scienze
  Esatte nell' Antica Grecia}, Modena, 1893--1900; by Cantor, chaps.~v--viii;
  by G.~J.~Allman in his \emph{Greek Geometry from Thales to Euclid}, Dublin, 1889;
  by J.~Gow, in his \emph{Greek Mathematics}, Cambridge, 1884; by C.~A.~Bretschneider
  in his \emph{Die Geometrie und die Geometer vor Eukleides}, Leipzig, 1870;
  and partially by H.~Hankel in his posthumous \emph{Geschichte der Mathematik,}
  Leipzig, 1874.}
\index{Loria, ref.\ to}\index{Cantor, M., ref.\ to}\index{Allman, G. J., ref.\ to}
\index{Gow, ref.\ to}\index{Hankel, ref.\ to}\index{Bretschneider, ref.\ to}\index{Cantor, M., ref.\ to}
\index{Geometry.!z2@Classical synthetic geometry, discussed or used by nearly all the mathematicians considered in the first period,
chapters \textsc{ii-v}|(}\nblabel{png034a}

\nblabel{png034b}\First{With} the foundation of the Ionian and Pythagorean schools we
emerge from the region of antiquarian research and conjecture
into the light of history. The materials at our disposal for
estimating the knowledge of the philosophers of these schools
previous to about the year 430~\textsc{b.c.}\ are, however, very scanty
Not only have all but fragments of the different mathematical
treatises then written been lost, but we possess no copy of the
history of mathematics written about 325~\textsc{b.c.}\ by \index{Eudemus}Eudemus (who
was a pupil of \index{Aristotle!ref.\ to}Aristotle). Luckily \index{Proclus!ref.\ to}Proclus, who about 450~\textsc{a.d.}\
wrote a commentary on the earlier part of Euclid's \emph{Elements},
was familiar with Eudemus's work, and freely utilised it in his
historical references. We have also a fragment of the \emph{General
View of Mathematics} written by \index{Geminus, ref.\ to}Geminus about 50~\textsc{b.c.}, in which
the methods of proof used by the early Greek geometricians are
compared with those current at a later date. In addition to
these general statements we have biographies of a few of the
leading mathematicians, and some scattered notes in various
writers in which allusions are made to the lives and works of
others. The original authorities are criticised and discussed at
length in the works mentioned in the footnote to the heading of
the chapter.

\section*{The Ionian School.\nblabel{png035a}}\index{Ionian School, the|(}

\nblabel{png035b}\nbparagraph{Thales.\footnote
  {See Loria, book~I, chap.~ii; Cantor, chap.~v; Allman, chap.~i.}}
\index{Loria, ref.\ to}\index{Cantor, M., ref.\ to}\index{Allman, G. J., ref.\ to}%
The founder of the earliest Greek school of mathematics
and philosophy was \index{Thales|(}\emph{Thales}, one of the seven sages of
Greece, who was born about 640~\textsc{b.c.}\ at Miletus, and died in the
same town about 550~\textsc{b.c.} The materials for an account of his
life consist of little more than a few anecdotes which have been
handed down by tradition.

During the early part of his life Thales was engaged partly
in commerce and partly in public affairs; and to judge by two
stories that have been preserved, he was then as distinguished
for shrewdness in business and readiness in resource as he was
subsequently celebrated in science. It is said that once when
transporting some salt which was loaded on mules, one of the
animals slipping in a stream got its load wet and so caused
some of the salt to be dissolved, and finding its burden thus
lightened it rolled over at the next ford to which it came; to
break it of this trick Thales loaded it with rags and sponges
which, by absorbing the water, made the load heavier and soon
effectually cured it of its troublesome habit. At another time,
according to \index{Aristotle!ref.\ to}Aristotle, when there was a prospect of an
unusually abundant crop of olives Thales got possession of all
the olive-presses of the district; and, having thus ``cornered''
them, he was able to make his own terms for lending them out,
or buying the olives, and thus realized a large sum. These
tales may be apocryphal, but it is certain that he must have
had considerable reputation as a man of affairs and as a good
engineer, since he was employed to construct an embankment so
as to divert the river Halys in such a way as to permit of the
construction of a ford.

Probably it was as a merchant that Thales first went to
Egypt, but during his leisure there he studied astronomy and
geometry. He was middle-aged when he returned to Miletus;
he seems then to have abandoned business and public life,
and to have devoted himself to the study of philosophy and
science---subjects which in the Ionian, Pythagorean, and
perhaps also the Athenian schools, were closely connected:
his views on philosophy do not here concern us. He continued
to live at Miletus till his death circ.~550~\textsc{b.c.}

\nblabel{png036a}We cannot form any exact idea as to how Thales presented
his geometrical teaching. We infer, however, from \index{Proclus!ref.\ to}Proclus that
it consisted of a number of isolated propositions which were
not arranged in a logical sequence, but that the proofs were
deductive, so that the theorems were not a mere statement of
an induction from a large number of special instances, as
probably was the case with the Egyptian geometricians. The
deductive character which he thus gave to the science is his
chief claim to distinction.

The following comprise the chief propositions that can now
with reasonable probability be attributed to him; they are
concerned with the geometry of angles and straight lines.

(i)~The angles at the base of an isosceles triangle are equal
\index{Eucn@Euc.!a05@\textsc{i}, 5. ref.\ to}(Euc.~\textsc{i}, 5). Proclus seems to imply that this was proved by
taking another exactly equal isosceles triangle, turning it over,
and then superposing it on the first---a sort of experimental
demonstration.

(ii)~If two straight lines cut one another, the vertically
opposite angles are equal \index{Eucn@Euc.!a15@\textsc{i}, 15. ref.\ to}(Euc.~\textsc{i}, 15). Thales may have
regarded this as obvious, for Proclus adds that Euclid was the
first to give a strict proof of it.

(iii)~A triangle is determined if its base and base angles be
given (\emph{cf.}\ \index{Eucn@Euc.!a26@\textsc{i}, 26. ref.\ to}Euc.~\textsc{i}, 26). Apparently this was applied to find the
distance of a ship at sea---the base being a tower, and the base
angles being obtained by observation.

(iv)~The sides of equiangular triangles are proportionals
(\index{Eucn@Euc.!e04@\textsc{vi}, 4. ref.\ to}Euc.~\textsc{vi}, 4, or perhaps rather \index{Eucn@Euc.!e02@\textsc{vi}, 2. ref.\ to}Euc.~\textsc{vi}, 2). This is said to
have been used by Thales when in Egypt to find the height of
a pyramid. In a dialogue given by \index{Plutarch, ref.\ to}Plutarch, the speaker,
addressing Thales, says, ``Placing your stick at the end of
the shadow of the pyramid, you made by the sun's rays two
triangles, and so proved that the [height of the] pyramid was
to the [length of the] stick as the shadow of the pyramid to
the shadow of the stick.'' It would seem that the theorem was
unknown to the Egyptians, and we are told that the king
\index{Amasis of Egypt, ref.\ to}Amasis, who was present, was astonished at this application of
abstract science.

(v)~A circle is bisected by any diameter. This may have
been enunciated by Thales, but it must have been recognised as
an obvious fact from the earliest times.

(vi)~The angle subtended by a diameter of a circle at any
point in the circumference is a right angle (\index{Eucn@Euc.!c31@\textsc{iii}, 31. ref.\ to}Euc.~\textsc{iii}, 31).
This appears to have been regarded as the most remarkable
of the geometrical achievements of Thales, and it is stated that
on inscribing a right-angled triangle in a circle he sacrificed an
ox to the immortal gods. It has been conjectured that he may
have come to this conclusion by noting that the diagonals of a
rectangle are equal and bisect one another, and that therefore a
rectangle can be inscribed in a circle. If so, and if he went on
to apply proposition (i), he would have discovered that the
sum of the angles of a right-angled triangle is equal to two
right angles, a fact with which it is believed that he was
acquainted. It has been remarked that the shape of the tiles
used in paving floors may have suggested these results.

On the whole it seems unlikely that he knew how to draw a
perpendicular from a point to a line; but if he possessed this
knowledge, it is possible he was also aware, as suggested by
some modern commentators, that the sum of the angles of any
triangle is equal to two right angles. As far as equilateral
and right-angled triangles are concerned, we know from
\index{Eudemus}Eudemus that the first geometers proved the general property
separately for three species of triangles, and it is not unlikely
that they proved it thus. The area about a point can be filled
by the angles of six equilateral triangles or tiles, hence the
proposition is true for an equilateral triangle. Again, any two
equal right-angled triangles can be placed in juxtaposition so
as to form a rectangle, the sum of whose angles is four right
angles; hence the proposition is true for a right-angled triangle.
Lastly, any triangle can be split into the sum of two right-angled
triangles by drawing a perpendicular from the biggest
angle on the opposite side, and therefore again the proposition
is true. The first of these proofs is evidently included in the
last, but there is nothing improbable in the suggestion that the
early Greek geometers continued to teach the first proposition
in the form above given.

\index{Thales|)}\nblabel{png038a}Thales wrote on \index{Astronomy.!z12@Early Greek theories of}astronomy, and among his contemporaries
was more famous as an astronomer than as a geometrician. A
story runs that one night, when walking out, he was looking so
intently at the stars that he tumbled into a ditch, on which an
old woman exclaimed, ``How can you tell what is going on in
the sky when you can't see what is lying at your own feet?''---an
anecdote which was often quoted to illustrate the unpractical
character of philosophers.

\index{Calendars}Without going into astronomical details, it may be mentioned
that he taught that a year contained about 365 days, and not
(as is said to have been previously reckoned) twelve months of
thirty days each. It is said that his predecessors occasionally
intercalated a month to keep the seasons in their customary
places, and if so they must have realized that the \index{Year, duration of}year contains,
on the average, more than 360 days. There is some reason to
think that he believed the earth to be a disc-like body floating
on water. \index{Eclipse foretold by Thales}He predicted a solar eclipse which took place at or
about the time he foretold; the actual date was either May~28,
585~\textsc{b.c.}, or September~30, 609~\textsc{b.c.} But though this prophecy
and its fulfilment gave extraordinary prestige to his teaching,
and secured him the name of one of the seven sages of Greece,
it is most likely that he only made use of one of the Egyptian
or Chaldaean registers which stated that solar eclipses recur
at intervals of about 18 years 11 days.

\nblabel{png039a}Among the pupils of Thales were \index{Anaximander}\index{Anaximenes}\index{Mamercus}\nbname{Anaximander}, \nbname{Anaximenes},
\nbname{Mamercus}, and \index{Mandryatus}\nbname{Mandryatus}. Of the three mentioned last we
know next to nothing. \emph{Anaximander} was born in 611~\textsc{b.c.},
and died in 545~\textsc{b.c.}, and succeeded Thales as head of the
school at Miletus. According to \index{Suidas, ref.\ to}Suidas he wrote a treatise on
geometry in which, tradition says, he paid particular attention
to the properties of spheres, and dwelt at length on the philosophical
ideas involved in the conception of infinity in space and
time. He constructed terrestrial and celestial globes.

Anaximander is alleged to have introduced the use of the
\index{Gnomon or style}\index{Style or gnomon}\emph{style} or \emph{gnomon} into Greece. This, in principle, consisted only
of a stick stuck upright in a horizontal piece of ground. It
was originally used as a \index{Sun-dials}sun-dial, in which case it was placed
at the centre of three concentric circles, so that every two
hours the end of its shadow passed from one circle to another.
Such sun-dials have been found at Pompeii and Tusculum. It
is said that he employed these styles to determine his meridian
(presumably by marking the lines of shadow cast by the style
at sunrise and sunset on the same day, and taking the plane
bisecting the angle so formed); and thence, by observing the
time of year when the noon-altitude of the sun was greatest
and least, he got the solstices; thence, by taking half the sum
of the noon-altitudes of the sun at the two solstices, he found
the inclination of the equator to the horizon (which determined
the altitude of the place), and, by taking half their difference,
he found the inclination of the ecliptic to the equator. There
seems good reason to think that he did actually determine the
\index{Latitude, introduction of}latitude of Sparta, but it is more doubtful whether he really
made the rest of these astronomical deductions.

We need not here concern ourselves further with the
successors of Thales. The school he established continued to
flourish till about 400~\textsc{b.c.}, but, as time went on, its members
occupied themselves more and more with philosophy and less
with mathematics. We know very little of the mathematicians
comprised in it, but they would seem to have devoted most of
their attention to astronomy. They exercised but slight influence
on the further advance of Greek mathematics, which
was made almost entirely under the influence of the Pythagoreans,
who not only immensely developed the science of
geometry, but created a science of numbers. If Thales was the
first to direct general attention to geometry, it was Pythagoras,
says \index{Proclus!ref.\ to}Proclus, quoting from \index{Eudemus}Eudemus, who ``changed the study
of geometry into the form of a liberal education, for he examined
its principles to the bottom and investigated its
theorems in an \dots intellectual manner''; and it is accordingly
to Pythagoras that we must now direct attention.\index{Ionian School, the|)}

\section*{The Pythagorean School.\nblabel{png040a}}\index{Pythagorean School, the|(}

\index{Iamblichus!ref.\ to}\index{Loria, ref.\ to}\index{Allman, G. J., ref.\ to}\index{Hoefer, ref.\ to}%
\index{Hankel, ref.\ to}\index{Schuré, E., ref.\ to}\index{Tannery, S. P., ref.\ to}%
\nbparagraph{Pythagoras.\footnote
  {See Loria, book~I, chap.~iii; Cantor, chaps.~vi,~vii; Allman, chap.~ii;
  Hankel, pp.~92--111; Hoefer, \emph{Histoire des mathématiques}, Paris, third edition,
  1886, pp.~87--130; and various papers by S.~P.~Tannery. For an account of
  Pythagoras's life, embodying the Pythagorean traditions, see the biography
  by Iamblichus, of which there are two or three English translations. Those
  who are interested in esoteric literature may like to see a modern attempt
  to reproduce the Pythagorean teaching in \emph{Pythagoras}, by E.~Schuré,
  Eng.\ trans., London, 1906.}}
\index{Pythagoras|(}\nblabel{png040b}\emph{Pythagoras} was born at Samos about 569~\textsc{b.c.},
perhaps of Tyrian parents, and died in 500~\textsc{b.c.} He was thus a
contemporary of Thales. The details of his life are somewhat
doubtful, but the following account is, I think, substantially
correct. He studied first under \index{Pherecydes of Syros}Pherecydes of Syros, and then
under Anaximander; by the latter he was recommended to go
to Thebes, and there or at Memphis he spent some years.
After leaving Egypt he travelled in Asia Minor, and then
settled at Samos, where he gave lectures but without much
success. About 529~\textsc{b.c.}\ he migrated to Sicily with his mother,
and with a single disciple who seems to have been the sole fruit
of his labours at Samos. Thence he went to Tarentum, but
very shortly moved to Croton, a Dorian colony in the south of
Italy. Here the schools that he opened were crowded with
enthusiastic audiences; citizens of all ranks, especially those
of the upper classes, attended, and even the women broke a law
which forbade their going to public meetings and flocked to hear
him. Amongst his most attentive auditors was \index{Theano, ref.\ to}Theano, the
young and beautiful daughter of his host \index{Milo of Tarentum}Milo, whom, in spite
of the disparity of their ages, he married. She wrote a biography
of her husband, but unfortunately it is lost.

\nblabel{png041a}Pythagoras divided those who attended his lectures into
two classes, whom we may term probationers and Pythagoreans.
The majority were probationers, but it was only to the Pythagoreans
that his chief discoveries were revealed. The latter
formed a brotherhood with all things in common, holding the
same philosophical and political beliefs, engaged in the same
pursuits, and bound by oath not to reveal the teaching or
secrets of the school; their food was simple; their discipline
severe; and their mode of life arranged to encourage self-command,
temperance, purity, and obedience. This strict
discipline and secret organisation gave the society a temporary
supremacy in the state which brought on it the hatred of various
classes; and, finally, instigated by his political opponents, the
mob murdered Pythagoras and many of his followers.

Though the political influence of the Pythagoreans was thus
destroyed, they seem to have re-established themselves at once
as a philosophical and mathematical society, with Tarentum as
their headquarters, and they continued to flourish for more than
a hundred years.

Pythagoras himself did not publish any books; the assumption
of his school was that all their knowledge was held in
common and veiled from the outside world, and, further, that the
glory of any fresh discovery must be referred back to their
founder. Thus \index{Hippasus}Hippasus (circ.~470~\textsc{b.c.})\ is said to have been
drowned for violating his oath by publicly boasting that he had
added the \index{Polyhedrons, regular}\index{Dodecahedron, discovery of}dodecahedron to the number of regular solids enumerated
by Pythagoras. Gradually, as the society became more
scattered, this custom was abandoned, and treatises containing
the substance of their teaching and doctrines were written.
The first book of the kind was composed, about 370~\textsc{b.c.}, by
\index{Philolaus}Philolaus, and we are told that \index{Plato!ref.\ to}Plato secured a copy of it. We
may say that during the early part of the fifth century before
Christ the Pythagoreans were considerably in advance of their
contemporaries, but by the end of that time their more
prominent discoveries and doctrines had become known to the
outside world, and the centre of intellectual activity was
transferred to Athens.

\index{Greek science|(}Though it is impossible to separate precisely the discoveries
of Pythagoras himself from those of his school of a later date,
we know from \index{Proclus!ref.\ to}Proclus that it was Pythagoras who gave
geometry that rigorous character of deduction which it still
bears, and made it the foundation of a liberal education; and
there is reason to believe that he was the first to arrange the
leading propositions of the subject in a logical order. It was
also, according to \index{Aristoxenus}Aristoxenus, the glory of his school that they
raised arithmetic above the needs of merchants. It was their
boast that they sought knowledge and not wealth, or in the
language of one of their maxims, ``a figure and a step forwards,
not a figure to gain three oboli.''

Pythagoras was primarily a moral reformer and philosopher,
but his system of morality and philosophy was built on a
mathematical foundation. His mathematical researches were,
however, designed to lead up to a system of philosophy whose
exposition was the main object of his teaching. The Pythagoreans
began by dividing the mathematical subjects with which
they dealt into four divisions: numbers absolute or arithmetic,
numbers applied or \index{Music, in the quadrivium}music, magnitudes at rest or geometry, and
magnitudes in motion or astronomy. This \index{Quadrivium}``quadrivium'' was
long considered as constituting the necessary and sufficient
course of study for a liberal education. Even in the case of
geometry and arithmetic (which are founded on inferences
unconsciously made and common to all men) the Pythagorean
presentation was involved with philosophy; and there is no
doubt that their teaching of the sciences of astronomy,
mechanics, and music (which can rest safely only on the
results of conscious observation and \index{Experiments, necessity of}experiment) was intermingled
with metaphysics even more closely. It will be convenient
to begin by describing their treatment of geometry and
arithmetic.

\nblabel{png043a}First, as to their geometry. Pythagoras probably knew and
taught the substance of what is contained in the first two books
of Euclid about parallels, triangles, and parallelograms, and was
acquainted with a few other isolated theorems including some
elementary propositions on irrational magnitudes; but it is
suspected that many of his proofs were not rigorous, and in
particular that the converse of a theorem was sometimes assumed
without a proof. It is hardly necessary to say that we are unable
to reproduce the whole body of Pythagorean teaching on
this subject, but we gather from the notes of Proclus on Euclid,
and from a few stray remarks in other writers, that it included
the following propositions, most of which are on the geometry
of areas.\index{Greek science|)}

(i)~It commenced with a number of definitions, which probably
were rather statements connecting mathematical ideas
with philosophy than explanations of the terms used. One
has been preserved in the definition of a \index{Point, Pythagorean def.\ of}point as unity having
position.

(ii)~The sum of the angles of a triangle was shown to be
equal to two right angles (\index{Eucn@Euc.!a32@\textsc{i}, 32. ref.\ to}Euc.~\textsc{i}, 32); and in the proof, which
has been preserved, the results of the propositions \index{Eucn@Euc.!a13@\textsc{i}, 13. ref.\ to}Euc.~\textsc{i}, 13 and
the first part of \index{Eucn@Euc.!a29@\textsc{i}, 29. ref.\ to}Euc.~\textsc{i}, 29 are quoted. The demonstration is
substantially the same as that in Euclid, and it is most likely
that the proofs there given of the two propositions last mentioned
are also due to Pythagoras himself.

(iii)~Pythagoras certainly proved the properties of right-angled
triangles which are given in \index{Eucn@Euc.!a47@\textsc{i}, 47. ref.\ to}Euc.~\textsc{i}, 47 and \index{Eucn@Euc.!a48@\textsc{i}, 48. ref.\ to}\textsc{i}, 48. We
know that the proofs of these propositions which are found in
Euclid were of Euclid's own invention; and a good deal of
curiosity has been excited to discover what was the demonstration
which was originally offered by Pythagoras of the first
of these theorems. It has been conjectured that not improbably
it may have been one of the two following.\index{Eucn@Euc.!a47@\textsc{i}, 47. ref.\ to}\footnote
  {A collection of a hundred proofs of Euc.~\textsc{i},~47 was published in
  the \emph{American Mathematical Monthly Journal}, vols.~iii.\ iv.\ v.\ vi. 1896--1899.}

\begin{center}
\labelincludegraphics{044a}
\end{center}

($\alpha$) Any square $ABCD$ can be split up, as in \index{Eucn@Euc.!b04@\textsc{ii}, 4. ref.\ to}Euc.~\textsc{ii}, 4, into
two squares $BK$ and $DK$ and two equal rectangles $AK$ and $CK$:
that is, it is equal to the square on $FK$, the square on $EK$, and
four times the triangle $AEF$.
But, if points be taken, $G$ on
$BC$, $H$ on $CD$, and $E$ on $DA$, so that $BG$, $CH$, and $DE$ are
each equal to $AF$, it can be easily shown that $EFGH$ is a
square, and that the triangles $AEF$, $BFG$, $CGH$, and $DHE$ are
equal: thus the square $ABCD$ is also equal to the square on
$EF$ and four times the triangle $AEF$. Hence the square on $EF$
is equal to the sum of the squares on $FK$ and $EK$.

\begin{center}
\labelincludegraphics{044b}
\end{center}

($\beta$) Let $ABC$ be a right-angled triangle, $A$ being the right
angle. Draw $AD$ perpendicular to $BC$. The triangles $ABC$
and $DBA$ are similar,
\begin{DPgather*}
                       \therefore BC : AB = AB : BD. \\
\lintertext{Similarly} BC : AC = AC : DC.\\
\lintertext{Hence}     AB^2 + AC^2 = BC(BD + DC) = BC^2.
\end{DPgather*}
This proof requires a knowledge of the results of \index{Eucn@Euc.!b02@\textsc{ii}, 2. ref.\ to}Euc.~\textsc{ii}, 2,
\index{Eucn@Euc.!e04@\textsc{vi}, 4. ref.\ to}\textsc{vi}, 4, and \index{Eucn@Euc.!e17@\textsc{vi}, 17. ref.\ to}\textsc{vi}, 17, with all of which Pythagoras was acquainted.

\index{Arithmetic.!z3@Pythagorean|(}(iv)~Pythagoras is credited by some writers with the discovery
of the theorems \index{Eucn@Euc.!a44@\textsc{i}, 44. ref.\ to}Euc.~\textsc{i}, 44, and \index{Eucn@Euc.!a45@\textsc{i}, 45. ref.\ to}\textsc{i}, 45, and with giving a solution
of the problem \index{Eucn@Euc.!b14@\textsc{ii}, 14. ref.\ to}Euc.~\textsc{ii}, 14. It is said that on the discovery of
the necessary construction for the problem last mentioned he
sacrificed an ox, but as his school had all things in common the
liberality was less striking than it seems at first. The Pythagoreans
of a later date were aware of the extension given in \index{Eucn@Euc.!e25@\textsc{vi}, 25. ref.\ to}Euc.~\textsc{vi}, 25,
and \index{Allman, G. J., ref.\ to}Allman thinks that Pythagoras himself was acquainted with
it, but this must be regarded as doubtful. It will be noticed that
\index{Eucn@Euc.!b14@\textsc{ii}, 14. ref.\ to}Euc.~\textsc{ii}, 14 provides a geometrical solution of the equation $x^2 = ab$.

(v)~Pythagoras showed that the plane about a point could be
completely filled by equilateral triangles, by squares, or by regular
hexagons\allowbreak---results that must have been familiar wherever tiles of
these shapes were in common use.

\index{Circle, quadrature of (or squaring the)}(vi)~The Pythagoreans were said to have attempted the quadrature
of the circle: they stated that the circle was the most
perfect of all plane figures.

(vii)~\index{Polyhedrons, regular}They knew that there were five regular solids inscribable
in a sphere, which was itself, they said, the most perfect
of all solids.

\index{Numbers, theory of.!Treatment of, by Pythagoras|(}(viii)~From their phraseology in the science of numbers and
from other occasional remarks, it would seem that they were
acquainted with the methods used in the second and fifth books
of Euclid, and knew something of \index{Irrational numbers|(}irrational magnitudes. \index{Incommensurables}In
particular, there is reason to believe that Pythagoras proved
that the side and the diagonal of a square were incommensurable,
and that it was this discovery which led the early Greeks
to banish the conceptions of number and measurement from
their geometry. A proof of this proposition which may be that
due to Pythagoras is given below.\footnote
  {See below, page~\pageref{xref:081}.}

\nblabel{png045a}Next, as to their theory of numbers.\footnote
  {See the appendix \emph{Sur l'arithmétique pythagorienne} to \index{Tannery, S. P., ref.\ to}S.~P.~Tannery's
  \emph{La science hellène}, Paris, 1887.}
In this Pythagoras
was chiefly concerned with four different groups of problems
which dealt respectively with polygonal numbers, with ratio and
proportion, with the factors of numbers, and with numbers in
series; but many of his arithmetical inquiries, and in particular
the questions on polygonal numbers and proportion, were treated
by geometrical methods.\index{Irrational numbers|)}

\begin{center}
\labelincludegraphics{046a}
\end{center}

Pythagoras commenced his theory of arithmetic by dividing
all numbers into even or odd: the \index{Gnomons or odd numbers}odd numbers being termed
\emph{gnomons}. An odd number, such as $2n + 1$, was regarded as the
difference of two square numbers $(n+1)^2$ and $n^2$; and the sum
of the gnomons from 1 to $2n+1$ was stated to be a \emph{square
number}, viz.\ $(n+1)^2$, its square root was termed a \emph{side}. Products
of two numbers were called \emph{plane}, and if a product had
no exact square root it
was termed an \emph{oblong}. A product of
three numbers was called a \emph{solid number}, and, if the three
numbers were equal, a \emph{cube}. All this has obvious reference to
geometry, and the opinion is confirmed by \index{Aristotle!ref.\ to}Aristotle's remark
that when a gnomon is put round a square the figure remains
a square though it is increased in dimensions. Thus, in the
\figureref{046a}{figure} given above in which $n$ is taken equal to 5, the
gnomon $AKC$ (containing 11 small squares) when put round the
square $AC$ (containing $5^2$ small squares) makes a square $HL$
(containing $6^2$ small squares). It is possible that several of
the numerical theorems due to Greek writers were discovered
and proved by an analogous method: the abacus\index{Abacus,!ref.\ to} can be used
for many of these demonstrations.

The numbers $(2n^2+2n+1)$, $(2n^2+2n)$, and $(2n+1)$ possessed
special importance as representing the hypotenuse and
two sides of a right-angled triangle: \index{Cantor, M., ref.\ to}Cantor thinks that
Pythagoras knew this fact before discovering the geometrical
proposition \index{Eucn@Euc.!a47@\textsc{i}, 47. ref.\ to}Euc.~\textsc{i}, 47. A more general expression for such
numbers is $(m^2+n^2)$, $2mn$, and $(m^2-n^2)$, or multiples of them:
it will be noticed that the result obtained by Pythagoras can be
deduced from these expressions by assuming $m=n+1$; at a
later time \index{Archytas!ref.\ to}Archytas and \index{Plato!ref.\ to}Plato gave rules which are equivalent
to taking $n=1$; \index{Diophantus!ref.\ to}Diophantus knew the general expressions.

After this preliminary discussion the Pythagoreans proceeded
to the four special problems already alluded to.
Pythagoras was himself acquainted with \index{Triangular numbers}triangular numbers;
\index{Polygonal numbers}\index{Numbers,!polygonal}polygonal numbers of a higher order were discussed by later
members of the school. A triangular number represents the
sum of a number of counters laid in rows on a plane; the
bottom row containing $n$, and each succeeding row one less:
it is therefore equal to the sum of the series
\[
 n + (n-1) + (n-2) + \ldots + 2 + 1,
\]
that is, to $\frac{1}{2} n(n+1)$. Thus the triangular number corresponding
to 4 is 10. This is the explanation of the language
of Pythagoras in the well-known passage in \index{Lucian, ref.\ to}Lucian where the
merchant asks Pythagoras what he can teach him. Pythagoras
replies ``I will teach you how to count.'' \emph{Merchant}, ``I know
that already.'' \emph{Pythagoras}, ``How do you count?'' \emph{Merchant},
``One, two, three, four---'' \emph{Pythagoras}, ``Stop! what you take
to be four is ten, a perfect triangle and our symbol.'' The
Pythagoreans are, on somewhat doubtful authority, said to have
classified numbers by comparing them with the sum of their
integral subdivisors or factors, calling a number \index{Numbers,!excessive}\index{Excessive numbers}excessive, \index{Numbers,!perfect}\index{Perfect numbers}perfect,
or \index{Numbers,!defective}\index{Defective numbers}defective, according as the sum of these subdivisors was greater
than, equal to, or less than the number: the classification at
first being restricted to even numbers. The third group of
problems which they considered dealt with numbers which
formed a proportion; presumably these were discussed with the
aid of geometry as is done in the fifth book of Euclid. Lastly,
the Pythagoreans were concerned with series of numbers in
\index{Progressions,!arithmetical}\index{Arithmetical!progressions}arithmetical, \index{Progressions,!geometrical}\index{Geometrical progressions}geometrical, \index{Harmonic series}harmonical, and \index{Progressions,!musical}\index{Musical progression}musical progressions.
The three progressions first mentioned are well known; four
integers are said to be in musical progression when they are in
the ratio $a : 2ab/(a+b) : \frac{1}{2}(a+b) : b$, for example, 6, 8, 9, and
12 are in musical progression.\index{Numbers, theory of.!Treatment of, by Pythagoras|)}

Of the Pythagorean treatment of the applied subjects of the
quadrivium, and the philosophical theories founded on them,
we know very little. It would seem that Pythagoras was much
impressed by certain numerical relations which occur in nature.
It has been suggested that he was acquainted with some of the
simpler facts of crystallography. It is thought that he was
aware that the notes sounded by a vibrating string depend on
the length of the string, and in particular that lengths which
gave a note, its fifth and its octave were in the ratio $2:3:4$,
forming terms in a musical progression. It would seem, too,
that he believed that the distances of the astrological planets
from the earth were also in musical progression, and that the
heavenly bodies in their motion through space gave out
harmonious sounds: hence the phrase the harmony of the
spheres. These and similar conclusions seem to have suggested
to him that the explanation of the order and harmony of the
universe was to be found in the science of numbers, and that
numbers are to some extent the cause of form as well as
essential to its accurate measurement. He accordingly proceeded
to attribute particular properties to particular numbers
and geometrical figures. For example, he taught that the cause
of colour was to be sought in properties of the number five,
that the explanation of fire was to be discovered in the nature
of the pyramid, and so on. I should not have alluded to this
were it not that the Pythagorean tradition strengthened, or
perhaps was chiefly responsible for the tendency of Greek
writers to found the study of nature on philosophical conjectures
and not on experimental observation---a tendency to
which the defects of Hellenic science must be largely attributed.

\nblabel{png049a}After the death of Pythagoras his teaching seems to have
been carried on by \index{Epicharmus}\nbname{Epicharmus} and \index{Hippasus}\nbname{Hippasus}, and subsequently
by \index{Philolaus}\nbname{Philolaus} (specially distinguished as an astronomer),
\index{Archippus}\nbname{Archippus}, and \index{Lysis}\nbname{Lysis}. About a century after the murder of
Pythagoras we find \index{Archytas|(}Archytas recognised as the head of the
school.\index{Pythagoras|)}\index{Arithmetic.!z3@Pythagorean|)}

\nbparagraph{Archytas.\footnote
  {See \index{Allman, G. J., ref.\ to}Allman, chap.~iv. A catalogue of the works of Archytas is given
  by \index{Fabricius on Archytas}Fabricius in his \emph{Bibliotheca Graeca}, vol.~i, p.~833: most of the fragments
  on philosophy were published by \index{Gale on Archytas}Thomas Gale in his \emph{Opuscula Mythologica}, %[**TN: 'Opuscula Mythologia' in original text]
  Cambridge, 1670; and by \index{Taylor, T., on Pythagoras}Thomas Taylor as an Appendix to his translation
  of \index{Iamblichus!ref.\ to}Iamblichus's \emph{Life of Pythagoras}, London, 1818. See also the references
  given by \index{Cantor, M., ref.\ to}Cantor, vol.~i, p.~203.}}
\nblabel{png049b} \emph{Archytas}, circ.~400~\textsc{b.c.}, was one of the most
influential citizens of Tarentum, and was made governor of
the city no less than seven times. His influence among his
contemporaries was very great, and he used it with \index{Dionysius of Tarentum}Dionysius
on one occasion to save the life of \index{Plato!ref.\ to}Plato. He was noted for the
attention he paid to the comfort and education of his slaves and
of children in the city. He was drowned in a shipwreck near
Tarentum, and his body washed on shore---a fit punishment, in
the eyes of the more rigid Pythagoreans, for his having departed
from the lines of study laid down by their founder. Several
of the leaders of the Athenian school were among his pupils
and friends, and it is believed that much of their work was due
to his inspiration.

The Pythagoreans at first made no attempt to apply their
knowledge to \index{Mechanics.!z1@Discussed by!x0@Archytas}mechanics, but Archytas is said to have treated it
with the aid of geometry. He is alleged to have invented and
worked out the \index{Pulley, theory of}theory of the pulley, and is credited with the
construction of a flying bird and some other ingenious mechanical
toys. He introduced various mechanical devices for constructing
curves and solving problems. These were objected to by
Plato, who thought that they destroyed the value of geometry
as an intellectual exercise, and later Greek geometricians confined
themselves to the use of two species of \index{Instruments, mathematical}instruments,
namely, rulers and compasses. Archytas was also interested in
astronomy; he taught that the earth was a sphere rotating
round its axis in twenty-four hours, and round which the
heavenly bodies moved.

\nblabel{png050a}Archytas was one of the first to give a solution of the
problem to duplicate a \index{Cube, duplication of}cube, that is, to find the side of a cube
whose volume is double that of a given cube. This was one of
the most famous problems of antiquity.\footnote
  {See below, pp.~\pageref{xref:058},~\pageref{xref:062},~\pageref{xref:063}.}
The construction
given by Archytas is equivalent to the following. On the
diameter $OA$ of the base of a right circular cylinder describe a
semicircle whose plane is perpendicular to the base of the
cylinder. Let the plane containing this semicircle rotate round
the generator through $O$, then the surface traced out by the
semicircle will cut the cylinder in a tortuous curve. This curve
will be cut by a right cone whose axis is $OA$ and semivertical
angle is (say) $60°$ in a point $P$, such that the projection of $OP$
on the base of the cylinder will be to the radius of the cylinder
in the ratio of the side of the required cube to that of the given
cube. The proof given by Archytas is of course geometrical;\footnote
  {It is printed by \index{Allman, G. J., ref.\ to}Allman, pp.~111--113.}
it will be enough here to remark that in the course of it he
shews himself acquainted with the results of the propositions
\index{Eucn@Euc.!c18@\textsc{iii}, 18. ref.\ to}\index{Eucn@Euc.!c35@\textsc{iii}, 35. ref.\ to}\index{Eucn@Euc.!h19@\textsc{xi}, 19. ref.\ to}Euc.~\textsc{iii}, 18, Euc.~\textsc{iii}, 35, and Euc.~\textsc{xi}, 19. To shew analytically
that the construction is correct, take $OA$ as the axis of $x$, and
the generator through $O$ as axis of $z$, then, with the usual
notation in polar co-ordinates, and if $a$ be the radius of
the cylinder, we have for the equation of the surface described
by the semicircle, $r = 2a\sin\theta$; for that of the cylinder,
$r\sin\theta = 2a\cos\phi$; and for that of the cone, $\sin\theta\cos\phi = \frac{1}{2}$. These
three surfaces cut in a point such that $\sin^3\theta = \frac{1}{2}$, and, therefore,
if $\rho$ be the projection of $OP$ on the base of the cylinder, then
$\rho^3 = (r\sin\theta)^3 = 2a^3$. Hence the volume of the cube whose side is
$\rho$ is twice that of a cube whose side is $a$. I mention the problem
and give the construction used by Archytas to illustrate how
considerable was the knowledge of the Pythagorean school at
the time.\index{Archytas|)}

\nblabel{xref:030}\nbparagraph{Theodorus.} Another Pythagorean of about the same date as\nblabel{png051a}
Archytas was \index{Theodorus of Cyrene}\emph{Theodorus of Cyrene}, who is said to have proved
geometrically that the numbers represented by $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$,
$\sqrt{7}$, $\sqrt{8}$, $\sqrt{10}$, $\sqrt{11}$, $\sqrt{12}$, $\sqrt{13}$, $\sqrt{14}$, $\sqrt{15}$, and $\sqrt{17}$ are \index{Incommensurables}incommensurable
with unity. Theaetetus was one of his pupils.

Perhaps \index{Timaeus of Locri}\nbname{Timaeus} of Locri and \nbname{Bryso} of Heraclea should be
mentioned as other distinguished Pythagoreans of this time. It
is believed that \index{Bryso}Bryso attempted to find the \index{Circle, quadrature of (or squaring the)}area of a circle by
inscribing and circumscribing squares, and finally obtained
polygons between whose areas the area of the circle lay; but it
is said that at some point he assumed that the area of the circle
was the arithmetic mean between an inscribed and a circumscribed
polygon.\index{Pythagorean School, the|)}

\section*{Other Greek Mathematical Schools in the Fifth Century~\textsc{b.c.}\nblabel{png051b}}

It would be a mistake to suppose that Miletus and Tarentum
were the only places where, in the fifth century, Greeks were
engaged in laying a scientific foundation for the study of mathematics.
These towns represented the centres of chief activity,
but there were few cities or colonies of any importance where
lectures on philosophy and geometry were not given. Among
these smaller schools I may mention those at Chios, Elea, and
Thrace.

\nblabel{png051c}The best known philosopher of the \index{Chios, School of}\emph{School of Chios} was
\index{Oenopides of Chios}\nbname{Oenopides}, who was born about 500~\textsc{b.c.}, and died about 430
\textsc{b.c.} He devoted himself chiefly to astronomy, but he had
studied geometry in Egypt, and is credited with the solution of
two problems, namely, to draw a straight line from a given
external point perpendicular to a given straight line \index{Eucn@Euc.!a12@\textsc{i}, 12. ref.\ to}(Euc.~\textsc{i}, 12),
and at a given point to construct an angle equal to a given angle
\index{Eucn@Euc.!a23@\textsc{i}, 23. ref.\ to}(Euc.~\textsc{i}, 23).

Another important centre was at Elea in Italy. This was
founded in Sicily by \index{Xenophanes}\nbname{Xenophanes}. He was followed by
\nblabel{png052a}\index{Parmenides}\nbname{Parmenides}, \nbname{Zeno}, and \index{Melissus}\nbname{Melissus}. The members of the \index{Eleatic School|(}\emph{Eleatic
School} were famous for the difficulties they raised in connection
with questions that required the use of infinite series, such, for
example, as the well-known \index{Achilles and tortoise, paradox}paradox of Achilles and the tortoise,
enunciated by \index{Zeno}\emph{Zeno}, one of their most prominent members.
Zeno was born in 495~\textsc{b.c.}, and was executed at Elea in 435~\textsc{b.c.}\
in consequence of some conspiracy against the state; he was a
pupil of Parmenides, with whom he visited Athens, circ.~455--450~\textsc{b.c.}

\index{Puzzles}Zeno argued that if Achilles ran ten times as fast as a
tortoise, yet if the tortoise had (say) 1000 yards start it could
never be overtaken: for, when Achilles had gone the 1000
yards, the tortoise would still be 100 yards in front of him; by
the time he had covered these 100 yards, it would still be 10
yards in front of him; and so on for ever: thus Achilles would
get nearer and nearer to the tortoise, but never overtake it. The
fallacy is usually explained by the argument that the time
required to overtake the tortoise, can be divided into an infinite
number of parts, as stated in the question, but these get smaller
and smaller in geometrical progression, and the sum of them all
is a finite time: after the lapse of that time Achilles would be
in front of the tortoise. \index{Infinite series, difficulties in connection with}Probably Zeno would have replied that
this argument rests on the assumption that space is infinitely
divisible, which is the question under discussion: he himself
asserted that magnitudes are not infinitely divisible.

These paradoxes made the Greeks look with suspicion on the
use of infinitesimals, and ultimately led to the invention of the
method of exhaustions.\index{Eleatic School|)}

\index{Atomistic School}The \emph{Atomistic School}, having its headquarters in Thrace, was
another important centre. This was founded by \index{Leucippus}\nbname{Leucippus},
who was a pupil of Zeno. He was succeeded by \index{Democritus}\nbname{Democritus}
and \index{Epicurus}\nbname{Epicurus}. Its most famous mathematician was \emph{Democritus},
born at Abdera in 460~\textsc{b.c.}, and said to have died in 370~\textsc{b.c.},
who, besides philosophical works, wrote on plane and solid
geometry, incommensurable lines, perspective, and numbers.
These works are all lost. From the Archimedean MS., discovered
by \index{Heiberg, ref.\ to}Heiberg in 1906, it would seem that Democritus enunciated,
but without a proof, the proposition that the volume of a
pyramid is equal to one-third that of a prism of an equal base
and of equal height.

But though several distinguished individual philosophers may
be mentioned who, during the fifth century, lectured at different
cities, they mostly seem to have drawn their inspiration from
Tarentum, and towards the end of the century to have looked to
Athens as the intellectual capital of the Greek world; and it is
to the Athenian schools that we owe the next great advance in
mathematics.

\nbChapter{III}{The Schools of Athens and Cyzicus}{}{Circ. 420~B.C.--300~B.C.}
  {The history of these schools is discussed at length in G.~Loria's \emph{Le
  Scienze Esatte nell' Antica Grecia}, Modena, 1893--1900; in G.~J.~Allman's
  \emph{Greek Geometry from Thales to Euclid}, Dublin, 1889; and in J.~Gow's \emph{Greek
  Mathematics}, Cambridge, 1884; it is also treated by Cantor, chaps.~ix, x,
  and~xi; by Hankel, pp.~111--156; and by C.~A.~Bretschneider in his \emph{Die
  Geometrie und die Geometer vor Eukleides}, Leipzig, 1870; a critical account
  of the original authorities is given by S.~P.~Tannery in his \emph{Géométrie Grecque},
  Paris, 1887, and other papers.}
\nblabel{xref:054}\index{Loria, ref.\ to}\index{Hankel, ref.\ to}\index{Tannery, S. P., ref.\ to}
\index{Athens,!School of, chapter \textsc{iii}}\index{Bretschneider, ref.\ to}\index{Cantor, M., ref.\ to}
\index{Cyzicus, School of, chapter \textsc{iii}}\nblabel{png054a}

\nblabel{png054b}\First{It} was towards the close of the fifth century before Christ that
Athens first became the chief centre of mathematical studies.
Several causes conspired to bring this about. During that
century she had become, partly by commerce, partly by appropriating
for her own purposes the contributions of her allies, the
most wealthy city in Greece; and the genius of her statesmen
had made her the centre on which the politics of the peninsula
turned. Moreover, whatever states disputed her claim to political
supremacy her intellectual pre-eminence was admitted by
all. There was no school of thought which had not at some
time in that century been represented at Athens by one or
more of its leading thinkers; and the ideas of the new science,
which was being so eagerly studied in Asia Minor and Graecia
Magna, had been brought before the Athenians on various
occasions.

\nblabel{png055a}\nblabel{png055b}\nbparagraph{Anaxagoras.} Amongst the most important of the philosophers
who resided at Athens and prepared the way for the
Athenian school I may mention \index{Anaxagoras of Clazomenae}\emph{Anaxagoras of Clazomenae},
who was almost the last philosopher of the \index{Ionian School, the}Ionian school. He
was born in 500~\textsc{b.c.}, and died in 428~\textsc{b.c.} He seems to have
settled at Athens about 440~\textsc{b.c.}, and there taught the results of
the Ionian philosophy. Like all members of that school he was
much interested in \index{Astronomy.!z12@Early Greek theories of}astronomy. He asserted that the \index{Sun, distance and radius of}sun was
larger than the Peloponnesus: this opinion, together with some
attempts he had made to explain various physical phenomena
which had been previously supposed to be due to the direct
action of the gods, led to a prosecution for impiety, and he was
convicted. While in prison he is said to have written a treatise
on the quadrature of the circle.

\index{Sophists, the}\nbparagraph{The Sophists.} The sophists can hardly be considered as
belonging to the Athenian school, any more than Anaxagoras
can; but like him they immediately preceded and prepared the
way for it, so that it is desirable to devote a few words to them.
One condition for success in public life at Athens was the power
of speaking well, and as the wealth and power of the city increased
a considerable number of ``sophists'' settled there who
undertook amongst other things to teach the art of oratory.
Many of them also directed the general education of their pupils,
of which geometry usually formed a part. We are told that two
of those who are usually termed sophists made a special study
of geometry---these were Hippias of Elis and \index{Antipho}Antipho, and one
made a special study of astronomy---this was \index{Meton}Meton, after whom
the metonic cycle is named.

\index{Hippias|(}\nbparagraph{Hippias.} The first of these geometricians, \emph{Hippias of Elis}
(circ.\ 420~\textsc{b.c.}), is described as an expert arithmetician, but he
is best known to us through his invention of a curve called the
\index{Angle,!trisection of}\index{Quadratrix}quadratrix, by means of which an \index{Trisection of angle}angle can be trisected, or
indeed divided in any given ratio. \index{Circle, quadrature of (or squaring the)}If the radius of a circle
rotate uniformly round the centre $O$ from the position $OA$
through a right angle to $OB$, and in the same time a straight
line drawn perpendicular to $OB$ move uniformly parallel to
itself from the position $OA$ to $BC$, the locus of their intersection
will be the \index{Quadratrix}quadratrix.

Let $OR$ and $MQ$ be the position of these lines at any time;
and let them cut in $P$, a point on the curve. Then
\begin{equation*}
\text{angle } AOP: \text{angle } AOB = OM:OB.
\end{equation*}
Similarly, if $OR'$ be another position of the radius,
\begin{eqnarray*}
& \text{angle } AOP': \text{angle } AOB = OM':OB \\
\therefore & \text{angle } AOP: \text{angle } AOP' = OM:OM'; \\%[**TN: OM:OM in original]
\therefore & \text{angle } AOP': \text{angle } P'OP = OM':MM.
\end{eqnarray*}
Hence, if the angle $AOP$ be given, and it be required to divide
it in any given ratio, it is sufficient to divide $OM$ in that ratio
at $M'$ and draw the line $M'P'$; %[**TN: MP' in original]
then $OP'$ will divide $AOP$ in
the required ratio.

\begin{center}
\labelincludegraphics{056a}
\end{center}

If $OA$ be taken as the initial line, $OP = r$, the angle $AOP = \theta$,
and $OA = a$, we have $\theta :\frac{1}{2}\pi = r \sin \theta :a$, and the equation of the
curve is $\pi r = 2a\theta \cosec \theta$.

Hippias devised an \index{Instruments, mathematical}instrument to construct the curve mechanically;
but constructions which involved the use of any mathematical
instruments except a ruler and a pair of compasses were
objected to by \index{Plato!ref.\ to}Plato, and rejected by most geometricians of a
subsequent date.\index{Hippias|)}

\nblabel{png057a}\index{Antipho}\nbparagraph{Antipho.} The second sophist whom I mentioned was
\emph{Antipho} (circ.~420 \textsc{b.c.}). He is one of the very few writers
among the ancients who attempted to find the area of a circle
by considering it as the limit of an inscribed regular polygon
with an infinite number of sides. He began by inscribing an
equilateral triangle (or, according to some accounts, a square);
on each side he inscribed in the smaller segment an isosceles
triangle, and so on \emph{ad infinitum}. This method of attacking the
quadrature problem is similar to that described above as used by
\index{Bryso}Bryso of Heraclea.

No doubt there were other cities in Greece besides Athens
where similar and equally meritorious work was being done,
though the record of it has now been lost; I have mentioned
here the investigations of these three writers, chiefly because they
were the immediate predecessors of those who created the
Athenian school.

\nbparagraph{The Schools of Athens and Cyzicus.} The history of the
Athenian school begins with the teaching of \index{Hippocrates of Chios!ref.\ to}Hippocrates about
420 \textsc{b.c.}; the school was established on a permanent basis by
the labours of Plato and Eudoxus; and, together with the
neighbouring school of Cyzicus, continued to extend on the lines
laid down by these three geometricians until the foundation
(about 300 \textsc{b.c.})\ of the university at Alexandria drew thither
most of the talent of Greece.

Eudoxus, who was amongst the most distinguished of the
Athenian mathematicians, is also reckoned as the founder of the
school at Cyzicus. The connection between this school and that
of Athens was very close, and it is now impossible to disentangle
their histories. It is said that \index{Hippocrates of Chios!ref.\ to}Hippocrates, Plato, and
Theaetetus belonged to the Athenian school; while \index{Eudoxus!ref.\ to}Eudoxus,
\index{Menaechmus!ref.\ to}Menaechmus, and Aristaeus belonged to that of Cyzicus. There
was always a constant intercourse between the two schools, the
earliest members of both had been under the influence either of
\index{Archytas!ref.\ to}Archytas or of his pupil \index{Theodorus of Cyrene!ref.\ to}Theodorus of Cyrene, and there was no
difference in their treatment of the subject, so that they may be
conveniently treated together.

\nblabel{xref:058}\nblabel{png058a}Before discussing the work of the geometricians of these
schools in detail I may note that they were especially interested
in three problems:\footnote
  {On these problems, solutions of them, and the authorities for their
  history, see my \emph{Mathematical Recreations and Problems}, London, ninth
  edition, 1920, chap.~xiv.}
namely (i),~the \index{Cube, duplication of}duplication of a cube, that
is, the determination of the side of a cube whose volume is
double that of a given cube; (ii)~the \index{Angle,!trisection of}\index{Trisection of angle}trisection of an angle; and
(iii)~the \index{Circle, quadrature of (or squaring the)}squaring of a circle, that is, the determination of a
square whose area is equal to that of a given circle.

Now the first two of these problems (considered analytically)
require the solution of a cubic equation; and, since a construction
by means of circles (whose equations are of the form
$x^{2} + y^{2} + ax + by + c = 0$) and straight lines (whose equations are
of the form $
 x + \beta y + \gamma = 0$) cannot be equivalent to the solution
of a cubic equation, the problems are insoluble if in our constructions
we restrict ourselves to the use of circles and straight
lines, that is, to Euclidean geometry. If the use of the conic
sections be permitted, both of these questions can be solved in
many ways. The third problem is equivalent to finding a
rectangle whose sides are equal respectively to the radius and to
the semiperimeter of the circle. \index{P1@$\pi$,!z2@incommensurability of}These lines have been long
known to be incommensurable, but it is only recently that it has
been shewn by \index{Lindemann}Lindemann that their ratio cannot be the root of
a rational algebraical equation. Hence this problem also is
insoluble by Euclidean geometry. The Athenians and Cyzicians
were thus destined to fail in all three problems, but the attempts
to solve them led to the discovery of many new theorems and
processes.

Besides attacking these problems the later Platonic school
collected all the geometrical theorems then known and arranged
them systematically. These collections comprised the bulk of
the propositions in Euclid's \emph{Elements}, books~\textsc{i--ix},~\textsc{xi}, and~\textsc{xii},
together with some of the more elementary theorems in conic
sections.

\nblabel{png058b}\nbparagraph{Hippocrates.} \index{Hippocrates of Chios|(}\emph{Hippocrates of Chios} (who must be carefully
distinguished from his contemporary, \index{Hippocrates of Cos}Hippocrates of Cos, the
celebrated physician) was one of the greatest of the Greek
geometricians. He was born about 470 \textsc{b.c.}\ at Chios, and began
life as a merchant. The accounts differ as to whether he was
swindled by the Athenian custom-house officials who were
stationed at the Chersonese, or whether one of his vessels was
captured by an Athenian pirate near Byzantium; but at any
rate somewhere about 430 \textsc{b.c.}\ he came to Athens to try to
recover his property in the law courts. A foreigner was not
likely to succeed in such a case, and the Athenians seem only to
have laughed at him for his simplicity, first in allowing himself
to be cheated, and then in hoping to recover his money. While
prosecuting his cause he attended the lectures of various
philosophers, and finally (in all probability to earn a livelihood)
opened a school of geometry himself. He seems to have been
well acquainted with the Pythagorean philosophy, though there
is no sufficient authority for the statement that he was ever
initiated as a Pythagorean.

He wrote the first elementary text-book of geometry, a text-book
on which probably Euclid's \emph{Elements} was founded; and
therefore he may be said to have sketched out the lines on
which geometry is still taught in English schools. \nblabel{png059a}It is supposed
that the use of \index{Letters in diagrams}letters in diagrams to describe a figure was made
by him or introduced about this time, as he employs expressions
such as ``the point on which the letter $A$ stands'' and ``the line
on which $AB$ is marked.'' \index{Cantor, M., ref.\ to}Cantor, however, thinks that the
Pythagoreans had previously been accustomed to represent the
five vertices of the pentagram-star by the letters $\upsilon\ \gamma\ \iota\ \theta\ \alpha$;
and though this was a single instance, perhaps they may have
used the method generally. The Indian geometers never
employed letters to aid them in the description of their figures.
Hippocrates also denoted the square on a line by the word
\mytextgreek{d'unamis}, and thus gave the technical meaning to the word
\index{Power, origin of term}\emph{power} which it still retains in algebra: there is reason to think
that this use of the word was derived from the Pythagoreans,
who are said to have enunciated the result of the proposition
\index{Eucn@Euc.!a47@\textsc{i}, 47. ref.\ to}Euc.~\textsc{i}, 47, in the form that ``the total power of the
sides of a right-angled triangle is the same as that of the
hypotenuse.''

\nblabel{png060a}In this text-book Hippocrates introduced the method of
``reducing'' one theorem to another, which being proved, the
thing proposed necessarily follows; of this method the \index{Reductio ad absurdum}\emph{reductio
ad absurdum} is an illustration. No doubt the principle had
been used occasionally before, but he drew attention to it as
a legitimate mode of proof which was capable of numerous
applications. \index{Reduction in geometry}He elaborated the geometry of the circle: proving,
among other propositions, that similar segments of a circle
contain equal angles; that the angle subtended by the chord of
a circle is greater than, equal to, or less than a right angle as
the segment of the circle containing it is less than, equal to, or
greater than a semicircle \index{Eucn@Euc.!c31@\textsc{iii}, 31. ref.\ to}(Euc.~\textsc{iii}, 31); and probably several
other of the propositions in the third book of Euclid. It is
most likely that he also established the propositions that [similar]
circles are to one another as the squares of their diameters
\index{Eucn@Euc.!j02@\textsc{xii}, 2. ref.\ to}(Euc.~\textsc{xii}, 2), and that similar segments are as the squares of
their chords. The proof given in Euclid of the first of these
theorems is believed to be due to Hippocrates.

\nblabel{png060b}The most celebrated discoveries of Hippocrates were, however,
in connection with the quadrature of the circle and the
duplication of the cube, and owing to his influence these
problems played a prominent part in the history of the Athenian
school.

The following propositions will sufficiently illustrate the
method by which he attacked the quadrature problem.

\begin{center}
\labelincludegraphics{061a}
\end{center}

($\alpha$) He commenced by finding the area of a \index{Lunes, quadrature of|(}\index{Quadrature of curves!lunes|(}lune contained
between a semicircle and a quadrantal arc standing on the same
chord. This he did as follows. Let $ABC$ be an isosceles right-angled
triangle inscribed in the semicircle $ABOC$, whose centre
is $O$. On $AB$ and $AC$ as diameters describe semicircles as in
the \figureref{061a}{figure}. Then, since by Euc.~\textsc{i}, 47,
\[
\text{sq.\ on } BC = \text{sq.\ on } AC + \text{sq.\ on } AB,
\]
therefore, by Euc.~\textsc{xii},~2,
\[
\text{area } \tfrac{1}{2}\odot \text{ on } BC = \text{area } \tfrac{1}{2}\odot \text{ on } AC + \text{area } \tfrac{1}{2}\odot \text{ on } AB.
\]
Take away the common parts
\[
\therefore \text{area } \triangle ABC = \text{sum of areas of lunes } AECD \text{ and } AFBG.
\]
Hence the area of the lune $AECD$ is equal to half that of the
triangle $ABC$.

\begin{center}
\labelincludegraphics{061b}
\end{center}

($\beta$) He next inscribed half a regular hexagon $ABCD$ in a semicircle whose
centre was $O$, and on $OA$, $AB$, $BC$, and $CD$ as
diameters described semicircles of which those on $OA$ and $AB$
are drawn in the \figureref{061b}{figure}. Then $AD$ is double any of the lines
$OA$, $AB$, $BC$, and $CD$,
\begin{gather*}
\therefore \text{sq.\ on } AD = \text{sum of sqs.\ on } OA, AB, BC, \text{ and } CD,\\
\therefore \text{area } \tfrac{1}{2}\odot ABCD = \text{sum of areas of } \tfrac{1}{2}\odot s \text{ on } OA, AB, BC, \text{ and } CD.
\end{gather*}
Take away the common parts
\[
\therefore \text{area trapezium } ABCD = 3 \text{ lune } AEBF + \tfrac{1}{2}\odot \text{ on } OA.
\]
If therefore the area of this latter lune be known, so is that of
the semicircle described on $OA$ as diameter. According to
Simplicius, Hippocrates assumed that the area of this lune was
the same as the area of the lune found in proposition ($\alpha$); if
this be so, he was of course mistaken, as in this case he is dealing
with a lune contained between a semicircle and a sextantal
arc standing on the same chord; but it seems more probable
that \index{Simplicius, ref.\ to}Simplicius misunderstood Hippocrates.\index{Lunes, quadrature of|)}\index{Quadrature of curves!lunes|)}

Hippocrates also enunciated various other theorems connected
with lunes (which have been collected by \index{Bretschneider, ref.\ to}Bretschneider and by
Allman) of which the theorem last given is a typical example.
I believe that they are the earliest instances in which areas
bounded by curves were determined by geometry.

\nblabel{xref:062}\nblabel{png062a}The other problem to which Hippocrates turned his attention
was the \index{Cube, duplication of|(}duplication of a cube, that is, the determination of
the side of a cube whose volume is double that of a given
cube.

\index{Cube, duplication of!origin of problem}This problem was known in ancient times as the Delian
problem, in consequence of a legend that the Delians had consulted
Plato on the subject. In one form of the story, which
is related by \index{Philoponus}Philoponus, it is asserted that the Athenians in
430~\textsc{b.c.}, when suffering from the plague of eruptive typhoid
fever, consulted the oracle at Delos as to how they could stop
it. Apollo replied that they must double the size of his altar
which was in the form of a cube. To the unlearned suppliants
nothing seemed more easy, and a new altar was constructed
either having each of its edges double that of the old one (from
which it followed that the volume was increased eightfold) or
by placing a similar cubic altar next to the old one. Whereupon,
according to the legend, the indignant god made the
pestilence worse than before, and informed a fresh deputation
that it was useless to trifle with him, as his new altar must
be a cube and have a volume exactly double that of his old
one. Suspecting a mystery the Athenians applied to Plato,
who referred them to the geometricians, and especially to
Euclid, who had made a special study of the problem. The
introduction of the names of \index{Plato|(}Plato and \index{Euclid!ref.\ to}Euclid is an obvious
anachronism. \index{Eratosthenes!ref.\ to}Eratosthenes gives a somewhat similar account
of its origin, but with \index{Minos, King, ref.\ to}king Minos as the propounder of the
problem.

\nblabel{xref:063}Hippocrates reduced the problem of duplicating the cube to
that of finding two means between one straight line ($a$), and
another twice as long ($2a$). If these means be $x$ and $y$, we
have $a : x = x : y = y : 2a$, from which it follows that $x^{3} = 2a^{3}$.
It is in this form that the problem is usually presented now.
Hippocrates did not succeed in finding a construction for these
means.\index{Hippocrates of Chios|)}\index{Cube, duplication of|)}

\nblabel{png063a}\nbparagraph{Plato.} The next philosopher of the Athenian school who
requires mention here was \emph{Plato}. He was born at Athens in
429~\textsc{b.c.}, and was, as is well known, a pupil for eight years of
\index{Socrates, ref.\ to}Socrates; much of the teaching of the latter is inferred from
Plato's dialogues. After the execution of his master in 399~\textsc{b.c.}\
Plato left Athens, and being possessed of considerable wealth
he spent some years in travelling; it was during this time that
he studied mathematics. He visited Egypt with \index{Eudoxus!ref.\ to}Eudoxus, and
\index{Strabo, ref.\ to}Strabo says that in his time the apartments they occupied at
Heliopolis were still shewn. Thence Plato went to Cyrene,
where he studied under \index{Theodorus of Cyrene!ref.\ to}Theodorus. Next he moved to Italy,
where he became intimate with \index{Archytas!ref.\ to}Archytas the then head of the
\index{Pythagorean School, the!ref.\ to}Pythagorean school, \index{Eurytas of Metapontum}Eurytas of Metapontum, and \index{Timaeus of Locri}Timaeus of
Locri. He returned to Athens about the year 380~\textsc{b.c.}, and
formed a school of students in a suburban gymnasium called
the ``Academy.'' He died in 348~\textsc{b.c.}

Plato, like Pythagoras, was primarily a philosopher, and
perhaps his philosophy should be regarded as founded on the
Pythagorean rather than on the Socratic teaching. At any
rate it, like that of the Pythagoreans, was coloured with the
idea that the secret of the universe is to be found in number
and in form; hence, as \index{Eudemus}Eudemus says, ``he exhibited on every
occasion the remarkable connection between mathematics and
philosophy.'' All the authorities agree that, unlike many later
philosophers, he made a study of geometry or some exact
science an indispensable preliminary to that of philosophy.
The inscription over the entrance to his school ran ``Let none
ignorant of geometry enter my door,'' and on one occasion an
applicant who knew no geometry is said to have been refused
admission as a student.

\index{Academy,!Plato's}Plato's position as one of the masters of the Athenian
mathematical school rests not so much on his individual discoveries
and writings as on the extraordinary influence he
exerted on his contemporaries and successors. Thus the objection
that he expressed to the use in the construction of curves
of any \index{Instruments, mathematical}instruments other than rulers and compasses was at once
accepted as a canon which must be observed in such problems.
It is probably due to Plato that subsequent geometricians
began the subject with a carefully compiled series of definitions,
postulates, and axioms. \nblabel{png064a}He also systematized the methods
which could be used in attacking mathematical questions, and
in particular directed attention to the value of analysis. The
analytical method of proof begins by assuming that the theorem
or problem is solved, and thence deducing some result: if the
result be false, the theorem is not true or the problem is incapable
of solution: if the result be true, and if the steps be
reversible, we get (by reversing them) a \index{Analysis,!in synthetic geometry}synthetic proof; but
if the steps be not reversible, no conclusion can be drawn.
Numerous illustrations of the method will be found in any
modern text-book on geometry. If the classification of the
methods of legitimate induction given by \index{Mill's \emph{Logic}, ref.\ to}Mill in his work on
logic had been universally accepted and every new discovery in
science had been justified by a reference to the rules there laid
down, he would, I imagine, have occupied a position in reference
to modern science somewhat analogous to that which Plato
occupied in regard to the mathematics of his time.

\nblabel{png065a}The following is the only extant theorem traditionally attributed
to Plato. If $CAB$ and $DAB$ be two right-angled
triangles, having one side, $AB$, common, their other sides, $AD$
and $BC$, parallel, and their hypotenuses, $AC$ and $BD$, at
right angles, then, if these hypotenuses cut in $P$, we have
$PC : PB = PB : PA = PA : PD$. This theorem was used in
\index{Cube, duplication of}duplicating the cube, for, if such triangles can be constructed
having $PD = 2PC$, the problem will be solved. It is easy
to make an instrument by which the triangles can be constructed.

\nblabel{png065b}\index{Eudoxus|(}\nbparagraph{Eudoxus.\footnote
  {The works of \index{Kunssberg@Künssberg on Eudoxus}Eudoxus were discussed in considerable detail by
  H.~Künssberg of Dinkelsbühl in 1888 and 1890; see also the authorities
  mentioned above in the footnote on p.~\pageref{xref:054}.}}
Of \emph{Eudoxus}, the third great mathematician of
the Athenian school and the founder of that at Cyzicus, we
know very little. He was born in Cnidus in 408~\textsc{b.c.} Like
Plato, he went to Tarentum and studied under \index{Archytas!ref.\ to}Archytas the
then head of the Pythagoreans. Subsequently he travelled
with \index{Plato|)}Plato to Egypt, and then settled at Cyzicus, where he
founded the school of that name. Finally he and his pupils
moved to Athens. There he seems to have taken some part in
public affairs, and to have practised medicine; but the hostility
of Plato and his own unpopularity as a foreigner made his
position uncomfortable, and he returned to Cyzicus or Cnidus
shortly before his death. He died while on a journey to Egypt
in 355~\textsc{b.c.}

His mathematical work seems to have been of a high order
of excellence. He discovered most of what we now know as
the \index{Eucn@Euc.!d@\textsc{v}. ref.\ to}fifth book of Euclid, and proved it in much the same form
as that in which it is there given.

\begin{wrapfigure}[3]{r}{0pt}
\labelincludegraphics{065a}
\end{wrapfigure}

\nblabel{png065c}He discovered some theorems on what was called \index{Section, the golden}\index{Golden section, the}``the
golden section.'' The problem to
cut a line $AB$ in the golden section,
that is, to divide it, say at $H$, in
extreme and mean ratio (that is, so that $AB : AH = AH : HB$) is
solved in \index{Eucn@Euc.!b11@\textsc{ii}, 11. ref.\ to}Euc.~\textsc{ii}, 11, and probably was known to the Pythagoreans
at an early date. If we denote $AB$ by $l$, $AH$ by $a$, and
$HB$ by $b$, the theorems that Eudoxus proved are equivalent
to the following algebraical identities. (i)~$(a + \frac{1}{2}l)^{2} = 5(\frac{1}{2}l)^{2}$.
(ii)~Conversely, if (i)~be true, and $AH$ be taken equal to $a$,
then $AB$ will be divided at $H$ in a golden section.
(iii)~$(b + \frac{1}{2}a)^{2} = 5(\frac{1}{2}a^{2})$.
(iv)~$l^{2} + b^{2} = 3a^{2}$.
(v)~$l + a : l = l : a$, which
gives another \index{Golden section, the}\index{Section, the golden}golden section. These propositions were subsequently
put by \index{Eucn@Euc.!k01@\textsc{xiii}, 1--5. ref.\ to}Euclid as the first five propositions of his
thirteenth book, but they might have been equally well placed
towards the end of the second book. All of them are obvious
algebraically, since $l = a + b$ and $a^{2} = bl$.

\nblabel{png066a}Eudoxus further established the \index{Exhaustions, method of}``method of exhaustions'';
which depends on the proposition that ``if from the greater
of two unequal magnitudes there be taken more than its half,
and from the remainder more than its half, and so on, there
will at length remain a magnitude less than the least of the
proposed magnitudes.'' This proposition was placed by \index{Eucn@Euc.!g01@\textsc{x}, 1. ref.\ to}Euclid
as the first proposition of the tenth book of his \emph{Elements}, but
in most modern school editions it is printed at the beginning of
the twelfth book. By the aid of this theorem the ancient
geometers were able to avoid the use of infinitesimals: the
method is rigorous, but awkward of application. A good illustration
of its use is to be found in the demonstration of \index{Eucn@Euc.!j02@\textsc{xii}, 2. ref.\ to}Euc.~\textsc{xii}, 2, %[**TN: "xii. 2" in original text]
namely, that the square of the radius of one circle is to
the square of the radius of another circle as the area of the first
circle is to an area which is neither less nor greater than the
area of the second circle, and which therefore must be exactly
equal to it: the proof given by Euclid is (as was usual) completed
by a \emph{reductio ad absurdum}. Eudoxus applied the
principle to shew that the \index{Pyramid,!volume of}\index{Spheres, volumes of}volume of a pyramid or a \index{Cone, sections of!volume of}cone is
one-third that of the prism or the cylinder on the same base and
of the same altitude \index{Eucn@Euc.!j07@\textsc{xii}, 7. ref.\ to}\index{Eucn@Euc.!j10@\textsc{xii}, 10. ref.\ to}(Euc.~\textsc{xii}, 7 and 10). It is believed that
he proved that the volumes of two spheres were to one another
as the cubes of their radii; some writers attribute the proposition
\index{Eucn@Euc.!j02@\textsc{xii}, 2. ref.\ to}Euc.~\textsc{xii}, 2 to him, and not to Hippocrates.

Eudoxus also considered certain curves other than the circle.
There is no authority for the statement made in some old books
that these were \index{Cone, sections of}conic sections, and recent investigations have
shewn that the assertion (which I repeated in the earlier editions
of this book) that they were plane sections of the \index{Anchor ring}anchor-ring is
also improbable. It seems most likely that they were tortuous
curves; whatever they were, he applied them in explaining the
apparent \index{Planetary motions}motions of the planets as seen from the earth.

\index{Astronomy.!z12@Early Greek theories of}Eudoxus constructed an \index{Orrery}orrery, and wrote a treatise on
practical astronomy, in which he supposed a number of moving
spheres to which the sun, moon, and stars were attached, and
which by their rotation produced the effects observed. In all
he required twenty-seven spheres. As observations became more
accurate, subsequent astronomers who accepted the theory had
continually to introduce fresh spheres to make the theory agree
with the facts. The work of \index{Aratus}Aratus on astronomy, which was
written about 300~\textsc{b.c.}\ and is still extant, is founded on that of
Eudoxus.

\nblabel{png067a}Plato and Eudoxus were contemporaries. Among Plato's
pupils were the mathematicians \index{Leodamas of Athens}\nbname{Leodamas}, \index{Neocleides of Athens}\nbname{Neocleides},
\index{Amyclas of Athens}\nbname{Amyclas}, and to their school also belonged \index{Leon of Athens}\nbname{Leon}, \index{Theudius of Athens}\nbname{Theudius}
(both of whom wrote text-books on plane geometry), \index{Cyzicenus of Athens}\nbname{Cyzicenus},
\index{Thasus of Athens}\nbname{Thasus}, \index{Hermotimus of Athens}\nbname{Hermotimus}, \index{Philippus of Athens}\nbname{Philippus}, and \index{Theaetetus!ref.\ to}\nbname{Theaetetus}. Among
the pupils of Eudoxus are reckoned \nbname{Menaechmus}, his brother
\index{Dinostratus of Cyzicus}\nbname{Dinostratus} (who applied the \index{Quadratrix}quadratrix to the duplication and
trisection problems), and \index{Aristaeus!ref.\ to}\nbname{Aristaeus}.\index{Eudoxus|)}

\index{Menaechmus|(}\nblabel{png067b}\nbname{Menaechmus.} Of the above-mentioned mathematicians
\emph{Menaechmus} requires special mention. He was born about
375~\textsc{b.c.}, and died about 325~\textsc{b.c.} Probably he succeeded
Eudoxus as head of the school at Cyzicus, where he acquired
great reputation as a teacher of geometry, and was for that
reason appointed one of the tutors of \index{Alexander the Great}Alexander the Great.
In answer to his pupil's request to make his proofs shorter,
Menaechmus made the well-known reply that though in the
country there are private and even royal roads, yet in geometry
there is only one road for all.

Menaechmus was the first to discuss the \index{Conic Sections (Geometrical).!Discussed by most of the Greek geometricians after Menaechmus}conic sections, which
were long called the \index{Menaechmian triads}Menaechmian triads. \nblabel{png068a}He divided them
into three classes, and investigated their properties, not by
taking different plane sections of a fixed cone, but by keeping
his plane fixed and cutting it by different cones. He shewed
that the section of a right cone by a plane perpendicular to
a generator is an ellipse, if the cone be acute-angled; a parabola,
if it be right-angled; and a hyperbola, if it be obtuse-angled;
and he gave a mechanical construction for curves of each class.
It seems almost certain that he was acquainted with the fundamental
properties of these curves; but some writers think that
he failed to connect them with the sections of the cone which he
had discovered, and there is no doubt that he regarded the
latter not as plane loci but as curves drawn on the surface of a
cone.

\nblabel{png068b}He also shewed how these curves could be used in either of
the two following ways to give a solution of the problem
to \index{Cube, duplication of}duplicate a cube. In the first of these, he pointed out that
two parabolas having a common vertex, axes at right angles,
and such that the latus rectum of the one is double that of the
other will intersect in another point whose abscissa (or ordinate)
will give a solution; for (using analysis) if the equations of the
parabolas be $y^{2} = 2ax$ and $x^{2} = ay$, they intersect in a point
whose abscissa is given by $x^{3} = 2a^{3}$. It is probable that this
method was suggested by the form in which Hippocrates had cast
the problem; namely, to find $x$ and $y$ so that $a : x = x : y = y : 2a$,
whence we have $x^{2} = ay$ and $y^{2} = 2ax$.

The second solution given by Menaechmus was as follows.
Describe a parabola of latus rectum $l$. Next describe a rectangular
hyperbola, the length of whose real axis is $4l$, and
having for its asymptotes the tangent at the vertex of the parabola
and the axis of the parabola. Then the ordinate and the
abscissa of the point of intersection of these curves are the
mean proportionals between $l$ and $2l$. This is at once obvious
by analysis. The curves are $x^{2} = ly$ and $xy = 2l^{2}$. These
cut in a point determined by $x^{3} = 2l^{3}$ and $y^{3} = 4l^{3}$. Hence
$l : x = x : y = y : 2l$.\index{Menaechmus|)}

{\nbstretchyspace\index{Aristaeus}\nblabel{png069a}\nbname{Aristaeus} and \index{Theaetetus}\nbname{Theaetetus}.\nbnamespace} Of the other members of these
schools, \emph{Aristaeus} and \emph{Theaetetus}, whose works are entirely lost,
were mathematicians of repute. We know that Aristaeus wrote
on the five regular solids and on conic sections, and that \index{Theodorus of Cyrene!ref.\ to}Theaetetus
developed the theory of \index{Incommensurables}incommensurable magnitudes.
The only theorem we can now definitely ascribe to the latter
is that given by \index{Eucn@Euc.!g00@\textsc{x}. ref.\ to}\index{Eucn@Euc.!g09@\textsc{x}, 9. ref.\ to}Euclid in the ninth proposition of the tenth
book of the \emph{Elements}, namely, that the squares on two commensurable
right lines have one to the other a ratio which a square
number has to a square number (and conversely); but the
squares on two incommensurable right lines have one to the
other a ratio which cannot be expressed as that of a square
number to a square number (and conversely). This theorem
includes the results given by Theodorus.\footnote
  {See above, p.~\pageref{xref:030}.}

The contemporaries or successors of these mathematicians
wrote some fresh text-books on the elements of geometry and
the conic sections, introduced problems concerned with finding
loci, and systematized the knowledge already acquired, but they
originated no new methods of research.

\index{Aristotle|(}\nblabel{png069b}\nbparagraph{Aristotle.} An account of the Athenian school would be
incomplete if there were no mention of \emph{Aristotle}, who was born
at Stagira in Macedonia in 384~\textsc{b.c.}\ and died at Chalcis in
Euboea in 322~\textsc{b.c.} Aristotle, however, deeply interested
though he was in natural philosophy, was chiefly concerned
with mathematics and mathematical physics as supplying illustrations
of correct reasoning. \nblabel{png069c}A small book containing a few
questions on \index{Mechanics.!z1@Discussed by!x1@Aristotle}mechanics which is sometimes attributed to him
is of doubtful authority; but, though in all probability it is
not his work, it is interesting, partly as shewing that the
principles of mechanics were beginning to excite attention, and
partly as containing the earliest known employment of \index{Letters in diagrams!to indicate magnitudes}letters
to indicate magnitudes.

The most instructive parts of the book are the dynamical
proof of the \index{Parallelogram of forces|(}\index{Forces!parallelogram of}parallelogram of forces for the direction of the
resultant, and the statement, in effect, that if $\alpha$ be a force, $\beta$ the
mass to which it is applied, $\gamma$ the distance through which it is
moved, and $\delta$ the time of the motion, then $\alpha$ will move $\frac{1}{2}\beta$
through $2\gamma$ in the time $\delta$, or through $\gamma$ in the time $\frac{1}{2}\delta$: but the
author goes on to say that it does not follow that $\frac{1}{2}\alpha$ will move
$\beta$ through $\frac{1}{2}\gamma$ in the time $\delta$, because $\frac{1}{2}\alpha$ may not be able to move
$\beta$ at all; for 100 men may drag a ship 100 yards, but it does
not follow that one man can drag it one yard. The first part
of this statement is correct and is equivalent to the statement
that an impulse is proportional to the momentum produced, but
the second part is wrong.\index{Parallelogram of forces|)}

The author also states the fact that what is gained in power
is lost in speed, and therefore that two weights which keep a
[weightless] lever in equilibrium are inversely proportional to
the arms of the lever; this, he says, is the explanation why it
is easier to extract teeth with a pair of pincers than with the
fingers. Among other questions raised, but not answered, are
why a projectile should ever stop, and why carriages with large
wheels are easier to move than those with small.

\index{Greek science}I ought to add that the book contains some gross blunders,
and as a whole is not as able or suggestive as might be inferred
from the above extracts. In fact, here as elsewhere, the Greeks
did not sufficiently realise that the fundamental facts on which
the mathematical treatment of mechanics must be based can
be established only by carefully devised observations and
experiments.\index{Aristotle|)}

\nbChapter{IV}{The First Alexandrian School}{}{Circ. 300~B.C.--30~B.C.}
  {The history of the Alexandrian Schools is discussed by G.~Loria in his
  \emph{Le Scienze Esatte nell' Antica Grecia}, Modena, 1893--1900; by Cantor,
  chaps.~xii--xxiii; and by J.~Gow in his \emph{History of Greek Mathematics},
  Cambridge, 1884. The subject of Greek algebra is treated by E.~H.~F.~Nesselmann
  in his \emph{Die Algebra der Griechen}, Berlin, 1842; see also L.~Matthiessen,
  \emph{Grundzüge der antiken und modernen Algebra der litteralen
  Gleichungen}, Leipzig, 1878. The Greek treatment of the conic sections forms
  the subject of \emph{Die Lehre von den Kegelschnitten in Altertum}, by H.~G.~Zeuthen,
  Copenhagen, 1886. The materials for the history of these schools
  have been subjected to a searching criticism by S.~P.~Tannery, and most of
  his papers are collected in his \emph{Géométrie Grecque}, Paris, 1887.}
\index{Alexandrian,!Schools, chapters \textsc{iv}, \textsc{v}}\nblabel{xref:050}\nblabel{png071a}
\index{Algebraa@Algebra,!histories of}\index{Gow, ref.\ to}\index{Loria, ref.\ to}
\index{Matthiessen}\index{Nesselmann, ref.\ to}\index{Tannery, S. P., ref.\ to}\index{Zeuthen}\index{Cantor, M., ref.\ to}

\nblabel{png071b}\First{The} earliest attempt to found a university, as we understand
the word, was made at Alexandria. Richly endowed, supplied
with lecture rooms, libraries, museums, laboratories, and
gardens, it became at once the intellectual metropolis of the
Greek race, and remained so for a thousand years. It was
particularly fortunate in producing within the first century of its
existence three of the greatest mathematicians of antiquity---Euclid,
Archimedes, and Apollonius. They laid down the lines
on which mathematics subsequently developed, and treated it as
a subject distinct from philosophy: hence the foundation of the
Alexandrian Schools is rightly taken as the commencement of a
new era. Thenceforward, until the destruction of the city by
the Arabs in 641~\textsc{a.d.}, the history of mathematics centres
more or less round that of Alexandria; for this reason the
Alexandrian Schools are commonly taken to include all Greek
mathematicians of their time.

\nblabel{png072a}The city and \index{Alexandria, university of}university of Alexandria were created under the
following circumstances. \index{Alexander the Great}Alexander the Great had ascended the
throne of Macedonia in 336~\textsc{b.c.}\ at the early age of twenty, and
by 332~\textsc{b.c.}\ he had conquered or subdued Greece, Asia Minor,
and Egypt. Following the plan he adopted whenever a commanding
site had been left unoccupied, he founded a new city
on the Mediterranean near one mouth of the Nile; and he himself
sketched out the ground-plan, and arranged for drafts of
Greeks, Egyptians, and Jews to be sent to occupy it. The city
was intended to be the most magnificent in the world, and, the
better to secure this, its erection was left in the hands of
\index{Dinocrates}Dinocrates, the architect of the temple of Diana at Ephesus.

After Alexander's death in 323~\textsc{b.c.}\ his empire was divided,
and Egypt fell to the lot of Ptolemy, who chose Alexandria
as the capital of his kingdom. A short period of confusion
followed, but as soon as \index{Ptolemies, dynasty of}Ptolemy was settled on the throne, say
about 306~\textsc{b.c.}, he determined to attract, so far as he was able,
learned men of all sorts to his new city; and he at once began
the erection of the university buildings on a piece of ground
immediately adjoining his palace. The university was ready to
be opened somewhere about 300~\textsc{b.c.}, and Ptolemy, who wished
to secure for its staff the most eminent philosophers of the time,
naturally turned to Athens to find them. \index{Alexandrian,!library}The great library
which was the central feature of the scheme was placed under
Demetrius \index{Phalereus}Phalereus, a distinguished Athenian, and so rapidly
did it grow that within forty years it (together with the
Egyptian annexe) possessed about 600,000 rolls. The mathematical
department was placed under Euclid, who was thus the
first, as he was one of the most famous, of the mathematicians
of the Alexandrian school.

It happens that contemporaneously with the foundation
of this school the information on which our history is based
becomes more ample and certain. Many of the works of the
Alexandrian mathematicians are still extant; and we have
besides an invaluable treatise by \index{Pappus!ref.\ to}Pappus, described below, in
which their best-known treatises are collated, discussed, and
criticized. It curiously turns out that just as we begin to be
able to speak with confidence on the subject-matter which was
taught, we find that our information as to the personality of
the teachers becomes vague; and we know very little of the
lives of the mathematicians mentioned in this and the next
chapter, even the dates at which they lived being frequently
in doubt.


\section*{The third century before Christ.\nblabel{png073a}}

\index{Demorgana@De~Morgan, A.!ref.\ to}\index{Euclid|(}\index{Heath, Sir T. L.}\index{Heiberg, ref.\ to}\index{Menge on Euclid}\index{Gow, ref.\ to}\nblabel{png073b}\nbname{Euclid.}\footnote
  {Besides Loria, book~ii, chap.~i; \index{Cantor, M., ref.\ to}Cantor, chaps.~xii,~xiii; and Gow, pp.~72--86,
  195--221; see the articles \emph{Eucleides} by A.~De~Morgan in Smith's
  \emph{Dictionary of Greek and Roman Biography}, London, 1849; the article on
  \emph{Irrational Quantity} by A.~De~Morgan in the \emph{Penny Cyclopaedia}, London,
  1839; \emph{Litterargeschichtliche Studien über Euklid}, by J.~L.~Heiberg, Leipzig,
  1882; and above all \emph{Euclid's Elements}, translated with an introduction and
  commentary by T.~L.~Heath, 3 volumes, Cambridge, 1908. The latest
  complete edition of all Euclid's works is that by J.~L.~Heiberg and H.~Menge,
  Leipzig, 1883--96.}%
---This century produced three of the greatest
mathematicians of antiquity, namely Euclid, \index{Archimedes!ref.\ to}Archimedes, and
\index{Apollonius!ref.\ to}Apollonius. The earliest of these was \emph{Euclid}. Of his life we
know next to nothing, save that he was of Greek descent, and
was born about 330~\textsc{b.c.}; he died about 275~\textsc{b.c.} It would
appear that he was well acquainted with the Platonic geometry,
but he does not seem to have read \index{Aristotle!ref.\ to}Aristotle's works; and these
facts are supposed to strengthen the tradition that he was
educated at Athens. Whatever may have been his previous
training and career, he proved a most successful teacher when
settled at Alexandria. He impressed his own individuality on
the teaching of the new university to such an extent that to
his successors and almost to his contemporaries the name
Euclid meant (as it does to us) the book or books he wrote,
and not the man himself. Some of the medieval writers went %[**TN: "mediaeval" changed to "medieval" for consistency]
so far as to deny his existence, and with the ingenuity of
philologists they explained that the term was only a corruption
of \mytextgreek{>ukli} a key, and \mytextgreek{dis} geometry. The former word was presumably
derived from \mytextgreek{kle'is}. I can only explain the meaning
assigned to \mytextgreek{dis} by the conjecture that as the \index{Pythagorean School, the!ref.\ to}Pythagoreans
said that the number two symbolized a line, possibly a schoolman
may have thought that it could be taken as indicative of
geometry.

From the meagre notices of Euclid which have come down
to us we find that the saying that there is no royal road in
geometry was attributed to Euclid as well as to \index{Menaechmus!ref.\ to}Menaechmus;
but it is an epigrammatic remark which has had many imitators.
According to tradition, Euclid was noticeable for his gentleness
and modesty. Of his teaching, an anecdote has been preserved.
\index{Stobaeus, ref.\ to}Stobaeus, who is a somewhat doubtful authority, tells us that,
when a lad who had just begun geometry asked, ``What do I
gain by learning all this stuff?'' Euclid insisted that knowledge
was worth acquiring for its own sake, but made his slave give
the boy some coppers, ``since,'' said he, ``he must make a profit
out of what he learns.''

Euclid was the author of several works, but his reputation
rests mainly on his \index{Euclid's \emph{Elements}|(}\nblabel{png074a}\emph{Elements}. This treatise contains a systematic
exposition of the leading propositions of elementary metrical
geometry (exclusive of conic sections) and of the theory of
numbers. It was at once adopted by the Greeks as the standard
text-book on the elements of pure mathematics, and it is probable
that it was written for that purpose and not as a philosophical
attempt to shew that the results of geometry and arithmetic are
necessary truths.

The modern text\footnote
  {Most of the modern text-books in English are founded on \index{Simson, Robert}Simson's
  edition, issued in 1758. \emph{Robert Simson}, who was born in 1687 and died in
  1768, was professor of mathematics at the University of Glasgow, and left
  several valuable works on ancient geometry.}
is founded on an edition or commentary
prepared by Theon, the father of Hypatia (circ.~380~\textsc{a.d.}).
There is at the Vatican a copy (circ.~1000~\textsc{a.d.})\ of an older text,
and we have besides quotations from the work and references to
it by numerous writers of various dates. From these sources we
gather that the definitions, axioms, and postulates were rearranged
and slightly altered by subsequent editors, but that
the propositions themselves are substantially as Euclid wrote
them.

As to the matter of the work. \nblabel{png075a}The geometrical part is to a
large extent a compilation from the works of previous writers.
Thus the substance of books~\textsc{i} and \textsc{ii} (except perhaps the treatment
of parallels) is probably due to Pythagoras; that of book
\textsc{iii} to \index{Hippocrates of Chios!ref.\ to}Hippocrates; that of book~\textsc{v} to \index{Eudoxus!ref.\ to}Eudoxus; and the bulk of
books~\textsc{iv}, \textsc{vi}, \textsc{xi}, and \textsc{xii} to the later Pythagorean or Athenian
schools. But this material was rearranged, obvious deductions
were omitted (for instance, the proposition that the perpendiculars
from the angular points of a triangle on the opposite sides meet
in a point was cut out), and in some cases new proofs substituted.
Book~X, which deals with irrational magnitudes, may
be founded on the lost book of \index{Theaetetus!ref.\ to}Theaetetus; but probably much
of it is original, for \index{Proclus!ref.\ to}Proclus says that while Euclid arranged
the propositions of Eudoxus he completed many of those of
Theaetetus. The whole was presented as a complete and
consistent body of theorems.

The form in which the propositions are presented, consisting
of enunciation, statement, construction, proof, and conclusion,
is due to Euclid: so also is the synthetical character of the
work, each proof being written out as a logically correct train of
reasoning but without any clue to the method by which it was
obtained.

The defects of Euclid's \emph{Elements} as a text-book of geometry
have been often stated; the most prominent are these. (i)~The
definitions and axioms contain many assumptions which are not
obvious, and in particular the postulate or axiom about parallel
lines is not self-evident.\footnote
  {We know, from the researches of \index{Lobatschewsky}Lobatschewsky and \index{Riemann!ref.\ to}Riemann, that it is
  incapable of proof.}
(ii)~No explanation is given as to
the reason why the proofs take the form in which they are
presented, that is, the synthetical proof is given but not the
analysis by which it was obtained. (iii)~There is no attempt
made to generalize the results arrived at; for instance, the idea
of an angle is never extended so as to cover the case where it is
equal to or greater than two right angles: the second half of
the thirty-third proposition in the sixth book, as now printed,
appears to be an exception, but it is due to \index{Theon of Alexandria!ref.\ to}Theon and not to
Euclid. (iv)~The principle of superposition as a method of
proof might be used more frequently with advantage. (v)~The
classification is imperfect. And (vi)~the work is unnecessarily
long and verbose. Some of those objections do not apply to
certain of the recent school editions of the \emph{Elements}.

On the other hand, the propositions in Euclid are arranged
so as to form a chain of geometrical reasoning, proceeding from
certain almost obvious assumptions by easy steps to results of
considerable complexity. The demonstrations are rigorous, often
elegant, and not too difficult for a beginner. Lastly, nearly all
the elementary metrical (as opposed to the graphical) properties
of space are investigated, while the fact that for two thousand
years it was the usual text-book on the subject raises a strong
presumption that it is not unsuitable for the purpose.

On the Continent rather more than a century ago, Euclid
was generally superseded by other text-books. In England
determined efforts have lately been made with the same purpose,
and numerous other works on elementary geometry have been
produced in the last decade. The change is too recent to enable
us to say definitely what its effect may be. But as far as I can
judge, boys who have learnt their geometry on the new system
know more facts, but have missed the mental and logical training
which was inseparable from a judicious study of Euclid's
treatise.

I do not think that all the objections above stated can fairly
be urged against Euclid himself. He published a collection of
problems, generally known as the \mytextgreek{Dedom'ena} or \emph{Data}. This
contains 95 illustrations of the kind of deductions which
frequently have to be made in analysis; such as that, if one
of the data of the problem under consideration be that one
angle of some triangle in the figure is constant, then it is
legitimate to conclude that the ratio of the area of the rectangle
under the sides containing the angle to the area of the triangle
is known [prop.~66]. \index{Pappus!ref.\ to}Pappus says that the work was written
for those ``who wish to acquire the power of solving problems.''
It is in fact a gradual series of exercises in geometrical analysis.
In short the \emph{Elements} gave the principal results, and were
intended to serve as a training in the science of reasoning, while
the \emph{Data} were intended to develop originality.

Euclid also wrote a work called \mytextgreek{Per`i Diair'esewn} or \emph{De
Divisionibus}, known to us only through an Arabic translation
which may be itself imperfect.\footnote
  {\index{Archibald, R. C.}R.~C.~Archibald, \emph{Euclid's Book on Divisions}, %[**TN: 'Book of Divisions' in original text]
   Cambridge, 1915.}
This is a collection of 36
problems on the division of areas into parts which bear to one
another a given ratio. It is not unlikely that this was only
one of several such collections of examples---possibly including
the \emph{Fallacies} and the \emph{Porisms}---but even by itself it shews that
the value of exercises and riders was fully recognized by Euclid.

I may here add a suggestion made by De~Morgan, whose
comments on Euclid's writings were notably ingenious and
informing. From internal evidence he thought it likely that
the \emph{Elements} were written towards the close of Euclid's life, and
that their present form represents the first draft of the proposed
work, which, with the exception of the tenth book, Euclid did
not live to revise. This opinion is generally discredited, and
there is no extrinsic evidence to support it.

The geometrical parts of the \emph{Elements} are so well known
that I need do no more than allude to them. Euclid admitted
only those constructions which could be made by the use of a
ruler and compasses.\footnote
  {The ruler must be of unlimited length and not graduated; the compasses
  also must be capable of being opened as wide as is desired. \index{Mascheroni}\emph{Lorenzo Mascheroni}
  (who was born at Castagneta on May~14, 1750, and died at Paris
  on July~30, 1800) set himself the task to obtain by means of constructions
  made only with a pair of compasses as many Euclidean results as possible.
  Mascheroni's treatise on the geometry of the compass, which was published
  at Pavia in 1795, is a curious \emph{tour de force}: he was professor first at
  Bergamo and afterwards at Pavia, and left numerous minor works. Similar
  limitations have been proposed by other writers.}
He also excluded practical work and
hypothetical constructions. The first four books and book~\textsc{vi}
deal with plane geometry; the theory of proportion (of
any magnitudes) is discussed in book~\textsc{v}; and books~\textsc{xi} and
\textsc{xii} treat of solid geometry. On the hypothesis that the
\emph{Elements} are the first draft of Euclid's proposed work, it is
possible that \index{Eucn@Euc.!k01@\textsc{xiii}, 1--5. ref.\ to}\index{Eucn@Euc.!k06@\textsc{xiii}, 6--12. ref.\ to}\index{Eucn@Euc.!k13@\textsc{xiii}, 13--18. ref.\ to}book~\textsc{xiii} is a sort of appendix containing some
additional propositions which would have been put ultimately
in one or other of the earlier books. Thus, as mentioned
above, the first five propositions which deal with a line cut
in \index{Golden section, the}\index{Section, the golden}golden section might be added to the second book. The
next seven propositions are concerned with the relations between
certain incommensurable lines in plane figures (such as
the radius of a circle and the sides of an inscribed regular
triangle, pentagon, hexagon, and decagon) which are treated by
the methods of the tenth book and as an illustration of them.
Constructions of the five regular solids are discussed in the last
six propositions, and it seems probable that Euclid and his
contemporaries attached great importance to this group of
problems. \index{Bretschneider, ref.\ to}Bretschneider inclined to think that the thirteenth
book is a summary of part of the lost work of \index{Aristaeus!ref.\ to}Aristaeus: but
the illustrations of the methods of the tenth book are due most
probably to \index{Theaetetus!ref.\ to}Theaetetus.

\nblabel{png078a}Books~\textsc{vii}, \textsc{viii}, \textsc{ix}, and \textsc{x} of the \emph{Elements} are given up
to the theory of numbers. \index{Arithmetic.!z4@Practical Greek}The mere art of calculation or
\mytextgreek{logistik'h} was taught to boys when quite young, it was stigmatized
by \index{Plato!ref.\ to}Plato as childish, and never received much attention
from Greek mathematicians; nor was it regarded as
forming part of a course of mathematics. We do not know
how it was taught, but the abacus\index{Abacus,!ref.\ to} certainly played a prominent
part in it. The scientific treatment of numbers was called
\index{Arj@\mytextgreek{>Arijmhtik'h} signification of}\mytextgreek{>arijmhtik'h}, which I have here generally translated as the
science of numbers. It had special reference to ratio, proportion,
and the theory of numbers. It is with this alone that
most of the extant Greek works deal.

\index{Algebra.!z1@Treated geometrically by Euclid and his School|(}In discussing Euclid's arrangement of the subject, we must
therefore bear in mind that those who attended his lectures
were already familiar with the art of calculation. The system
of numeration adopted by the Greeks is described later,\footnote
  {See below, chapter~\hyperref[chapter:VII]{vii}.}
but it was so clumsy that it rendered the scientific treatment of
numbers much more difficult than that of geometry; hence
Euclid commenced his mathematical course with plane geometry.
At the same time it must be observed that the results of the
second book, though geometrical in form, are capable of
expression in algebraical language, and the fact that numbers
could be represented by lines was probably insisted on at an
early stage, and illustrated by concrete examples. This
\index{Graphical methods}graphical method of using lines to represent numbers possesses
the obvious advantage of leading to proofs which are true for
all numbers, rational or irrational. \index{Arithmetic.!z5@Theory of,!x1@ treated geometrically by most of the Greek mathematicians to the end of the first Alexandrian School}It will be noticed that
among other propositions in the second book we get geometrical
proofs of the distributive and commutative laws, of rules for
multiplication, and finally geometrical solutions of the \index{Quadratic equations}equations
$a(a-x)=x^2$, that is $x^2+ax-a^2=0$ \index{Eucn@Euc.!b11@\textsc{ii}, 11. ref.\ to}(Euc.~\textsc{ii}, 11), and $x^2-ab=0$
\index{Eucn@Euc.!b14@\textsc{ii}, 14. ref.\ to}(Euc.~\textsc{ii}, 14): the solution of the first of these equations is
given in the form $\sqrt{\smash[b]{a^2+(\frac{1}{2}a)^2}}-\frac{1}{2}a$. The solutions of the
equations $ax^2-bx+c=0$ and $ax^2+bx-c=0$ are given later in
\index{Eucn@Euc.!e28@\textsc{vi}, 28. ref.\ to}Euc.~\textsc{vi}, 28 and \index{Eucn@Euc.!e29@\textsc{vi}, 29. ref.\ to}\textsc{vi}, 29; the cases when $a=1$ can be deduced
from the identities proved in \index{Eucn@Euc.!b05@\textsc{ii}, 5. ref.\ to}\index{Eucn@Euc.!b06@\textsc{ii}, 6. ref.\ to}Euc.~\textsc{ii}, 5 and 6, but it is doubtful
if Euclid recognized this.

The results of the fifth book, in which the theory of \index{Proportion, symbols for!treatment by Euclid}proportion
is considered, apply to any magnitudes, and therefore are
true of numbers as well as of geometrical magnitudes. In the
opinion of many writers this is the most satisfactory way of
treating the theory of proportion on a scientific basis; and it
was used by Euclid as the foundation on which he built the
theory of numbers. The theory of proportion given in this
book is believed to be due to \index{Eudoxus!ref.\ to}Eudoxus. The treatment of the
same subject in the seventh book is less elegant, and is supposed
to be a reproduction of the Pythagorean teaching. This double
discussion of proportion is, as far as it goes, in favour of the
conjecture that Euclid did not live to revise the work.

\index{Numbers, theory of.!Treatment of, by Pythagoras!q1@by Euclid|(}In books~\textsc{vii}, \textsc{viii}, and \textsc{ix} Euclid discusses the theory of
\index{Rational numbers, Euclid on}rational numbers. He commences the seventh book with some
definitions founded on the Pythagorean notation. In propositions
1 to 3 he shews that if, in the usual process for finding
the \index{Greatest common measure}greatest common measure of two numbers, the last divisor
be unity, the numbers must be \index{Primes}prime; and he thence deduces
the rule for finding their G.C.M. Propositions 4 to 22 include
the theory of fractions, which he bases on the theory of proportion;
among other results he shews that $ab = ba$ [prop.~16].
In propositions 23 to 34 he treats of prime numbers, giving
many of the theorems in modern text-books on algebra. In
propositions 35 to 41 he discusses the \index{Least common multiple}least common multiple of
numbers, and some miscellaneous problems.

The eighth book is chiefly devoted to numbers in continued
proportion, that is, in a \index{Geometrical progressions}\index{Progressions,!geometrical}geometrical progression; and the cases
where one or more is a product, square, or cube are specially
considered.

In the ninth book Euclid continues the discussion of geometrical
progressions, and in proposition 35 he enunciates the
rule for the summation of a series of $n$ terms, though the proof
is given only for the case where $n$ is equal to 4. He also
develops the theory of primes, shews that the number of primes
is infinite [prop.~20], and discusses the properties of odd and
even numbers. He concludes by shewing that a number of the
form $2^{n-1} (2^{n}-1)$, where $2^{n}-1$ is a prime, is a \index{Numbers,!perfect}\index{Perfect numbers}``perfect''
number [prop.~36].

\index{Commensurables, Euclid on}In the tenth book Euclid deals with certain \index{Irrational numbers|(}irrational
magnitudes; and, since the Greeks possessed no symbolism for
surds, he was forced to adopt a geometrical representation.
Propositions~1 to 21 deal generally with \index{Incommensurables}incommensurable
magnitudes. The rest of the book, namely, propositions~22 to
117, is devoted to the discussion of every possible variety of
lines which can be represented by $\surd{(\surd{a} ± \surd{b})}$, where $a$ and $b$
denote commensurable lines. There are twenty-five species of
such lines, and that Euclid could detect and classify them all
is in the opinion of so competent an authority as \index{Nesselmann, ref.\ to}Nesselmann
the most striking illustration of his genius. No further advance
in the \index{Incommensurables}theory of incommensurable magnitudes was made until
the subject was taken up by \index{Leonardo of Pisa!ref.\ to}Leonardo and \index{Cardan!ref.\ to}Cardan after an
interval of more than a thousand years.\index{Irrational numbers|)}\index{Algebra.!z1@Treated geometrically by Euclid and his School|)}

\nblabel{xref:081}In the last proposition of the \index{Eucn@Euc.!g117@\textsc{x}, 117. ref.\ to}tenth book [prop.~117] the
side and diagonal of a square are proved to be incommensurable.
The proof is so short and easy that I may quote it. If possible
let the side be to the diagonal in a commensurable ratio,
namely, that of two integers, $a$ and $b$. Suppose this ratio
reduced to its lowest terms so that $a$ and $b$ have no common
divisor other than unity, that is, they are \index{Primes}prime to one another.
Then (by Euc.~\textsc{i}, 47) $b^2 = 2a^{2}$; therefore $b^{2}$ is an even number;
therefore $b$ is an even number; hence, since $a$ is prime to $b$, $a$
must be an odd number. Again, since it has been shewn that
$b$ is an even number, $b$ may be represented by $2n$; therefore
$(2n)^{2} = 2a^{2}$; therefore $a^{2} = 2n^{2}$; therefore $a^{2}$ is an even number;
therefore $a$ is an even number. Thus the same number $a$ must
be both odd and even, which is absurd; therefore the side and
diagonal are incommensurable. \index{Hankel, ref.\ to}Hankel believes that this proof
was due to \index{Pythagoras!ref.\ to}Pythagoras, and this is not unlikely. This proposition
is also proved in another way in Euc.~\textsc{x}, 9, and for this
and other reasons it is now usually believed to be an interpolation
by some commentator on the \emph{Elements}.\index{Euclid's \emph{Elements}|)}\index{Numbers, theory of.!Treatment of, by Pythagoras!q1@by Euclid|)}

\nblabel{png081a}In addition to the \emph{Elements} and the two collections of riders
above mentioned (which are extant) Euclid wrote the following
books on geometry: (i)~an elementary treatise on \index{Conic Sections (Geometrical).!Discussed by most of the Greek geometricians after Menaechmus!x1@especially by Euclid}\emph{conic sections}
in four books; (ii)~a book on \emph{surface loci}, probably confined
to curves on the cone and cylinder; (iii)~a collection of \emph{geometrical
fallacies}, which were to be used as exercises in the
detection of errors; and (iv)~a treatise on \index{Porisms of Euclid}\emph{porisms} arranged in
three books. All of these are lost, but the work on porisms
was discussed at such length by \index{Pappus!ref.\ to}Pappus, that some writers
have thought it possible to restore it. In particular, \index{Chasles, M., ref.\ to}Chasles
in 1860 published what he considered to be substantially a reproduction
of it. In this will be found the conceptions of cross
ratios and projection, and those ideas of modern geometry which
were used so extensively by Chasles and other writers of the
nineteenth century. It should be realized, however, that the
statements of the classical writers concerning this book are
either very brief or have come to us only in a mutilated
form, and \index{Demorgana@De~Morgan, A.!ref.\ to}De~Morgan frankly says that he found them unintelligible,
an opinion in which most of those who read them
will, I think, concur.

Euclid published a book on \index{Optics (geometrical).!Discussed by (among others) Euclid}\index{Optics (geometrical).!z11@(physical)}optics, treated geometrically,
which contains 61 propositions founded on 12 assumptions. \index{Light,!physical theories of}It
commences with the assumption that objects are seen by rays
emitted from the eye in straight lines, ``for if light proceeded
from the object we should not, as we often do, fail to perceive a
needle on the floor.'' A work called \emph{Catoptrica} is also attributed
to him by some of the older writers; the text is corrupt and the
authorship doubtful; it consists of 31 propositions dealing with
reflexions in plane, convex, and concave mirrors. The geometry
of both books is Euclidean in form.

Euclid has been credited with an ingenious demonstration\index{Woepcke, ref.\ to}\footnote
  {It is given (from the Arabic) by F.~Woepcke in the \emph{Journal Asiatique},
  series 4, vol.~xviii, October 1851, pp.~225--232.}
of the principle of the \index{Lever, principle of}lever, but its authenticity is doubtful.
\index{Astronomy.!z12@Early Greek theories of}He also wrote the \emph{Phaenomena}, a treatise on geometrical astronomy.
It contains references to the work of \index{Autolycus}Autolycus\index{Hultsch, ref.\ to}\footnote
  {\emph{Autolycus} lived at Pitane in Aeolis and flourished about 330~\textsc{b.c.} His
  two works on astronomy, containing 43 propositions, are said to be the oldest
  extant Greek mathematical treatises. They exist in manuscript at Oxford.
  They were edited, with a Latin translation, by \index{Hultsch, ref.\ to}F.~Hultsch, Leipzig, 1885.}
and to
some book on spherical geometry by an unknown writer. \index{Pappus!ref.\ to}Pappus
asserts that Euclid also composed a book on the elements of
music: this may refer to the \emph{Sectio Canonis}, which is by Euclid,
and deals with musical intervals.

\index{Puzzles|(}\index{Arithmetical!problems}To these works I may add the following little problem, which
occurs in the \index{Anthology, Palatine}\index{Palatine Anthology}Palatine Anthology and is attributed by tradition
to Euclid. ``A mule and a donkey were going to market laden
with wheat. The mule said, `If you gave me one measure I
should carry twice as much as you, but if I gave you one we
should bear equal burdens.' Tell me, learned geometrician,
what were their burdens.'' It is impossible to say whether the
question is due to Euclid, but there is nothing improbable in
the suggestion.\index{Puzzles|)}

It will be noticed that Euclid dealt only with magnitudes,
and did not concern himself with their numerical measures, but
it would seem from the works of \index{Aristarchus|(}Aristarchus and \index{Archimedes!ref.\ to}Archimedes
that this was not the case with all the Greek mathematicians
of that time. As one of the works of the former is extant it
will serve as another illustration of Greek mathematics of this
period.\index{Euclid|)}

\nblabel{png083a}\nbparagraph{Aristarchus.} \emph{Aristarchus} of Samos, born in 310~\textsc{b.c.}\ and
died in 250~\textsc{b.c.}, was an astronomer rather than a mathematician.
\index{Planetary motions}\index{Astronomy.!z12@Early Greek theories of}He asserted, at any rate as a working hypothesis, that the sun
was the centre of the universe, and that the earth revolved
round the sun. This view, in spite of the simple explanation
it afforded of various phenomena, was generally rejected by his
contemporaries. But his propositions\index{Nizze, ref.\ to}\index{Wallis!ref.\ to}\index{Urban, d', on Aristarchus}\index{Commandino!ref.\ to}\footnote
  {\mytextgreek{Per`i meg'ejwn ka`i >aposthm'atwn `Hl'iou ka`i Sel'hnhs}, edited by E.~Nizze,
  Stralsund, 1856. Latin translations were issued by F.~Commandino in 1572
  and by J.~Wallis in 1688; and a French translation was published by
  F.~d'Urban in 1810 and 1823.}
on the measurement of
\index{Distance of sun}\index{Sun, distance and radius of}\nblabel{png083b}the sizes and distances of the sun and moon were accurate in
principle, and his results were accepted by Archimedes in his
\mytextgreek{Yamm'iths}, mentioned below, as approximately correct. There
are 19 theorems, of which I select the seventh as a typical
illustration, because it shews the way in which the Greeks
evaded the difficulty of finding the numerical value of surds.

Aristarchus observed the angular distance between the moon
when dichotomized and the sun, and found it to be twenty-nine
thirtieths of a right angle. It is actually about $89° 21'$, but of
course his instruments were of the roughest description. He
then proceeded to shew that the distance of the sun is greater
than eighteen and less than twenty times the distance of the
moon in the following manner.

Let $S$ be the sun, $E$ the earth, and $M$ the moon. Then when
the moon is dichotomized, that is, when the bright part which
we see is exactly a half-circle, the angle between $MS$ and $ME$ is
a right angle. With $E$ as centre,
and radii $ES$ and $EM$ describe
circles, as in the \figureref{084a}{figure} below. %[**TN: "figure above" in original text]
Draw $EA$ perpendicular to $ES$.
Draw $EF$ bisecting the angle $AES$, and $EG$ bisecting the angle
$AEF$, as in the \figureref{084a}{figure}. Let $EM$ (produced) cut $AF$ in $H$.
The angle $AEM$ is by hypothesis $\frac{1}{30}$th of a right angle. Hence
we have
\begin{gather*}
\text{angle } AEG : \text{angle }AEH = \tfrac{1}{4}\text{ rt. }\angle : \tfrac{1}{30}\text{ rt. }\angle = 15 : 2,\\
\therefore AG : AH \text{ [}=\tan AEG : \tan AEH\text{] } > 15 : 2 \tag{$\alpha$}.
\end{gather*}
\begin{center}
\labelincludegraphics{084a}
\end{center}
Again $FG^{2} : AG^{2} = EF^{2} : EA^{2}$ (Euc.~\textsc{vi}, 3) $= 2 : 1$ (Euc.~\textsc{i}, 47),
\begin{align*}
\therefore FG^{2} : AG^{2} & > 49 : 25,\\
\therefore FG : AG & > 7 : 5, \\
\therefore AF : AG & > 12 : 5, \\
\therefore AE : AG & > 12 : 5. \tag{$\beta$}
\end{align*}
Compounding the ratios ($\alpha$) and ($\beta$), we have
\[
AE : AH > 18 : 1.
\]
But the triangles $EMS$ and $EAH$ are similar,
\[
\therefore ES : EM > 18 : 1.
\]

I will leave the second half of the proposition to amuse any
reader who may care to prove it: the analysis is straightforward.
In a somewhat similar way Aristarchus found the ratio of the
radii of the sun, earth, and moon.\index{Aristarchus|)}

\nblabel{png085a}We know very little of \index{Conon of Alexandria}\nbname{Conon} and \index{Dositheus}\nbname{Dositheus}, the immediate
successors of Euclid at Alexandria, or of their contemporaries
\index{Zeuxippus}\nbname{Zeuxippus} and \index{Nicoteles of Alexandria}\nbname{Nicoteles}, who most likely also lectured there,
except that \index{Archimedes|(}Archimedes, who was a student at Alexandria
probably shortly after Euclid's death, had a high opinion of
their ability and corresponded with the three first mentioned.
Their work and reputation has been completely overshadowed
by that of Archimedes.

\nblabel{png085b}\nbparagraph{Archimedes.\footnote
  {Besides Loria, book~ii, chap.~iii, \index{Cantor, M., ref.\ to}Cantor, chaps.~xiv,~xv, and Gow,
  pp.~221--244, see \emph{Quaestiones Archimedeae}, by J.~L.~Heiberg, Copenhagen,
  1879; and Marie, vol.~i, pp.~81--134. The best editions of the extant works
  of Archimedes are those by J.~L.~Heiberg, in 3 vols., Leipzig, 1880--81,
  and by Sir Thomas L.~Heath, Cambridge, 1897. In 1906 a manuscript,
  previously unknown, was discovered at Constantinople, containing propositions
  on hydrostatics and on methods; see \emph{Eine neue Schrift des Archimedes},
  by J.~L.~Heiberg and H.~G.~Zeuthen, Leipzig, 1907, and the \emph{Method of
  Archimedes}, by Sir Thomas L.~Heath, Cambridge, 1912.}}
\index{Zeuthen}\index{Heiberg, ref.\ to}\index{Heath, Sir T. L.}\index{Marie, ref.\ to}\emph{Archimedes}, who probably was related to
the royal family at Syracuse, was born there in 287 \textsc{b.c.}\ and
died in 212 \textsc{b.c}. He went to the university of Alexandria
and attended the lectures of Conon, but, as soon as he had
finished his studies, returned to Sicily where he passed the
remainder of his life. He took no part in public affairs, but
his mechanical ingenuity was astonishing, and, on any difficulties
which could be overcome by material means arising, his advice
was generally asked by the government.

Archimedes, like \index{Plato!ref.\ to}Plato, held that it was undesirable for a
philosopher to seek to apply the results of science to any practical
use; but in fact he did introduce a large number of new
inventions. The stories of the detection of the fraudulent
goldsmith and of the use of burning-glasses to destroy the
ships of the Roman blockading squadron will recur to most
readers. Perhaps it is not as well known that \index{Hiero of Syracuse}Hiero, who had
built a ship so large that he could not launch it off the slips,
applied to Archimedes. The difficulty was overcome by means
of an apparatus of cogwheels worked by an endless \index{Screw, the Archimedean}screw, but
we are not told exactly how the machine was used. \nblabel{xref:086}It is said
that it was on this occasion, in acknowledging the compliments
of \index{Hiero of Syracuse}Hiero, that Archimedes made the well-known remark that had
he but a fixed fulcrum he could move the earth.

Most mathematicians are aware that the \index{Archimedean!screw}Archimedean screw
was another of his inventions. It consists of a tube, open at
both ends, and bent into the form of a spiral like a corkscrew.
If one end be immersed in water, and the axis of the instrument
(\emph{i.e.} the axis of the cylinder on the surface of which the tube
lies) be inclined to the vertical at a sufficiently big angle, and
the instrument turned round it, the water will flow along the
tube and out at the other end. In order that it may work, the
inclination of the axis of the instrument to the vertical must
be greater than the pitch of the screw. It was used in Egypt
to drain the fields after an inundation of the Nile, and was
also frequently applied to take water out of the hold of a
ship.

The story that Archimedes set fire to the Roman ships by
means of burning-glasses and concave \index{Archimedean!mirrors}mirrors is not mentioned
till some centuries after his death, and is generally rejected.
The mirror of Archimedes is said to have been made in the
form of a hexagon surrounded by rings of polygons; and \index{Buffon on Archimedes}Buffon\footnote
  {See \emph{Mémoires de l'académie royale des sciences} for 1747, Paris, 1752,
  pp.~82--101.}
in 1747 contrived, by the use of a single composite mirror made
on this model, to set fire to wood at a distance of 150 feet,
and to melt lead at a distance of 140 feet. This was in April
and as far north as Paris, so in a Sicilian summer the use
of several such mirrors might be a serious annoyance to a
blockading fleet, if the ships were sufficiently near. It is
perhaps worth mentioning that a similar device is said to have
been used in the defence of Constantinople in 514~\textsc{a.d.}, and is
alluded to by writers who either were present at the siege or
obtained their information from those who were engaged in it.

But whatever be the truth as to this story, there is no doubt
that Archimedes devised the catapults which kept the Romans,
who were then besieging Syracuse, at bay for a considerable
time. These were constructed so that the range could be made
either short or long at pleasure, and so that they could be discharged
through a small loophole without exposing the artillery-men
to the fire of the enemy. So effective did they prove that
the siege was turned into a blockade, and three years elapsed
before the town was taken.

Archimedes was killed during the sack of the city which
followed its capture, in spite of the orders, given by the consul
\index{Marcellus}Marcellus who was in command of the Romans, that his house
and life should be spared. It is said that a soldier entered his
study while he was regarding a geometrical diagram drawn in
sand on the floor, which was the usual way of drawing figures
in classical times. Archimedes told him to get off the diagram,
and not spoil it. The soldier, feeling insulted at having orders
given to him and ignorant of who the old man was, killed him.
According to another and more probable account, the cupidity
of the troops was excited by seeing his instruments, constructed
of polished brass which they supposed to be made of gold.

The Romans erected a splendid tomb to Archimedes, on which
was engraved (in accordance with a wish he had expressed) the
figure of a \index{Sphere, surface and volume of}sphere inscribed in a cylinder, in commemoration of
the proof he had given that the volume of a sphere was equal
to two-thirds that of the circumscribing right cylinder, and its
surface to four times the area of a great circle. \index{Cicero, ref.\ to}Cicero\footnote
  {See his \emph{Tusculanarum Disputationum}, v.~23.}
gives
a charming account of his efforts (which were successful) to
rediscover the tomb in 75~\textsc{b.c.}

It is difficult to explain in a concise form the works or discoveries
of Archimedes, partly because he wrote on nearly all
the mathematical subjects then known, and partly because his
writings are contained in a series of disconnected monographs.
Thus, while \index{Euclid!ref.\ to}Euclid aimed at producing systematic treatises
which could be understood by all students who had attained
a certain level of education, Archimedes wrote a number of
brilliant essays addressed chiefly to the most educated mathematicians
of the day. The work for which he is perhaps now
best known is his treatment of the mechanics of solids and
fluids; but he and his contemporaries esteemed his geometrical
discoveries of the \index{Quadrature of curves!parabolas|(}\index{Parabola,!quadrature of|(}\index{Quadrature of curves!sphere}quadrature of a parabolic area and of a
spherical surface, and his rule for finding the volume of a sphere
as more remarkable; while at a somewhat later time his numerous
mechanical inventions excited most attention.

\nblabel{png088a}(i)~On \emph{plane geometry} the extant works of Archimedes are
three in number, namely, (a) the \emph{Measure of the Circle}, (b) the
\emph{Quadrature of the Parabola}, and (c) one on \emph{Spirals}.

(\emph{a}) The \emph{Measure of the Circle} contains three propositions.
In the first proposition Archimedes proves that the area is the
same as that of a right-angled triangle whose sides are equal
respectively to the radius $a$ and the circumference of the circle,
\emph{i.e.} the area is equal to $\frac{1}{2}a (2\pi a)$. In the second proposition
he shows that $\pi a^{2} : (2a)^{2} = 11 : 14$ very nearly; and next, in
the third proposition, that \index{P1@$\pi$,!z1@value of}$\pi$ is less than $3\frac{1}{7}$ and greater than
$3\frac{10}{71}$. These theorems are of course proved geometrically. To
demonstrate the two latter propositions, he inscribes in and
circumscribes about a circle regular polygons of ninety-six
sides, calculates their perimeters, and then assumes the circumference
of the circle to lie between them: this leads to the
result $6336 / 2017\frac{1}{4} < \pi < 14688 / 4673\frac{1}{2}$, from which he deduces
the limits given above. It would seem from the proof that he
had some (at present unknown) method of extracting the square
roots of numbers approximately. The table which he formed
of the numerical values of the chords of a circle is essentially a
\index{Sines, table of}table of natural sines, and may have suggested the subsequent
work on these lines of \index{Hipparchus!ref.\ to}Hipparchus and \index{Ptolemy!ref.\ to}Ptolemy.

(\emph{b}) The \emph{Quadrature of the Parabola} contains twenty-four
propositions. Archimedes begins this work, which was sent to
\index{Dositheus}Dositheus, by establishing some properties of conics [props.~1--5].
He then states correctly the area cut off from a parabola by any
chord, and gives a proof which rests on a preliminary mechanical
experiment of the ratio of areas which balance when suspended
from the arms of a lever [props.~6--17]; and, lastly, he gives a
geometrical demonstration of this result [props.~18--24]. The
latter is, of course, based on the method of exhaustions, but for
brevity I will, in quoting it, use the method of limits.\index{Archimedes|)}

\begin{center}
\labelincludegraphics{089a}
\end{center}

Let the area of the parabola (see \figureref{089a}{figure} above) be bounded %[**TN: Replaced 'below' with 'above']
by the chord $PQ$. Draw $VM$ the diameter to the chord
$PQ$, then (by a previous proposition), $V$ is more remote from
$PQ$ than any other point in the arc $PVQ$. Let the area of
the triangle $PVQ$ be denoted by $\triangle$. In the segments bounded
by $VP$ and $VQ$ inscribe triangles in the same way as the triangle
$PVQ$ was inscribed in the given segment. Each of these triangles
is (by a previous proposition of his) equal to $\frac{1}{8}\triangle$, and
their sum is therefore $\frac{1}{4}\triangle$. Similarly in the four segments left
inscribe triangles; their sum will be $\frac{1}{16}\triangle$. Proceeding
in this
way the area of the given segment is shown to be equal to the
limit of
\[
\triangle + \frac{\triangle}{4} + \frac{\triangle}{16} +\cdots + \frac{\triangle}{4^{n}} + \cdots,
\]
when $n$ is indefinitely large.

\index{Geometrical progressions}\index{Progressions,!geometrical}The problem is therefore reduced to finding the sum of a
geometrical series. This he effects as follows. Let $A, B, C,
\ldots, J, K$ be a series of magnitudes such that each is one-fourth
of that which precedes it. Take magnitudes $b, c, \ldots, k$ equal
respectively to $\frac{1}{3}B, \frac{1}{3}C, \ldots, \frac{1}{3}K$. Then
\[
B + b = \tfrac{1}{3}A, \quad C + c = \tfrac{1}{3}B, \quad \ldots, \quad K + k = \tfrac{1}{3}J.
\]
Hence $(B + C + \ldots + K) + (b + c + \ldots + k) = \frac{1}{3}(A + B + \ldots + J);$
but, by hypothesis, $(b + c + \ldots + j + k) = \frac{1}{3}(B + C + \ldots +J) + \frac{1}{3}K;$
\[
\therefore (B + C + \ldots + K) + \tfrac{1}{3}K = \tfrac{1}{3}A.
\]
\[
\therefore A + B + C + \ldots + K = \tfrac{4}{3}A - \tfrac{1}{3}K.
\]
Hence the sum of these magnitudes exceeds four times the third
of the largest of them by one-third of the smallest of them.

Returning now to the problem of the quadrature of the
parabola $A$ stands for $\Delta$, and ultimately $K$ is indefinitely
small; therefore the area of the parabolic segment is four-thirds
that of the triangle $PVQ$, or two-thirds that of a rectangle
whose base is $PQ$ and altitude the distance of $V$ from $PQ$.\index{Parabola,!quadrature of|)}\index{Quadrature of curves!parabolas|)}

\index{Quadrature of curves!ellipse}While discussing the question of quadratures it may be
added that in the fifth and sixth propositions of his work on
\index{Conicoids}conoids and \index{Spheroids, Archimedes on}spheroids he determined the \index{Ellipse, area of}area of an ellipse.

(\emph{c}) The work on \index{Spiral of Archimedes}\emph{Spirals} contains twenty-eight propositions
on the properties of the curve now known as the spiral of
Archimedes. It was sent to \index{Dositheus}Dositheus at Alexandria accompanied
by a letter, from which it appears that Archimedes had
previously sent a note of his results to \index{Conon of Alexandria}Conon, who had died
before he had been able to prove them. The spiral is defined by
saying that the vectorial angle and radius vector both increase
uniformly, hence its equation is $r = c\theta$. Archimedes finds most
of its properties, and determines the area inclosed between the
curve and two radii vectores. This he does (in effect) by
saying, in the language of the infinitesimal calculus, that an
element of area is $> \frac{1}{2}r^2 d\theta$ and
$< \frac{1}{2}(r + dr)^2 d\theta$: to effect the
sum of the elementary areas he gives two lemmas in which he
sums (geometrically) the series $a^{2} + (2a)^{2} + (3a)^{2} + \ldots + (na)^{2}$
[prop.~10], and $a + 2a + 3a + \ldots + na$ [prop.~11].

(\emph{d}) In addition to these he wrote a small treatise on
\emph{geometrical methods}, and works on \emph{parallel lines}, \emph{triangles}, \emph{the
properties of right-angled triangles}, \emph{data}, \emph{the heptagon inscribed
in a circle}, and \emph{systems of circles touching one another}; possibly
he wrote others too. These are all lost, but it is probable that
fragments of four of the propositions in the last-mentioned work
are preserved in a Latin translation from an Arabic manuscript
entitled \emph{Lemmas of Archimedes}.

\nblabel{png091a}(ii)~On \emph{geometry of three dimensions} the extant works of
Archimedes are two in number, namely (\emph{a}), the \emph{Sphere and
Cylinder}, and (\emph{b}) \index{Spheroids, Archimedes on}\index{Conicoids}\emph{Conoids and Spheroids}.

(\emph{a}) The \emph{Sphere and Cylinder} contains sixty propositions
arranged in two books. Archimedes sent this like so many
of his works to Dositheus at Alexandria; but he seems to
have played a practical joke on his friends there, for he purposely
misstated some of his results ``to deceive those vain
geometricians who say they have found everything, but never
give their proofs, and sometimes claim that they have discovered
what is impossible.'' He regarded this work as his
masterpiece. It is too long for me to give an analysis of its
contents, but I remark in passing that in it he finds expressions
for the surface and volume of a \index{Pyramid,!surface of}\index{Pyramid,!volume of}pyramid, of a \index{Cone, sections of!surface of}\index{Cone, sections of!volume of}\index{Quadrature of circle!cone}cone, and of
a sphere, as well as of the figures produced by the revolution
of polygons inscribed in a circle about a diameter of the circle.
There are several other propositions on areas and volumes of
which perhaps the most striking is the tenth proposition of
the second book, namely, that ``of all spherical segments whose
surfaces are equal the hemisphere has the greatest volume.''
In the second proposition of the second book he enunciates the
remarkable theorem that a line of length $a$ can be divided
so that $a - x : b = 4a^{2} : 9x^{2}$ (where $b$ is a given length), only
if $b$ be less than $\frac{1}{3}a$; that is to say, the \index{Cubic equations}cubic equation
$x^{3} - ax^{2} + \frac{4}{9}a^{2}b = 0$ can have a real and positive root only if
$a$ be greater than $3b$. This proposition was required to complete
his solution of the problem to divide a given sphere by
a plane so that the volumes of the segments should be in a given
ratio. One very simple cubic equation occurs in the \emph{Arithmetic}
of \index{Diophantus!ref.\ to}Diophantus, but with that exception no such equation appears
again in the history of European mathematics for more than a
thousand years.

(\emph{b}) The \index{Conicoids}\emph{Conoids and Spheroids} contains forty propositions
on \index{Quadrature of curves!sphere}\index{Diophantus!ref.\ to}quadrics of revolution (sent to \index{Dositheus}Dositheus in Alexandria)
mostly concerned with an investigation of their volumes.

(\emph{c}) Archimedes also wrote a treatise on certain \index{Polyhedrons, regular!semi-regular}\emph{semi-regular
polyhedrons}, that is, solids contained by regular but dissimilar
polygons. This is lost, but references to it are given by
Pappus.

\nblabel{png092a}(iii)~On \emph{arithmetic} Archimedes wrote two papers. One
(addressed to Zeuxippus) was on the principles of numeration;
this is now lost. The other (addressed to \index{Gelon of Syracuse}Gelon) was called
\mytextgreek{Yamm'iths} (\emph{the sand-reckoner}), and in this he meets an objection
which had been urged against his first paper.

\index{Numeration, systems of|(}\index{Decimal numeration|(}The object of the first paper had been to suggest a convenient
system by which numbers of any magnitude could
be represented; and it would seem that some philosophers at
Syracuse had doubted whether the system was practicable.
Archimedes says people talk of the sand on the Sicilian shore
as something beyond the power of calculation, but he can
estimate it; and, further, he will illustrate the power of his
method by finding a superior limit to the number of grains of
sand which would fill the whole universe, \emph{i.e.} a sphere whose
centre is the earth, and radius the distance of the sun. He
begins by saying that in ordinary Greek nomenclature it was
only possible to express numbers from 1 up to $10^8$: these are
expressed in what he says he may call units of the first order.
If $10^8$ be termed a unit of the second order, any number from
$10^8$ to $10^{16}$ can be expressed as so many units of the second
order plus so many units of the first order. If $10^{16}$ be a unit
of the third order any number up to $10^{24}$ can be then expressed,
and so on. Assuming that 10,000 grains of sand occupy a
sphere whose radius is not less than $\frac{1}{80}$th of a finger-breadth,
and that the diameter of the universe is not greater than $10^{10}$
stadia, he finds that the number of grains of sand required to
fill the solar universe is less than $10^{51}$.

Probably this system of numeration was suggested merely
as a scientific curiosity. The Greek system of numeration
with which we are acquainted had been only recently introduced,
most likely at Alexandria, and was sufficient for all the
purposes for which the Greeks then required numbers; and
Archimedes used that system in all his papers. On the other
hand, it has been conjectured that Archimedes and Apollonius
had some symbolism based on the decimal system for their
own investigations, and it is possible that it was the one here
sketched out. The units suggested by Archimedes form a
\index{Geometrical progressions}\index{Progressions,!geometrical}geometrical progression, having $10^{8}$ for the radix. He incidentally
adds that it will be convenient to remember that the
product of the $m$th and $n$th terms of a geometrical progression,
whose first term is unity, is equal to the $(m + n)$th term of the
series, that is, that $r^{m} × r^{n} = r^{m+n}$.\index{Decimal numeration|)}\index{Numeration, systems of|)}

To these two arithmetical papers I may add the following
celebrated \index{Arithmetical!problems}problem\index{Amthor, A.}\index{Krumbiegel, B.}\footnote
  {See a memoir by B.~Krumbiegel and A.~Amthor, \emph{Zeitschrift für Mathematik,
  Abhandlungen zur Geschichte der Mathematik}, Leipzig, vol.~xxv, 1880,
  pp.~121--136, 153--171.}
which he sent to the Alexandrian mathematicians.
\index{Bull problem, the|(}The sun had a herd of bulls and cows, all of
which were either white, grey, dun, or piebald: the number
of piebald bulls was less than the number of white bulls by
$5/6$ths of the number of grey bulls, it was less than the
number of grey bulls by $9/20$ths of the number of dun bulls,
and it was less than the number of dun bulls by $13/42$nds
of the number of white bulls; the number of white cows was
$7/12$ths of the number of grey cattle (bulls and cows), the
number of grey cows was $9/20$ths of the number of dun
cattle, the number of dun cows was $11/30$ths of the number of
piebald cattle, and the number of piebald cows was $13/42$nds
of the number of white cattle. The problem was to find the
composition of the herd. The problem is indeterminate, but
the solution in lowest integers is
\begin{center}
\begin{tabular}{p{0.25\textwidth}@{\;}r@{\qquad}p{0.25\textwidth}@{\;}r}
white bulls,\dotfill  &10,366,482; &white cows,\dotfill  &7,206,360; \\
grey bulls,\dotfill   & 7,460,514; &grey cows,\dotfill   &4,893,246; \\
dun bulls,\dotfill    & 7,358,060; &dun cows,\dotfill    &3,515,820; \\
piebald bulls,\dotfill& 4,149,387; &piebald cows,\dotfill&5,439,213.
\end{tabular}
\end{center}
In the classical solution, attributed to Archimedes, these numbers
are multiplied by 80.

Nesselmann believes, from internal evidence, that the problem
has been falsely attributed to Archimedes. It certainly
is unlike his extant work, but it was attributed to him among
the ancients, and is generally thought to be genuine, though
possibly it has come down to us in a modified form. It is
in verse, and a later copyist has added the additional conditions
that the sum of the white and grey bulls shall be a
square number, and the sum of the piebald and dun bulls a
triangular number.\index{Bull problem, the|)}

\index{Fractions, continued!treatment of}It is perhaps worthy of note that in the enunciation the
fractions are represented as a sum of fractions whose numerators
are unity: thus Archimedes wrote $1/7 + 1/6$ instead of
$13/42$, in the same way as \index{Ahmes!ref.\ to}Ahmes would have done.

\nblabel{png094a}(iv)~On \emph{mechanics} the extant works of Archimedes are
two in number, namely, (\emph{a}) his \emph{Mechanics}, and (\emph{c}) his
\emph{Hydrostatics}.

(\emph{a}) The \index{Mechanics.!z1@Discussed by!x2@Archimedes}\emph{Mechanics} is a work on statics with special reference
to the equilibrium of plane laminas and to properties of
their centres of gravity; it consists of twenty-five propositions
in two books. In the first part of book~\textsc{i}, most of the elementary
properties of the \index{Centres of mass}\index{Gravity,!centres of}\index{Mass, centres of}centre of gravity are proved [props.~1--8]; and in
the remainder of book~\textsc{i}, [props.~9--15] and in
book~\textsc{ii} the centres of gravity of a variety of plane areas, such
as parallelograms, triangles, trapeziums, and parabolic areas
are determined.

As an illustration of the influence of Archimedes on the
history of mathematics, I may mention that the science of
statics rested on his theory of the lever until 1586, when
\index{Stevinus!ref.\ to}Stevinus published his treatise on statics.

His reasoning is sufficiently illustrated by an outline of his
proof for the case of two weights, $P$ and $Q$, placed at their centres
of gravity, $A$ and $B$, on a weightless bar $AB$. He wants to shew
that the \index{Centres of mass}\index{Gravity,!centres of}\index{Mass, centres of}centre of gravity of $P$ and $Q$ is at a point $O$ on the bar
such that $P.OA = Q.OB$.

\begin{center}
\labelincludegraphics{095a}
\end{center}

On the line $AB$ (produced if necessary) take points $H$ and $K$,
so that $HB = BK = AO$; and a point $L$ so that $LA = OB$. It
follows that $LH$ will be bisected at $A$, $HK$ at $B$, and $LK$ at $O$;
also $LH : HK = AH : HB = OB : AO = P : Q$. Hence, by a
previous proposition, we may consider that the effect of $P$ is the
same as that of a heavy uniform bar $LH$ of weight $P$, and the
effect of $Q$ is the same as that of a similar heavy uniform bar
$HK$ of weight $Q$. Hence the effect of the weights is the same
as that of a heavy uniform bar $LK$. But the centre of gravity
of such a bar is at its middle point $O$.

(\emph{b}) Archimedes also wrote a treatise on \index{Lever, principle of}\emph{levers} and perhaps,
on all the mechanical machines. The book is lost, but we
know from \index{Pappus!ref.\ to}Pappus that it contained a discussion of how a
given weight could be moved with a given power. It was in
this work probably that Archimedes discussed the \index{Pulley, theory of}theory of
a certain compound pulley consisting of three or more simple
pulleys which he had invented, and which was used in some
public works in Syracuse. It is well known\footnote
  {See above, p.~\pageref{xref:086}.}
that he boasted
that, if he had but a fixed fulcrum, he could move the whole
earth; and a commentator of later date says that he added
he would do it by using a compound pulley.

(\emph{c}) \index{Hydrostatics. Developed by!z1@Archimedes|(}His work on \emph{floating bodies} contains nineteen propositions
in two books, and was the first attempt to apply mathematical
reasoning to hydrostatics. The story of the manner in which
his attention was directed to the subject is told by \index{Vitruvius, ref.\ to}Vitruvius.
\index{Hiero of Syracuse}Hiero, the king of Syracuse, had given some gold to a goldsmith
to make into a crown. The crown was delivered, made up, and
of the proper weight, but it was suspected that the workman
had appropriated some of the gold, replacing it by an equal
weight of silver. Archimedes was thereupon consulted. Shortly
afterwards, when in the public baths, he noticed that his body
was pressed upwards by a force which increased the more completely
he was immersed in the water. Recognising the value
of the observation, he rushed out, just as he was, and ran home
through the streets, shouting \mytextgreek{e<'urhka}, \mytextgreek{e<'urhka}, ``I have found it, I
have found it.'' There (to follow a later account) on making
accurate experiments he found that when equal weights of gold
and silver were weighed in water they no longer appeared equal:
each seemed lighter than before by the weight of the water it
displaced, and as the silver was more bulky than the gold its
weight was more diminished. Hence, if on a balance he weighed
the crown against an equal weight of gold and then immersed
the whole in water, the gold would outweigh the crown if any
silver had been used in its construction. Tradition says that
the goldsmith was found to be fraudulent.

Archimedes began the work by proving that the surface of
a fluid at rest is spherical, the centre of the sphere being at the
centre of the earth. He then proved that the pressure of the
fluid on a body, wholly or partially immersed, is equal to the
weight of the fluid displaced; and thence found the position
of equilibrium of a floating body, which he illustrated by
spherical segments and paraboloids of revolution floating on a
fluid. Some of the latter problems involve geometrical reasoning
of considerable complexity.\index{Hydrostatics. Developed by!z1@Archimedes|)}

The following is a fair specimen of the questions considered.
A solid in the shape of a paraboloid of revolution of height $h$
and latus rectum $4a$ floats in water, with its vertex immersed
and its base wholly above the surface. If equilibrium be
possible when the axis is not vertical, then the density of the
body must be less than $(h-3a)^2/h^3$ [book~\textsc{ii}, prop.~4]. When
it is recollected that Archimedes was unacquainted with trigonometry
or analytical geometry, the fact that he could discover
and prove a proposition such as that just quoted will serve as an
illustration of his powers of analysis.

It will be noticed that the mechanical investigations of
Archimedes were concerned with statics. It may be added that
though the Greeks attacked a few problems in dynamics, they
did it with but indifferent success: some of their remarks were
acute, but they did not sufficiently realise that the fundamental
facts on which the theory must be based can be established only
by carefully devised observations and \index{Experiments, necessity of}experiments. It was not
until the time of \index{Galileo!ref.\ to}Galileo and \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton that this was done.

\nblabel{png097a}(v)~\index{Astronomy.!z12@Early Greek theories of}We know, both from occasional references in his works
and from remarks by other writers, that Archimedes was largely
occupied in \emph{astronomical observations}. He wrote a book, \mytextgreek{Per`i
Sfeiropoi'ias}, on the construction of a celestial sphere, which is
lost; and he constructed a sphere of the stars, and an \index{Orrery}orrery.
These, after the capture of Syracuse, were taken by \index{Marcellus}Marcellus to
Rome, and were preserved as curiosities for at least two or three
hundred years.

\begin{wrapfigure}[8]{l}{75mm}
\begin{center}
\labelincludegraphics{097a}
\end{center}
\end{wrapfigure}

This mere catalogue of his works will show how wonderful
were his achievements; but no one who has not actually read
some of his writings can form a just appreciation of his extraordinary
ability. This will be still further increased if we
\nblabel{png097b}recollect that the only principles used by Archimedes, in
addition to those contained in \index{Euclid!ref.\ to}Euclid's \emph{Elements} and \emph{Conic
sections}, are that
of all lines like
$ACB$, $ADB$, $\ldots$ connecting two
points $A$ and $B$, the straight line
is the shortest, and of the curved
lines, the inner one $ADB$ is
shorter than the outer one $ACB$; together with two similar
statements for space of three dimensions.

In the old and medieval world Archimedes was reckoned
as the first of mathematicians, but possibly the best tribute to
his fame is the fact that those writers who have spoken most
highly of his work and ability are those who have been themselves
the most distinguished men of their own generation.

\nblabel{png098a}\index{Apollonius|(}\nbparagraph{Apollonius.\footnote
  {In addition to Zeuthen's work and the other authorities mentioned in
  the footnote on p.~\pageref{png071b}, see \emph{Litterargeschichtliche Studien über Euklid}, by
  J.~L.~Heiberg, Leipzig, 1882. Editions of the extant works of Apollonius
  were issued by J.~L.~Heiberg in two volumes, Leipzig, 1890, 1893; and by
  E.~Halley, Oxford, 1706 and 1710: an edition of the conics was published by
  T.~L.~Heath, Cambridge, 1896.}}
\index{Zeuthen}\index{Halley!ref.\ to}\index{Heiberg, ref.\ to}The third great mathematician of this century
was \index{Conic Sections (Geometrical).!Discussed by most of the Greek geometricians after Menaechmus!x2@and Apollonius|(}\emph{Apollonius of Perga}, who is chiefly celebrated for having
\nblabel{png098b}produced a systematic treatise on the conic sections which not
only included all that was previously known about them, but
immensely extended the knowledge of these curves. This work
was accepted at once as the standard text-book on the subject,
and completely superseded the previous treatises of \index{Menaechmus!ref.\ to}Menaechmus,
\index{Aristaeus!ref.\ to}Aristaeus, and \index{Euclid!ref.\ to}Euclid which until that time had been in
general use.

We know very little of Apollonius himself. He was born
about 260~\textsc{b.c.}, and died about 200~\textsc{b.c.} He studied in Alexandria
for many years, and probably lectured there; he is
represented by \index{Pappus!ref.\ to}Pappus as ``vain, jealous of the reputation of
others, and ready to seize every opportunity to depreciate them.''
It is curious that while we know next to nothing of his life, or
of that of his contemporary Eratosthenes, yet their nicknames,
which were respectively \emph{epsilon} and \emph{beta}, have come down to us.
\index{Gow, ref.\ to}Dr.~Gow has ingeniously suggested that the lecture rooms at
Alexandria were numbered, and that they always used the rooms
numbered 5 and 2 respectively.

Apollonius spent some years at Pergamum in Pamphylia,
where a university had been recently established and endowed
in imitation of that at Alexandria. There he met \index{Eudemus}Eudemus and
\index{Attalus}Attalus, to whom he subsequently sent each book of his conics
as it came out with an explanatory note. He returned to
Alexandria, and lived there till his death, which was nearly
contemporaneous with that of Archimedes.

In his great work on \emph{conic sections} Apollonius so thoroughly
investigated the properties of these curves that he left but little
for his successors to add. But his proofs are long and involved,
and I think most readers will be content to accept a short
analysis of his work, and the assurance that his demonstrations
are valid. \index{Zeuthen}Dr.~Zeuthen believes that many of the properties
enunciated were obtained in the first instance by the use of
co-ordinate geometry, and that the demonstrations were translated
subsequently into geometrical form. If this be so, we
must suppose that the classical writers were familiar with some
branches of analytical geometry---Dr.~Zeuthen says the use of
orthogonal and oblique co-ordinates, and of transformations
depending on abridged notation---that this knowledge was
confined to a limited school, and was finally lost. This is a
mere conjecture and is unsupported by any direct evidence, but
it has been accepted by some writers as affording an explanation
of the extent and arrangement of the work.

The treatise contained about four hundred propositions, and
was divided into eight books; we have the Greek text of the
first four of these, and we also possess copies of the commentaries
by \index{Pappus!ref.\ to}Pappus and \index{Eutocius!ref.\ to}Eutocius on the whole work. In the ninth
century an Arabic translation was made of the first seven books,
which were the only ones then extant; we have two manuscripts
of this version. The eighth book is lost.

In the letter to \index{Eudemus}Eudemus which accompanied the first book
Apollonius says that he undertook the work at the request of
\index{Naucrates}Naucrates, a geometrician who had been staying with him at
Alexandria, and, though he had given some of his friends a
rough draft of it, he had preferred to revise it carefully before
sending it to Pergamum. In the note which accompanied the
next book, he asks Eudemus to read it and communicate it to
others who can understand it, and in particular to \index{Philonides}Philonides,
a certain geometrician whom the author had met at Ephesus.

The first four books deal with the elements of the subject,
and of these the first three are founded on Euclid's previous
work (which was itself based on the earlier treatises by
\index{Menaechmus!ref.\ to}Menaechmus and \index{Aristaeus!ref.\ to}Aristaeus). \index{Heracleides}Heracleides asserts that much
of the matter in these books was stolen from an unpublished
work of Archimedes, but a critical examination by \index{Heiberg, ref.\ to}Heiberg
has shown that this is improbable.

\begin{center}
\labelincludegraphics{100a}
\end{center}

Apollonius begins by defining a cone on a circular base.
He then investigates the different plane sections of it, and
shows that they are divisible into three kinds of curves which
he calls ellipses, parabolas, and hyperbolas. He proves the
proposition that, if $A$, $A'$ be the vertices of a conic, and if $P$ be
any point on it, and $PM$ the perpendicular drawn from $P$ on
$AA'$, then (in the usual notation) the ratio $MP^2: AM \ctrdot MA'$ is
constant in an ellipse or hyperbola, and the ratio $MP^2: AM$
is constant in a parabola. These are the characteristic properties
on which almost all the rest of the work is based. He next
shows that, if $A$ be the vertex, $l$ the latus rectum, and if $AM$
and $MP$ be the abscissa and ordinate of any point on a conic
(see above \figureref{100a}{figure}), then $MP^2$ is less than, equal to, or greater
than $l \ctrdot AM$ according as the conic is an ellipse, parabola, or
hyperbola; hence the names which he gave to the curves and
by which they are still known.

He had no idea of the \index{Directrix in conics}directrix, and was not aware that
the parabola had a \index{Focus of a conic}focus, but, with the exception of the propositions
which involve these, his first three books contain most
of the propositions which are found in modern text-books.
In the fourth book he develops the theory of lines cut
harmonically, and treats of the points of intersection of
systems of conics. In the fifth book he commences with the
theory of maxima and minima; applies it to find the centre of
curvature at any point of a conic, and the evolute of the curve;
and discusses the number of normals which can be drawn from
a point to a conic. In the sixth book he treats of similar
conics. The seventh and eighth books were given up to a
discussion of conjugate diameters; the latter of these was
conjecturally restored by \index{Halley!ref.\ to}E.~Halley in his edition of 1710.

The verbose explanations make the book repulsive to most
modern readers; but the arrangement and reasoning are
unexceptional, and it has been not unfitly described as the
crown of Greek geometry. It is the work on which the
reputation of Apollonius rests, and it earned for him the name
of ``the great geometrician.''\index{Conic Sections (Geometrical).!Discussed by most of the Greek geometricians after Menaechmus!x2@and Apollonius|)}

\nblabel{png101a}Besides this immense treatise he wrote numerous shorter
works; of course the books were written in Greek, but they
are usually referred to by their Latin titles: those about which
we now know anything are enumerated below. He was the
author of a work on the problem ``given two co-planar straight
lines $Aa$ and $Bb$, drawn through fixed points $A$ and $B$; to draw
a line $Oab$ from a given point $O$ outside them cutting them in
$a$ and $b$, so that $Aa$ shall be to $Bb$ in a given ratio.'' He reduced
the question to seventy-seven separate cases and gave an
appropriate solution, with the aid of conics, for each case; this
was published by E.~Halley (translated from an Arabic copy) in
1706. He also wrote a treatise \emph{De Sectione Spatii} (restored by
E.~Halley in 1706) on the same problem under the condition
that the rectangle $Aa \ctrdot Bb$ was given. He wrote another entitled
\emph{De Sectione Determinata} (restored by \index{Simson, Robert!ref.\ to}R.~Simson in 1749),
dealing with problems such as to find a point $P$ in a given
straight line $AB$, so that $PA^2$ shall be to $PB$ in a given ratio.
He wrote another \emph{De Tactionibus} (restored by \index{Vieta!ref.\ to}Vieta in 1600)
on the construction of a circle which shall touch three given
circles. Another work was his \emph{De Inclinationibus} (restored by
\index{Ghetaldi on Apollonius}M.~Ghetaldi in 1607) on the problem to draw a line so that the
intercept between two given lines, or the circumferences of two
given circles, shall be of a given length. He was also the
author of a treatise in three books on plane loci, \emph{De Locis
Planis} (restored by \index{Fermat!ref.\ to}Fermat in 1637, and by \index{Simson, Robert!ref.\ to}R.~Simson in
1746), and of another on the \emph{regular solids}. And, lastly, he
wrote a treatise on \emph{unclassed incommensurables}, being a commentary
on the tenth book of \index{Eucn@Euc.!g00@\textsc{x}. ref.\ to}Euclid. It is believed that in
one or more of the lost books he used the method of conical
projections.

Besides these geometrical works he wrote on the \emph{methods of
arithmetical calculation}. All that we know of this is derived
from some remarks of \index{Pappus!ref.\ to}Pappus. \index{Friedlein, G., ref.\ to}Friedlein thinks that it was
merely a sort of ready-reckoner. It seems, however, more
probable that Apollonius here suggested a \index{Numeration, systems of}system of numeration
similar to that proposed by \index{Archimedes!ref.\ to}Archimedes, but proceeding
by tetrads instead of octads, and described a notation for it.
It will be noticed that our modern notation goes by hexads,
a million $= 10^{6}$, a billion $= 10^{12}$, a trillion $= 10^{18}$, etc. It is not
impossible that Apollonius also pointed out that a \index{Decimal numeration}decimal
system of notation, involving only nine symbols, would facilitate
numerical multiplications.

Apollonius was interested in astronomy, and wrote a book
on the \index{Planetary motions}\emph{stations and regressions of the planets} of which \index{Ptolemy!ref.\ to}Ptolemy
made some use in writing the \index{Almagest@\emph{Almagest}, the!ref.\ to}\emph{Almagest}. He also wrote a
treatise on the use and theory of the screw in statics.

This is a long list, but I should suppose that most of these
works were short tracts on special points.

Like so many of his predecessors, he too gave a construction
for finding two mean proportionals between two given lines, and
\nblabel{png102a}thereby \index{Cube, duplication of}duplicating the cube. It was as follows. Let $OA$ and
$OB$ be the given lines. Construct a rectangle $OADB$, of
which they are adjacent sides. Bisect $AB$ in $C$. Then, if
with $C$ as centre we can describe a circle cutting $OA$ produced
in $a$, and cutting $OB$ produced in $b$, so that $aDb$ shall be a
straight line, the problem is effected. For it is easily shewn
that
{\setlength{\abovedisplayskip}{2pt plus 2pt minus 1pt} % reduce vertical space before equations
\begin{DPgather*}
                        Oa \ctrdot Aa + CA^{2} = Ca^{2}.\\
\lintertext{Similarly}  Ob \ctrdot Bb + CB^2 = Cb^2.\\
\lintertext{Hence}      Oa \ctrdot Aa = Ob \ctrdot Bb.\\
\lintertext{That is,}   Oa : Ob = Bb : Aa.
\end{DPgather*}
} % end of vertical space adjustment
\begin{center}
\labelincludegraphics{103a}
\end{center}
But, by similar triangles,
\begin{DPgather*}
                        BD : Bb = Oa : Ob = Aa : AD.\\
\lintertext{Therefore } Oa : Bb = Bb : Aa = Aa : OB,
\end{DPgather*}
that is, $Bb$ and $Oa$ are the two mean proportionals between
$OA$ and $OB$. It is impossible to construct the circle whose
centre is $C$ by Euclidean geometry, but Apollonius gave a
mechanical way of describing it. This construction is quoted
by several Arabic writers.

\nblabel{png103a}In one of the most brilliant passages of his \emph{Aperçu historique}
\index{Chasles, M., ref.\ to}Chasles remarks that, while \index{Archimedes!ref.\ to}Archimedes and Apollonius were
the most able geometricians of the old world, their works are
distinguished by a contrast which runs through the whole subsequent
history of geometry. Archimedes, in attacking the
problem of the quadrature of curvilinear areas, established the
principles of the geometry which rests on measurements; this
naturally gave rise to the infinitesimal calculus, and in fact the
\index{Exhaustions, method of}method of exhaustions as used by Archimedes does not differ
in principle from the method of limits as used by \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton.
Apollonius, on the other hand, in investigating the properties of
conic sections by means of transversals involving the ratio of
rectilineal distances and of perspective, laid the foundations of
the geometry of form and position.\index{Apollonius|)}

\nblabel{png104a}\index{Eratosthenes}\nbparagraph{Eratosthenes.\footnote
  {The works of Eratosthenes exist only in fragments. A collection of these
  was published by G.~Bernhardy at Berlin in 1822: some additional fragments
  were printed by E.~Hillier, Leipzig, 1872.}}
\index{Hillier on Eratosthenes}\index{Bernhardy on Eratosthenes}Among the contemporaries of Archimedes
and Apollonius I may mention \emph{Eratosthenes}. Born at Cyrene
in 275~\textsc{b.c.}, he was educated at Alexandria---perhaps at the
same time as Archimedes, of whom he was a personal friend---and
Athens, and was at an early age entrusted with the care of
the \index{Alexandrian,!library}university library at Alexandria, a post which probably he
occupied till his death. He was the Admirable Crichton of his
age, and distinguished for his athletic, literary, and scientific
attainments: he was also something of a poet. He lost his
sight by ophthalmia, then as now a curse of the valley of the
Nile, and, refusing to live when he was no longer able to read,
he committed suicide in 194~\textsc{b.c.}

\index{Astronomy.!z12@Early Greek theories of}In science he was chiefly interested in astronomy and geodesy,
and he constructed various astronomical instruments which were
used for some centuries at the university. He suggested the
\index{Julian calendar}\index{Calendars}calendar (now known as Julian), in which every \index{Year, duration of}fourth year
contains 366 days; and he determined the obliquity of the
\index{Ecliptic, obliquity of}ecliptic as $23° 51' 20''$. He measured the \index{Degree, length of}length of a degree on
the earth's surface, making it to be about 79 miles, which is too
long by nearly 10 miles, and thence \index{Earth,!dimensions of}calculated the circumference
of the earth to be 252,000 stadia. If we take the Olympic
stadium of $202\frac{1}{4}$ yards, this is equivalent to saying that the
radius is about 4600 miles, but there was also an Egyptian
stadium, and if he used this he estimated the radius as 3925
miles, which is very near the truth. The principle used in the
determination is correct.

Of Eratosthenes's work in mathematics we have two extant
illustrations: one in a description of an instrument to \index{Cube, duplication of|(}duplicate
a cube, and the other in a rule he gave for constructing a table
of prime numbers. The former is given in many books. The
\nblabel{png104b}latter, called the ``sieve of Eratosthenes,'' was as follows: write
down all the numbers from 1 upwards; then every second
number from 2 is a multiple of 2 and may be cancelled; every
third number from 3 is a multiple of 3 and may be cancelled;
every fifth number from 5 is a multiple of 5 and may be
cancelled; and so on. It has been estimated that it would
involve working for about 300 hours to thus find the primes in
the numbers from 1 to 1,000,000. The labour of determining
whether any particular number is a prime may be, however,
much shortened by observing that if a number can be expressed
as the product of two factors, one must be less and the
other greater than the square root of the number, unless the
number is the square of a prime, in which case the two factors
are equal. Hence every composite number must be divisible by
a prime which is not greater than its square root.\index{Cube, duplication of|)}

\section*{The second century before Christ.\nblabel{png105a}}

The third century before Christ, which opens with the career
of Euclid and closes with the death of Apollonius, is the most
brilliant era in the history of Greek mathematics. But the
great mathematicians of that century were geometricians, and
under their influence attention was directed almost solely to that
branch of mathematics. With the methods they used, and to
which their successors were by tradition confined, it was hardly
possible to make any further great advance: to fill up a few
details in a work that was completed in its essential parts was
all that could be effected. It was not till after the lapse of
nearly 1800 years that the genius of \index{Descartes!ref.\ to}Descartes opened the way
to any further progress in geometry, and I therefore pass over
the numerous writers who followed Apollonius with but slight
mention. Indeed it may be said roughly that during the next
thousand years \index{Pappus!ref.\ to}Pappus was the sole geometrician of great original
ability; and during this long period almost the only other pure
mathematicians of exceptional genius were \index{Hipparchus!ref.\ to}Hipparchus and
\index{Ptolemy!ref.\ to}Ptolemy, who laid the foundations of trigonometry, and \index{Diophantus!ref.\ to}Diophantus,
who laid those of algebra.

Early in the second century, circ.~180~\textsc{b.c.}, we find the names
of three mathematicians---\index{Hypsicles}Hypsicles, \index{Nicomedes}Nicomedes, and \index{Diocles}Diocles---who
in their own day were famous.

\nblabel{png106a}\nbparagraph{Hypsicles.} The first of these was \emph{Hypsicles}, who added a
\index{Eucn@Euc.!m14@\textsc{xiv}. ref.\ to}fourteenth book to Euclid's \emph{Elements} in which the regular solids
were discussed. In another small work, entitled \emph{Risings}, we
find for the first time in Greek mathematics a \index{Degrees, angular}right angle
divided in the Babylonian manner into ninety degrees; possibly
\index{Eratosthenes!ref.\ to}Eratosthenes may have previously estimated angles by the
number of degrees they contain, but this is only a matter of
conjecture.

\nbparagraph{Nicomedes.} The second was \emph{Nicomedes}, who invented the
curve known as the \index{Conchoid}\emph{conchoid} or the shell-shaped curve. If from
a fixed point $S$ a line be drawn cutting a given fixed straight
line in $Q$, and if $P$ be taken on $SQ$ so that the length $QP$ is
constant (say $d$), then the locus of $P$ is the conchoid. Its
equation may be put in the form $r = a \sec \theta ± d$. It is easy
with its aid to \index{Trisection of angle}\index{Angle,!trisection of}trisect a given angle or to \index{Cube, duplication of}duplicate a cube; and
this no doubt was the cause of its invention.

\index{Diocles|(}\nbparagraph{Diocles.} The third of these mathematicians was \emph{Diocles}, the
inventor of the curve known as the \index{Cissoid}\emph{cissoid} or the ivy-shaped
curve, which, like the conchoid, was used to give a solution of
the duplication problem. He defined it thus: let $AOA'$ and
$BOB'$ be two fixed diameters of a circle at right angles to one
another. Draw two chords $QQ'$ and $RR'$ parallel to $BOB'$ and
equidistant from it. Then the locus of the intersection of $AR$
and $QQ'$ will be the cissoid. Its equation can be expressed in
the form $y^{2}(2a-x) = x^{3}$. The curve may be used to duplicate
the cube. For, if $OA$ and $OE$ be the two lines between which
it is required to insert two geometrical means, and if, in the
figure constructed as above, $A'E$ cut the cissoid in $P$, and $AP$
cut $OB$ in D, we have $OD^{3} = OA^{2} \ctrdot OE$. Thus $OD$ is one
of the means required, and the other mean can be found at
once.

Diocles also solved (by the aid of conic sections) a problem
which had been proposed by \index{Archimedes!ref.\ to}Archimedes, namely, to draw a
plane which will divide a sphere into two parts whose volumes
shall bear to one another a given ratio.\index{Diocles|)}

\nblabel{png107a}\index{Perseus}\index{Zenodorus}\nbparagraph{Perseus. Zenodorus.} About a quarter of a century later,
say about 150~\textsc{b.c.}, \emph{Perseus} investigated the various plane sections
of the \index{Anchor ring}anchor-ring, and \emph{Zenodorus} wrote \index{Isoperimetrical problem}a treatise on isoperimetrical
figures. Part of the latter work has been preserved;
one proposition which will serve to show the nature of the
problems discussed is that ``of segments of circles, having equal
arcs, the semicircle is the greatest.''

Towards the close of this century we find two mathematicians
who, by turning their attention to new subjects, gave a fresh
stimulus to the study of mathematics. These were Hipparchus
and Hero.

\nblabel{png107b}\index{Hipparchus|(}\nbparagraph{Hipparchus.\footnote
  {See C.~Manitius, \emph{Hipparchi in Arati et Eudoxi Phaenomena Commentarii}, %[**TN: "phaenomena" changed to "Phaenomena"]
  Leipzig, 1894, and J.~B.~J.~Delambre, \emph{Histoire de l'astronomie ancienne}, Paris,
  1817, vol.~i, pp.~106--189. S.~P.~Tannery in his \emph{Recherches sur l'histoire de
  l'astronomie ancienne}, Paris, 1893, argues that the work of Hipparchus has
  been overrated, but I have adopted the view of the majority of writers on the
  subject.}}
\index{Manitius on Hipparchus}\index{Delambre}\index{Tannery, S. P., ref.\ to}\emph{Hipparchus} was the most eminent of Greek
\index{Astronomy.!z13@Scientific astronomy founded by Hipparchus|(}astronomers\allowbreak---his chief predecessors being \index{Eudoxus!ref.\ to}Eudoxus, \index{Aristarchus!ref.\ to}Aristarchus,
\index{Archimedes!ref.\ to}Archimedes, and \index{Eratosthenes!ref.\ to}Eratosthenes. Hipparchus is said to have been
born about 160~\textsc{b.c.} at Nicaea in Bithynia; it is probable that
he spent some years at Alexandria, but finally he took up his
abode at Rhodes where he made most of his observations.
Delambre has obtained an ingenious confirmation of the tradition
which asserted that Hipparchus lived in the second century
before Christ. Hipparchus in one place says that the longitude
of a certain star $\eta$ Canis observed by him was exactly $90°$, and
it should be noted that he was an extremely careful observer.
Now in 1750 it was $116° 4' 10''$, and, as the first point of Aries
regredes at the rate of $50.2''$ a year, the observation was made
about 120~\textsc{b.c.}

Except for a short commentary on a poem of \index{Aratus}Aratus dealing
with astronomy all his works are lost, but \index{Ptolemy!ref.\ to}Ptolemy's great
treatise, the \index{Almagest@\emph{Almagest}, the!ref.\ to}\emph{Almagest}, described below, was founded on the
observations and writings of Hipparchus, and from the notes
there given we infer that the chief discoveries of Hipparchus
were as follows. He determined the \index{Year, duration of}duration of the year to
within six minutes of its true value. He calculated the inclination
of the \index{Ecliptic, obliquity of}ecliptic and equator as $23° 51'$; it was actually at
that time $23° 46'$. He estimated the annual precession of the
equinoxes as $59''$; it is $50.2''$. He stated the lunar parallax as
$57'$, which is nearly correct. He worked out the \index{Eccentrics}eccentricity of
the solar orbit as $1/24$; it is very approximately $1/30$. \index{Evection}He
determined the perigee and mean motion of the sun and of the
moon, and he calculated the extent of the shifting of the plane
of the moon's motion. Finally he obtained the synodic periods
of the five planets then known. I leave the details of his
observations and calculations to writers who deal specially with
astronomy such as \index{Delambre}Delambre; but it may be fairly said that
this work placed the subject for the first time on a scientific
basis.

To account for the lunar motion Hipparchus supposed the
moon to move with uniform velocity in a circle, the earth
occupying a position near (but not at) the centre of this circle.
This is equivalent to saying that the orbit is an \index{Epicycles}epicycle of the
first order. The longitude of the moon obtained on this
hypothesis is correct to the first order of small quantities for a
few revolutions. To make it correct for any length of time
Hipparchus further supposed that the apse line moved forward
about $3°$ a month, thus giving a correction for eviction. He
explained the motion of the sun in a similar manner. This
theory accounted for all the facts which could be determined
with the instruments then in use, and in particular enabled him
to calculate the details of eclipses with considerable accuracy.

\nblabel{png108a}\index{Planetary motions}He commenced a series of planetary observations to enable
his successors to frame a theory to account for their motions;
and with great perspicacity he predicted that to do this it
would be necessary to introduce epicycles of a higher order,
that is, to introduce three or more circles the centre of each
successive one moving uniformly along the circumference of the
preceding one.

He also formed a \index{Stars, lists of}list of 1080 of the fixed stars. It is said
that the sudden appearance in the heavens of a new and
brilliant star called his attention to the need of such a catalogue;
and the appearance of such a star during his lifetime is confirmed
by Chinese records.\index{Astronomy.!z13@Scientific astronomy founded by Hipparchus|)}

No further advance in the theory of astronomy was made
until the time of \index{Copernicus!ref.\ to}Copernicus, though the principles laid down
by \index{Hipparchus!ref.\ to}Hipparchus were extended and worked out in detail by
\index{Ptolemy!ref.\ to}Ptolemy.\index{Hipparchus|)}

\nblabel{png109a}Investigations such as these naturally led to \index{Trigonometry.!z2@Created by Hipparchus}\emph{trigonometry},
and Hipparchus must be credited with the invention of that
subject. It is known that in \index{Trigonometrical functions}plane trigonometry he constructed
a table of chords of arcs, which is practically the same as one of
natural \index{Sine}sines; and that in spherical trigonometry he had some
method of solving triangles: but his works are lost, and we can
give no details. It is believed, however, that the elegant
theorem, printed as \index{Eucn@Euc.!ed@\textsc{vi}, D. ref.\ to}Euc.~\textsc{vi}, \textsc{d}, and generally known as
Ptolemy's Theorem, is due to Hipparchus and was copied from
him by Ptolemy. It contains implicitly the \index{Trigonometry, addition formulae}addition formulae
for $\sin(A ± B)$ and $\cos(A ± B)$; and \index{Carnot, Lazare!ref.\ to}Carnot showed how the
whole of elementary plane trigonometry could be deduced
from it.

I ought also to add that Hipparchus was the first to indicate
the position of a place on the earth by means of its \index{Latitude, introduction of}\index{Longitude}latitude and
longitude.

\index{Hero of Alexandria|(}\nblabel{png109b}\nbparagraph{Hero.\footnote
  {See \emph{Recherches sur la vie et les ouvrages d'Héron d'Alexandrie} by T.~H.~Martin
  in vol.~iv of \emph{Mémoires présentés \dots à l'académie d'inscriptions}, Paris,
  1854; see also Loria, book~iii, chap.~v, pp.~107--128, and \index{Cantor, M., ref.\ to}Cantor, chaps.~xviii,
  xix. On the work entitled \emph{Definitions}, which is attributed to Hero,
  see S.~P.~Tannery, chaps.~xiii,~xiv, and an article by G.~Friedlein in
  Boncompagni's \emph{Bulletino di bibliografia} March 1871, vol.~iv, pp.~93--126.
  Editions of the extant works of Hero were published in Teubner's series,
  Leipzig, 1899, 1900, 1903. An English translation of the \mytextgreek{Pneumatik'a} was
  published by B.~Woodcroft and J.~G.~Greenwood, London, 1851: drawings
  of the apparatus are inserted.}}
\index{Martin, ref.\ to}\index{Loria, ref.\ to}\index{Greenwood on Hero}\index{Friedlein, G., ref.\ to}\index{Tannery, S. P., ref.\ to}\index{Woodcroft on Hero}\nblabel{png109c} The second of these mathematicians was \emph{Hero} of
Alexandria, who placed engineering and land-surveying on a
scientific basis. He was a pupil of \index{Ctesibus}Ctesibus, who invented
several ingenious machines, and is alluded to as if he were a
mathematician of note. It is not likely that Hero flourished
before 80~\textsc{b.c.}, but the precise period at which he lived is
uncertain.

In pure mathematics Hero's principal and most characteristic
work consists of (i)~some elementary geometry, with applications
to the determination of the areas of fields of given shapes; (ii)
propositions on finding the volumes of certain solids, with
applications to theatres, baths, banquet-halls, and so on; (iii)~a
rule to find the height of an inaccessible object; and (iv)~tables
of weights and measures. He invented a solution of the
\index{Cube, duplication of}duplication problem which is practically the same as that which
\index{Apollonius!ref.\ to}Apollonius had already discovered. Some commentators think
that he knew how to solve a \index{Quadratic equations}quadratic equation even when the
coefficients were not numerical; but this is doubtful. He
\nblabel{png110a}proved the formula that the \index{Triangle, area of}\index{Area of triangle|(}area of a triangle is equal to
$\{s(s-a) (s-b) (s-c)\}^{1/2}$, where $s$ is the semiperimeter, and $a$, $b$, $c$,
the lengths of the sides, and gave as an illustration a triangle
whose sides were in the ratio 13:14:15. He seems to have
been acquainted with the trigonometry of \index{Hipparchus!ref.\ to}Hipparchus, and the
values of \index{Cotangent}$\cot 2\pi/n$ are computed for various values of $n$, but he
nowhere quotes a formula or expressly uses the value of the
sine; it is probable that like the later Greeks he regarded
trigonometry as forming an introduction to, and being an
integral part of, astronomy.

\begin{center}
\labelincludegraphics[width=4.5in]{111a}
\end{center}

The following is the manner in which he solved\footnote
  {In his \emph{Dioptra}, \index{Hultsch, ref.\ to}Hultsch, part viii, pp.~235--237. It should be stated
  that some critics think that this is an interpolation, and is not due to Hero.}
the problem
to find the area of a triangle $ABC$ the length of whose sides are
$a$, $b$, $c$. Let $s$ be the semiperimeter of the triangle. Let the
inscribed circle touch the sides in $D$, $E$, $F$, and let $O$ be its
centre. On $BC$ produced take $H$ so that $CH=AF$, therefore
$BH = s$. Draw $OK$ at right angles to $OB$, and $CK$ at right
angles to $BC$; let them meet in $K$. The area $ABC$ or $\triangle$ is equal
to the sum of the areas $OBC$, $OCA$, $OAB = \frac{1}{2}ar + \frac{1}{2}br + \frac{1}{2}cr = sr$,
that is, is equal to $BH \ctrdot OD$. He then shews that the angle
$OAF$ = angle $CBK$; hence the triangles $OAF$ and $CBK$ are
similar.\index{Area of triangle|)}
\begin{gather*}
  \begin{aligned}
    \therefore BC:CK &= AF:OF = CH:OD,\\
    \therefore BC:CH &= CK:OD = CL:LD,\\
    \therefore BH:CH &= CD:LD,
  \end{aligned}\\
  \therefore BH^2:CH \ctrdot BH = CD \ctrdot BD:LD \ctrdot BD = CD \ctrdot BD:OD^2.
\end{gather*}

Hence
\[
\triangle = BH \ctrdot OD = \{CH \ctrdot BH \ctrdot CD \ctrdot BD\}^\frac{1}{2} = \{(s-a)s(s-c)(s-b)\}^\frac{1}{2}.
\]

In applied mathematics Hero discussed the centre of gravity,
the five simple machines, and the problem of moving a given
weight with a given power; and in one place he suggested a
way in which the power of a catapult could be tripled. He
also wrote on the theory of hydraulic machines. He described a
theodolite and cyclometer, and pointed out various problems in
surveying for which they would be useful. But the most
interesting of his smaller works are his \mytextgreek{Pneumatik'a} and
\mytextgreek{A>ut'omata}, containing descriptions of about 100 small machines
and mechanical toys, many of which are ingenious. In the
former there is an account of a \index{Avery's steam-engine}small stationary steam-engine
which is of the form now known as Avery's patent: it was in
common use in Scotland at the beginning of this century, but is
not so economical as the form introduced by \index{Watt, ref.\ to}Watt. There is
also an account of a double forcing pump to be used as a \index{Fire engine invented by Hero}\index{Steam-engine, Hero's}fire-engine.
It is probable that in the hands of Hero these instruments
never got beyond models. It is only recently that
general attention has been directed to his discoveries, though
\index{Arago!ref.\ to}Arago had alluded to them in his \emph{éloge} on Watt.\index{Hero of Alexandria|)}

\nblabel{png112a}All this is very different from the classical geometry and
arithmetic of \index{Euclid!ref.\ to}Euclid, or the mechanics of \index{Archimedes!ref.\ to}Archimedes. Hero
did nothing to extend a knowledge of abstract mathematics; he
learnt all that the text-books of the day could teach him, but he
was interested in science only on account of its practical applications,
and so long as his results were true he cared nothing
for the logical accuracy of the process by which he arrived at
them. Thus, in finding the \index{Triangle, area of}area of a triangle, he took the
square root of the product of four lines. The classical Greek
geometricians permitted the use of the square and the cube of
a line because these could be represented geometrically, but a
figure of four dimensions is inconceivable, and certainly they
would have rejected a proof which involved such a conception.


\section*{The first century before Christ.\nblabel{png112b}}

The successors of Hipparchus and Hero did not avail themselves
of the opportunity thus opened of investigating new
subjects, but fell back on the well-worn subject of geometry.
Amongst the more eminent of these later geometricians were
\index{Theodosius|(}Theodosius and Dionysodorus, both of whom flourished about
50~\textsc{b.c.}

\nblabel{png112c}\nbparagraph{Theodosius.} \emph{Theodosius} was the author of a complete
treatise on the geometry of the sphere, and of two works on
astronomy.\index{Barrow!ref.\ to}\index{Nizze, ref.\ to}\index{Dasypodius on Theodosius}\footnote
  {The work on the sphere was edited by I.~Barrow, Cambridge, 1675,
  and by E.~Nizze, Berlin, 1852. The works on astronomy were published by
  Dasypodius in 1572.}

\nblabel{png113a}\index{Dionysodorus}\nbparagraph{Dionysodorus.} \emph{Dionysodorus} is known to us only by his
solution\index{Suter on Dionysodorus}\footnote
  {It is reproduced in H.~Suter's \emph{Geschichte der mathematischen Wissenschaften},
  second edition, Zürich, 1873, p.~101.} %[**TN: Zurich in original]
of the problem to divide a hemisphere by a plane
parallel to its base into two parts, whose volumes shall be in a
given ratio. Like the solution by \index{Diocles!ref.\ to}Diocles of the similar problem
for a sphere above alluded to, it was effected by the aid of conic
sections. \index{Pliny, ref.\ to}Pliny says that Dionysodorus determined the length
of the radius of the \index{Earth,!dimensions of}earth approximately as 42,000 stadia,
which, if we take the Olympic stadium of $202\frac{1}{4}$ yards, is a little
less than 5000 miles; we do not know how it was obtained.
This may be compared with the result given by \index{Eratosthenes!ref.\ to}Eratosthenes
and mentioned above.\index{Theodosius|)}


\section*{End of the First Alexandrian School.\nblabel{png113b}}

\nblabel{png113c}The administration of Egypt was definitely undertaken
by Rome in 30~\textsc{b.c.} The closing years of the dynasty of the
\index{Ptolemies, dynasty of}Ptolemies and the earlier years of the Roman occupation of
the country were marked by much disorder, civil and political.
The studies of the university were naturally interrupted, and
it is customary to take this time as the close of the first
Alexandrian school.

\nbChapter{V}{The Second Alexandrian School}{}{30~B.C.--641~A.D.}
  {For authorities, see footnote above on p.~\pageref{xref:050}. All dates given hereafter
  are to be taken as \emph{anno domini} unless the contrary is expressly stated.}
\index{Geometry.!z2@Classical synthetic geometry, discussed or used by nearly all the mathematicians
considered in the first period, chapters \textsc{ii-v}|)}
\index{Alexandrian,!Schools, chapters \textsc{iv}, \textsc{v}}\nblabel{png114a}

\nblabel{png114b}\First{I concluded} the last chapter by stating that the first school of
Alexandria may be said to have come to an end at about the
same time as the country lost its nominal independence. But,
although the schools at Alexandria suffered from the disturbances
which affected the whole Roman world in the transition,
in fact if not in name, from a republic to an empire, there was
no break of continuity; the teaching in the university was
never abandoned; and as soon as order was again established,
students began once more to flock to Alexandria. This time of
confusion was, however, contemporaneous with a change in the
prevalent views of philosophy which thenceforward were mostly
neo-platonic or neo-pythagorean, and it therefore fitly marks the
commencement of a new period. These mystical opinions
reacted on the mathematical school, and this may partially
account for the paucity of good work.

Though Greek influence was still predominant and the
Greek language always used, Alexandria now became the intellectual
centre for most of the Mediterranean nations which
were subject to Rome. It should be added, however, that
the direct connection with it of many of the mathematicians
of this time is at least doubtful, but their knowledge was
ultimately obtained from the Alexandrian teachers, and they
are usually described as of the second Alexandrian school.
Such mathematics as were taught at Rome were derived from
Greek sources, and we may therefore conveniently consider
their extent in connection with this chapter.

\section*{The first century after Christ.\nblabel{png115a}}

There is no doubt that throughout the first century after
Christ geometry continued to be that subject in science to
which most attention was devoted. But by this time it was
evident that the geometry of Archimedes and Apollonius was not
capable of much further extension; and such geometrical treatises
as were produced consisted mostly of commentaries on the
writings of the great mathematicians of a preceding age. In
this century the only original works of any ability of which we
know anything were two by \index{Serenus}Serenus and one by \index{Menelaus}Menelaus.

\nblabel{png115b}\nbparagraph{Serenus.} \nbparagraph{Menelaus.} Those by \emph{Serenus} of Antissa or of
Antinoe, circ.~70, are on the \emph{plane sections of the cone and
cylinder},\index{Halley!ref.\ to}\index{Heiberg, ref.\ to}\footnote
  {These have been edited by J.~L.~Heiberg, Leipzig, 1896; and by
  E.~Halley, Oxford, 1710.}
in the course of which he lays down the fundamental
proposition of \index{Transversals}transversals. That by \emph{Menelaus} of Alexandria,
circ.~98, is on \emph{spherical trigonometry}, investigated in the %[**TN 'circ 98' in original]
Euclidean method.\footnote
  {This was translated by E.~Halley, Oxford, 1758.}
The fundamental theorem on which the
subject is based is the relation between the six segments of the
sides of a spherical triangle, formed by the arc of a great circle
which cuts them [book~\textsc{iii}, prop.~1]. Menelaus also wrote on
the calculation of chords, that is, on plane trigonometry; this
is lost.

\nblabel{png115c}\index{Nicomachus|(}\nbparagraph{Nicomachus.} Towards the close of this century, circ.~100,
a Jew, \emph{Nicomachus}, of Gerasa, published an \emph{Arithmetic},\index{Hoche on Nicomachus}\footnote
  {The work has been edited by R.~Hoche, Leipzig, 1866.}
which (or rather the Latin translation of it) remained for a
thousand years a standard authority on the subject. Geometrical
demonstrations are here abandoned, and the work is a
mere classification of the results then known, with numerical
illustrations: the evidence for the truth of the propositions
enunciated, for I cannot call them proofs, being in general an
\index{Inductive arithmetic}induction from numerical instances. The object of the book
is the study of the properties of numbers, and particularly of
their ratios. Nicomachus commences with the usual distinctions
between even, odd, prime, and perfect numbers; he next
discusses fractions in a somewhat clumsy manner; he then
turns to polygonal and to solid numbers; and finally treats of
\nblabel{png116a}ratio, proportion, and the progressions. Arithmetic of this kind
is usually termed Boethian, and the work of \index{Boethius!ref.\ to}Boethius on it was
a recognised text-book in the middle ages.\index{Arithmetic.!z5@Theory of,!x2@and thenceforward treated empirically (Boethian arithmetic) by most of the Greek and European mathematicians to the end of the fourteenth century after Christ}
\index{Nicomachus|)}


\section*{The second century after Christ.\nblabel{png116b}}

\nblabel{png116c}\index{Theon of Smyrna}\nbparagraph{Theon.} Another text-book on arithmetic on much the
same lines as that of Nicomachus was produced by \emph{Theon} of
Smyrna, circ.~130. It formed the first book of his work\index{Dupuis on Theon}\footnote
  {The Greek text of those parts which are now extant, with a French
  translation, was issued by J.~Dupuis, Paris, 1892.}
on mathematics, written with the view of facilitating the study
of Plato's writings.

\index{Thymaridas}\nbparagraph{Thymaridas.} Another mathematician, reckoned by some
writers as of about the same date as Theon, was \emph{Thymaridas},
who is worthy of notice from the fact that he is the \index{Algebraa@Algebra,!earliest theorem in|(}earliest
known writer who explicitly enunciates an algebraical theorem.
He states that, if the sum of any number of quantities be
given, and also the sum of every pair which contains one of
them, then this quantity is equal to one $(n - 2)$th part of the
difference between the sum of these pairs and the first given
sum. Thus, if
\begin{DPgather*}
x_{1} + x_{2} + \ldots + x_{n} = S,\\
\lintertext{and if} x_{1} + x_{2} = s_{2},\quad x_{1} + x_{3} = s_{3}, \ldots,\text{ and } x_{1} + x_{n} = s_{n},\\
\lintertext{then}   x_{1} = (s_{2} + s_{3} + \ldots + s_{n} - S)/(n - 2).
\end{DPgather*}
He does not seem to have used a symbol to denote the unknown
quantity, but he always represents it by the same word, which
is an approximation to symbolism.\index{Algebraa@Algebra,!earliest theorem in|)}

\nblabel{png117a}\index{Ptolemy|(}\nbparagraph{Ptolemy.\footnote
  {See the article \emph{Ptolemaeus Claudius}, by A.~De~Morgan in Smith's
  \emph{Dictionary of Greek and Roman Biography}, London, 1849; S.~P.~Tannery,
  \emph{Recherches sur l'histoire de l'astronomie ancienne}, Paris, 1893; and
  J.~B.~J.~Delambre, \emph{Histoire de l'astronomie ancienne}, Paris, 1817, vol.~ii. %[**TN: "ii," in original text]
  An edition of all the works of Ptolemy which are now extant was
  published at Bâle in 1551. The \emph{Almagest} with various minor works
  was edited by M.~Halma, 12 vols. Paris, 1813--28, and a new edition,
  in two volumes, by J.~L.~Heiberg, Leipzig, 1898, 1903, 1907.}}
\index{Heiberg, ref.\ to}\index{Tannery, S. P., ref.\ to}\index{Delambre}\index{Almagest@\emph{Almagest}, the|(}\index{Demorgana@De~Morgan, A.!ref.\ to}\index{Halma, M., ref.\ to}About the same time as these writers \index{Astronomy.!z13@Scientific astronomy founded by Hipparchus!x3@and developed by Ptolemy in the \emph{Almagest}|(}\emph{Ptolemy}
of Alexandria, who died in 168, produced his great work on
astronomy, which will preserve his name as long as the history
of science endures. This treatise is usually known as the
\nblabel{png117b}\emph{Almagest}: the name is derived from the Arabic title \emph{al midschisti},
which is said to be a corruption of \mytextgreek{meg'isth} [\mytextgreek{majhmatik'h}]
\mytextgreek{s'untaxis}. The work is founded on the writings of \index{Hipparchus!ref.\ to}Hipparchus,
and, though it did not sensibly advance the theory of the
subject, it presents the views of the older writer with a completeness
and elegance which will always make it a standard
\nblabel{png117c}treatise. We gather from it that Ptolemy made observations
at Alexandria from the years 125 to 150; he, however, was
but an indifferent practical astronomer, and the observations
of Hipparchus are generally more accurate than those of his
expounder.

The work is divided into thirteen books. In the first book
Ptolemy discusses various preliminary matters; treats of \index{Trigonometry.!z2@Created by Hipparchus!x1@and by Ptolemy}\index{Trigonometrical functions}trigonometry,
plane or spherical; gives a table of chords, that is,
of \index{Sine}natural sines (which is substantially correct and is probably
taken from the lost work of Hipparchus); and explains the
obliquity of the ecliptic; in this book he uses degrees, minutes,
and seconds as measures of angles. The second book is devoted
chiefly to phenomena depending on the spherical form of the
earth: he remarks that the explanations would be much
simplified if the earth were supposed to rotate on its axis
once a day, but states that this hypothesis is inconsistent with
known facts. In the third book he explains the motion of the
sun round the earth by means of excentrics and epicycles: and
in the fourth and fifth books he treats the motion of the moon
in a similar way. \nblabel{xref:097}The sixth book is devoted to the theory of
eclipses; and in it he gives $3° 8' 30''$, that is $3\frac{17}{120}$, as the
approximate value of \index{P1@$\pi$,!z1@value of}$\pi$, which is equivalent to taking it equal
to 3.1416. The seventh and eighth books contain a \index{Stars, lists of}catalogue
(probably copied from Hipparchus) of 1028 fixed stars determined
by indicating those, three or more, that appear to be in
a plane passing through the observer's eye: and in another
work Ptolemy added a list of annual sidereal phenomena. The
remaining books are given up to the theory of the planets.

This work is a splendid testimony to the ability of its
author. \index{Planetary motions}It became at once the standard authority on astronomy,
and remained so till \index{Copernicus!ref.\ to}Copernicus and Kepler shewed
that the sun and not the earth must be regarded as the centre
of the solar system.

The idea of \index{Excentrics}excentrics and \index{Epicycles}epicycles on which the theories
of Hipparchus and Ptolemy are based has been often ridiculed
in modern times. No doubt at a later time, when more accurate
observations had been made, the necessity of introducing
epicycle on epicycle in order to bring the theory into accordance
with the facts made it very complicated. But \index{Demorgana@De~Morgan, A.!ref.\ to}De~Morgan
has acutely observed that in so far as the ancient astronomers
supposed that it was necessary to resolve every celestial motion
into a series of uniform circular motions they erred greatly,
but that, if the hypothesis be regarded as a convenient way
of expressing known facts, it is not only legitimate but
convenient. The theory suffices to describe either the angular
motion of the heavenly bodies or their change in distance. The
ancient astronomers were concerned only with the former question,
and it fairly met their needs; for the latter question it is
less convenient. In fact it was as good a theory as for their
purposes and with their instruments and knowledge it was
possible to frame, and corresponds to the expression of a given
function as a sum of sines or cosines, a method which is of
frequent use in modern analysis.

In spite of the trouble taken by \index{Delambre}Delambre it is almost
impossible to separate the results due to \index{Hipparchus!ref.\ to}Hipparchus from
those due to Ptolemy. But Delambre and \index{Demorgana@De~Morgan, A.!ref.\ to}De~Morgan agree
in thinking that the observations quoted, the fundamental
ideas, and the explanation of the apparent solar motion are due
to Hipparchus; while all the detailed explanations and calculations
of the lunar and planetary motions are due to Ptolemy.

\begin{center}
\labelincludegraphics{119a}
\end{center}

\nblabel{png119a}The \emph{Almagest} shews that Ptolemy was a geometrician of
the first rank, though it is with the application of geometry
to astronomy that he is chiefly concerned. He was also the
author of numerous other treatises. \index{Eucm@Euc.\ post.\ 12, Ptolemy's proof of}Amongst these is one on
\emph{pure geometry} in which he proposed to cancel Euclid's postulate
on \index{Parallel lines|(}parallel lines, and to prove it in the following manner. Let
the straight line $EFGH$ meet the two straight lines $AB$ and
$CD$ so as to make the sum of the angles $BFG$ and $FGD$ equal
to two right angles. It is required to prove that $AB$ and $CD$
are parallel. If possible let them not be parallel, then they will
meet when produced say at $M$ (or $N$). But the angle $AFG$ is
the supplement of $BFG$, and is therefore equal to $FGD$:
similarly the angle $FGC$ is equal to the angle $BFG$. Hence
the sum of the angles $AFG$ and $FGC$ is equal to two right
angles, and the lines $BA$ and $DC$ will therefore if produced
meet at $N$ (or $M$). But two straight lines cannot enclose a
space, therefore $AB$ and $CD$ cannot meet when produced, that
is, they are parallel. Conversely, if $AB$ and $CD$ be parallel,
then $AF$ and $CG$ are not less parallel than $FB$ and $GD$; and
therefore whatever be the sum of the angles $AFG$ and $FGC$
such also must be the sum of the angles $FGD$ and $BFG$. But
the sum of the four angles is equal to four right angles, and
therefore the sum of the angles $BFG$ and $FGD$ must be equal
to two right angles.\index{Almagest@\emph{Almagest}, the|)}\index{Astronomy.!z13@Scientific astronomy founded by Hipparchus!x3@and developed by Ptolemy in the \emph{Almagest}|)}\index{Parallel lines|)}

Ptolemy wrote another work to shew that there could not
be more than three dimensions in space: he also discussed
\emph{orthographic} and \emph{stereographic projections} with special reference
to the construction of sun-dials. He wrote on geography,
and stated that the length of one degree of latitude is 500
stadia. A book on \emph{sound} is sometimes attributed to him, but
on doubtful authority.\index{Ptolemy|)}


\section*{The third century after Christ.\nblabel{png120a}}

\nblabel{png120b}\index{Pappus|(}\nbparagraph{Pappus.} Ptolemy had shewn not only that geometry
could be applied to astronomy, but had indicated how new
methods of analysis like trigonometry might be thence developed.
He found however no successors to take up the
work he had commenced so brilliantly, and we must look
forward 150 years before we find another geometrician of any
eminence. That geometrician was \emph{Pappus} who lived and
taught at Alexandria about the end of the third century. We
know that he had numerous pupils, and it is probable that he
temporarily revived an interest in the study of geometry.

\nblabel{png120c}Pappus wrote several books, but the only one which has
come down to us is his \mytextgreek{Sunagwg'h},\index{Hultsch, ref.\ to}\footnote
  {It has been published by F.~Hultsch, Berlin, 1876--8.}
a collection of mathematical
papers arranged in eight books of which the first and
part of the second have been lost. This collection was intended
to be a synopsis of Greek mathematics together with comments
and additional propositions by the editor. A careful comparison
of various extant works with the account given of
them in this book shews that it is trustworthy, and we rely
largely on it for our knowledge of other works now lost. It
is not arranged chronologically, but all the treatises on the
same subject are grouped together, and it is most likely that
it gives roughly the order in which the classical authors were
read at Alexandria. Probably the first book, which is now
lost, was on arithmetic. The next four books deal with
geometry exclusive of conic sections; the sixth with astronomy
including, as subsidiary subjects, \index{Optics (geometrical).!Discussed by (among others) Euclid!z1@Pappus}optics and trigonometry; the
seventh with analysis, conics, and porisms; and the eighth with
mechanics.

The last two books contain a good deal of original work by
Pappus; at the same time it should be remarked that in two or
three cases he has been detected in appropriating proofs from
earlier authors, and it is possible he may have done this in other
cases.

Subject to this suspicion we may say that Pappus's best
work is in geometry. He discovered the \index{Directrix in conics}directrix in the conic
sections, but he investigated only a few isolated properties:
the earliest comprehensive account was given by \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton and
\index{Boscovich}Boscovich. As an illustration of his power I may mention
that he solved [book~\textsc{vii}, prop.~107] the problem to inscribe in
a given circle a triangle whose sides produced shall pass
through three collinear points. This question was in the
eighteenth century generalised by Cramer by supposing the
three given points to be anywhere; and was considered a
difficult problem.\footnote
  {For references to this problem see a note by H.~Brocard in \emph{L'Intermédiaire
  des mathématiciens}, Paris, 1904, vol.~xi, pp.~219--220.}
It was sent in 1742 as a challenge to
\index{Castillon on Pappus's problem}Castillon, and in 1776 he published a solution. \index{Lagrange!ref.\ to}Lagrange,
\index{Euler!ref.\ to}Euler, \index{Lhulier}Lhulier, \index{Fuss, ref.\ to}Fuss, and \index{Lexell on Pappus's problem}Lexell also gave solutions in 1780.
A few years later the problem was set to a Neapolitan lad
\index{Giordano on Pappus's problem}A.~Giordano, who was only 16 but who had shewn marked
mathematical ability, and he extended it to the case of a
polygon of $n$ sides which pass through $n$ given points, and gave
a solution both simple and elegant. \index{Poncelet!ref.\ to}Poncelet extended it to
conics of any species and subject to other restrictions.

\index{Mechanics.!z1@Discussed by!x3@and Pappus|(}In mechanics Pappus shewed that the \index{Centres of mass}\index{Gravity,!centres of}\index{Mass, centres of}centre of mass of a
triangular lamina is the same as that of an inscribed triangular
lamina whose vertices divide each of the sides of the original
triangle in the same ratio. He also discovered the two
theorems on the surface and volume of a solid of revolution
which are still quoted in text-books under his name: these
are that the volume generated by the revolution of a curve
about an axis is equal to the product of the area of the curve
and the length of the path described by its \index{Centres of mass}\index{Gravity,!centres of}centre of mass;
and the surface is equal to the product of the perimeter of
the curve and the length of the path described by its centre of
mass.\index{Mechanics.!z1@Discussed by!x3@and Pappus|)}

\nblabel{xref:101}The problems above mentioned are but samples of many
brilliant but isolated theorems which were enunciated by
Pappus. His work as a whole and his comments shew that he
was a geometrician of power; but it was his misfortune to
live at a time when but little interest was taken in geometry,
and when the subject, as then treated, had been practically
exhausted.\index{Pappus|)}

Possibly a small tract\index{Arithmetic.!z4@Practical Greek}\index{Henry C., ref.\ to}\footnote
  {It was edited by C.~Henry, Halle, 1879, and is valuable as an illustration
  of practical Greek arithmetic.}
on multiplication and division of
\index{Sexagesimal fractions}sexagesimal fractions, which would seem to have been written
about this time, is due to Pappus.

\section*{The fourth century after Christ.\nblabel{png122a}}

Throughout the second and third centuries, that is, from
the time of Nicomachus, interest in geometry had steadily
decreased, and more and more attention had been paid to the
theory of numbers, though the results were in no way commensurate
with the time devoted to the subject. It will
be remembered that \index{Euclid!ref.\ to}Euclid used lines as symbols for any
magnitudes, and investigated a number of theorems about
numbers in a strictly scientific manner, but he confined himself
to cases where a \index{Algebra.!z1@Treated geometrically by Euclid and his School}geometrical representation was possible.
There are indications in the works of \index{Archimedes!ref.\ to}Archimedes that he was
prepared to carry the subject much further: he introduced
numbers into his geometrical discussions and divided lines by
lines, but he was fully occupied by other researches and had
no time to devote to arithmetic. \index{Hero of Alexandria!ref.\ to}Hero abandoned the geometrical
representation of numbers, but he, Nicomachus, and
other later writers on arithmetic did not succeed in creating
any other symbolism for numbers in general, and thus when
they enunciated a theorem they were content to verify it by
a large number of numerical examples. They doubtless knew
how to solve a \index{Quadratic equations}quadratic equation with numerical coefficients---for,
as pointed out above, geometrical solutions of the equations
$ax^{2} - bx + c = 0$ and $ax^{2} + bx -c = 0$ are given in \index{Eucn@Euc.!e28@\textsc{vi}, 28. ref.\ to}\index{Eucn@Euc.!e29@\textsc{vi}, 29. ref.\ to}Euc.~\textsc{vi},
28 and 29---but probably this represented their highest attainment.

\index{Algebraa@Algebra,!earliest problems in}\index{Algebraa@Algebra,!historical development|(}It would seem then that, in spite of the time given to their
study, arithmetic and algebra had not made any sensible
advance since the time of \index{Archimedes!ref.\ to}Archimedes. The problems of this
kind which excited most interest in the third century may be
illustrated from a collection of questions, printed in the
\index{Anthology, Palatine}\index{Palatine Anthology}\nblabel{png123a}Palatine Anthology, which was made by \index{Metrodorus}\nbname{Metrodorus} at the
beginning of the next century, about 310. Some of them are
due to the editor, but some are of an anterior date, and they
fairly illustrate the way in which arithmetic was leading up
to algebraical methods. The following are typical examples.
``Four pipes discharge into a cistern: one fills it in one day;
another in two days; the third in three days; the fourth in
four days: if all run together how soon will they fill the
cistern?'' ``Demochares has lived a fourth of his life as a
boy; a fifth as a youth; a third as a man; and has spent
thirteen years in his dotage: how old is he?'' ``Make a crown
of gold, copper, tin, and iron weighing 60 minae: gold and
copper shall be two-thirds of it; gold and tin three-fourths of
it; and gold and iron three-fifths of it: find the weights of
the gold, copper, tin, and iron which are required.'' The
last is a numerical illustration of \index{Thymaridas!ref.\ to}Thymaridas's theorem quoted
above.

\index{Rhetorical algebra|(}\index{Algebra.!z2@Development of rhetorical and syncopated algebra in the fourth century after Christ|(}It is believed that these problems were solved by \emph{rhetorical
algebra}, that is, by a process of algebraical reasoning expressed
in words and without the use of any symbols. This, according
\nblabel{png124a}to \index{Nesselmann, ref.\ to}Nesselmann, is the first stage in the development of algebra,
and we find it used both by Ahmes and by the earliest Arabian,
Persian, and Italian algebraists: examples of its use in the
solution of a geometrical problem and in the rule for the solution
of a quadratic equation are given later.\footnote
  {See below, pp.~\pageref{xref:203},~\pageref{xref:210}.}
On this view then a
rhetorical algebra had been gradually evolved by the Greeks,
or was then in process of evolution. Its development was
however very imperfect. \index{Hankel, ref.\ to}Hankel, who is no unfriendly critic,
says that the results attained as the net outcome of the work
of six centuries on the theory of numbers are, whether we
look at the form or the substance, unimportant or even childish,
and are not in any way the commencement of a science.\index{Rhetorical algebra|)}

In the midst of this decaying interest in geometry and these
feeble attempts at algebraic arithmetic, a single algebraist of
marked originality suddenly appeared who created what was
practically a new science. This was \index{Diophantus|(}Diophantus who introduced
a system of abbreviations for those operations and quantities
which constantly recur, though in using them he observed all
the rules of grammatical syntax. The resulting science is called
by Nesselmann \emph{syncopated algebra}: it is a sort of shorthand.
Broadly speaking, it may be said that European algebra did
not advance beyond this stage until the close of the sixteenth
century.

Modern algebra has progressed one stage further and is
entirely \emph{symbolic}; that is, it has a language of its own and a
system of notation which has no obvious connection with the
things represented, while the operations are performed according
to certain rules which are distinct from the laws of grammatical
construction.\index{Algebraa@Algebra,!historical development|)}

\nblabel{png124b}\nbparagraph{Diophantus.\footnote
  {A critical edition of the collected works of Diophantus was edited by
  S.~P.~Tannery, 2~vols., Leipzig, 1893; see also \emph{Diophantos of Alexandria},
  by T.~L.~Heath, Cambridge, 1885; and Loria, book~\textsc{v}, chap.~\textsc{v}, pp.~95--158.}}
\index{Heath, Sir T. L.}All that we know of \emph{Diophantus} is that
he lived at Alexandria, and that most likely he was not a
Greek. Even the date of his career is uncertain; it cannot
reasonably be put before the middle of the third century, and
it seems probable that he was alive in the early years of the
fourth century, that is, shortly after the death of \index{Pappus!ref.\ to}Pappus. He
was 84 when he died.

In the above sketch of the lines on which algebra has developed
\nblabel{png125a}I credited Diophantus with the invention of \index{Syncopated algebra}syncopated
algebra. This is a point on which opinions differ, and some
writers believe that he only systematized the knowledge which
was familiar to his contemporaries. In support of this latter
opinion it may be stated that \index{Cantor, M., ref.\ to}Cantor thinks that there are traces
of the use of \index{Symbolic algebra}algebraic symbolism in Pappus, and \index{Friedlein, G., ref.\ to}Freidlein
mentions a \index{Rhind papyrus, the!ref.\ to}Greek papyrus in which the signs $/$ and \index{Subtraction,!symbols for}\raisebox{-.6ex}{\labelincludegraphics[width=1.1em]{125a}} are used
for \index{Addition, processes of!symbols for}addition and \index{Negative sign}subtraction respectively; but no other direct
evidence for the non-originality of Diophantus has been produced,
and no ancient author gives any sanction to this opinion.

Diophantus wrote a short essay on \index{Numbers,!polygonal}\index{Polygonal numbers}polygonal numbers; a
treatise on algebra which has come down to us in a mutilated
condition; and a work on porisms which is lost.

The \emph{Polygonal Numbers} contains ten propositions, and
was probably his earliest work. In this he reverts to the
classical system by which numbers are represented by lines, a
construction is (if necessary) made, and a strictly deductive
proof follows: it may be noticed that in it he quotes propositions,
such as \index{Eucn@Euc.!b03@\textsc{ii}, 3. ref.\ to}Euc.~\textsc{ii}, 3, and \index{Eucn@Euc.!b08@\textsc{ii}, 8. ref.\ to}\textsc{ii}, 8, as referring to numbers
and not to magnitudes.

His chief work is his \emph{Arithmetic}. This is really a treatise
on algebra; algebraic symbols are used, and the problems are
treated analytically. Diophantus tacitly assumes, as is done
in nearly all modern algebra, that the steps are reversible. He
applies this algebra to find solutions (though frequently only
particular ones) of several problems involving numbers. I
propose to consider successively the notation, the methods of
analysis employed, and the subject-matter of this work.

\nblabel{png125b}First, as to the notation. Diophantus always employed a
symbol to represent the \index{Unknown quantity, word or symbol for}unknown quantity in his equations,
but as he had only one symbol he could not use more than
one unknown at a time.\footnote
  {See, however, below, page~\pageref{xref:108}, example~(iii), for an instance of how
  he treated a problem involving two unknown quantities.}
The unknown quantity is called
\mytextgreek{<o >arijm'os}, and is represented by $\Mysigma'$ or $\Mysigma^{o\prime}$. It is usually printed
as $\varsigma$. In the plural it is denoted by $\varsigma\varsigma$ or $\overline{\varsigma\varsigma}^{o\grave{\iota}}$. This symbol
may be a corruption of $\alpha^\rho$, or perhaps it may be the final
sigma of this word, or possibly it may stand for the word \mytextgreek{swr'os}
a \index{Heap for unknown number}heap.\footnote
  {See above, page~\pageref{xref:005}.}
The square of the unknown is called \mytextgreek{d'unamis}, and
denoted by $\delta^{\bar{\upsilon}}$: the cube \mytextgreek{k'ubos}, and denoted by $\kappa^\upsilon$; and so on
up to the sixth power.

\index{Multiplication, processes of}The coefficients of the unknown quantity and its powers are
numbers, and a numerical coefficient is written immediately after
the quantity it multiplies: thus $\varsigma'\bar{\alpha} = x$, and $\varsigma\varsigma^{o\iota}\,\overline{\iota\alpha} =
\overline{\varsigma\varsigma}\, \overline{\iota\alpha} = 11x$.
An absolute term is regarded as a certain number of units or
\mytextgreek{mon'ades} which are represented by $\mu^{\hat{\omicron}}$: thus $\mu^{\hat{\omicron}}\bar{\alpha} = 1$, $\mu^{\hat{\omicron}} \overline{\iota\alpha} = 11$.

\index{Addition, processes of!symbols for}\index{Plus!symbols for}There is no sign for addition beyond juxtaposition. \index{Negative sign}\index{Subtraction,!symbols for}\index{Minus!z1@symbols for}Subtraction
is represented by \inlinegraphics{1.5ex}{-0.5ex}{subtraction}, and this symbol affects all the
symbols that follow it. \index{Equality, symbols for}Equality is represented by $\iota$. Thus
\begin{DPgather*}
\kappa^{\hat{\upsilon}} \bar{\alpha}\ \overline{\varsigma\varsigma} \bar{\eta}\ \inlinegraphics{1.5ex}{-0.5ex}{subtraction}\ \delta^{\hat{\omicron}} \bar{\epsilon}\ \mu^{\hat{\omicron}}\bar{\alpha}\ \iota\ \varsigma \bar{\alpha}\\
\lintertext{represents}    (x^{3} + 8x) - (5x^{2} + 1) = x.
\end{DPgather*}

Diophantus also introduced a somewhat similar notation
for fractions involving the unknown quantity, but into the
details of this I need not here enter.

It will be noticed that all these symbols are mere abbreviations
for words, and Diophantus reasons out his proofs,
writing these abbreviations in the middle of his text. In
most manuscripts there is a marginal summary in which the
symbols alone are used and which is really symbolic algebra;
but probably this is the addition of some scribe of later times.

This introduction of a contraction or a symbol instead of a
word to represent an unknown quantity marks a greater advance
than anyone not acquainted with the subject would imagine,
and those who have never had the aid of some such abbreviated
symbolism find it almost impossible to understand complicated
algebraical processes. It is likely enough that it might have
been introduced earlier, but for the unlucky system of numeration
adopted by the Greeks by which they used all the letters
of the alphabet to denote particular numbers and thus made it
impossible to employ them to represent any number.

Next, as to the knowledge of algebraic methods shewn in
the book. Diophantus commences with some definitions which
include an explanation of his notation, and in giving the \index{Minus!z1@symbols for}symbol
for \emph{minus} he \index{Signs, rule of}states that a subtraction multiplied by a
subtraction gives an addition; by this he means that the
product of $-b$ and $-d$ in the expansion of $(a-b)(c-d)$ is
$+bd$, but in applying the rule he always takes care that the
numbers $a$, $b$, $c$, $d$ are so chosen that $a$ is greater than $b$ and $c$
is greater than $d$.

The whole of the work itself, or at least as much as is now
extant, is devoted to solving problems which lead to equations.
\index{Simple equations}It contains rules for solving a simple equation of the first
degree and a \index{Quadratic equations}binomial quadratic. Probably the rule for solving
any quadratic equation was given in that part of the work which
is now lost, but where the equation is of the form $ax^2 + bx + c = 0$
he seems to have multiplied by $a$ and then ``completed the
square'' in much the same way as is now done: when the
roots are negative or irrational the equation is rejected as
``impossible,'' and even when both roots are positive he never
gives more than one, always taking the positive value of the
square root. Diophantus solves one \index{Cubic equations}cubic equation, namely,
$x^3 + x = 4x^2 + 4$ [book~\textsc{vi}, prob.~19].

The greater part of the work is however given up to indeterminate
equations between two or three variables. When
the equation is between two variables, then, if it be of the first
degree, he assumes a suitable value for one variable and solves
the equation for the other. Most of his equations are of the
form $y^2 = Ax^2 + Bx + C$. Whenever $A$ or $C$ is equal to zero,
he is able to solve the equation completely. When this is not
the case, then, if $A$ = $a^2$, he assumes $y = ax+m$; if $C = c^2$, he
assumes $y = mx+c$; and lastly, if the equation can be put in the
form $y^2 = (ax ± b)^2 + c^2$, he assumes $y = mx$: where in each case
$m$ has some particular numerical value suitable to the problem
under consideration. A few particular equations of a higher
order occur, but in these he generally alters the problem so as
to enable him to reduce the equation to one of the above
forms.

\index{Equations, differential!indeterminate}The simultaneous indeterminate equations involving three
variables, or ``double equations'' as he calls them, which he
considers are of the forms $y^2 = Ax^2 + Bx + C$ and $z^2 = ax^2 + bx + c$.
If $A$ and $a$ both vanish, he solves the equations in one of two
ways. It will be enough to give one of his methods which is
as follows: he subtracts and thus gets an equation of the form
$y^2 - z^2 = mx + n$; hence, if $y ± z = \lambda$, then $y \mp z = (mx + n) / \lambda$; and
solving he finds $y$ and $z$. His treatment of ``double equations''
of a higher order lacks generality and depends on the particular
numerical conditions of the problem.

Lastly, as to the matter of the book. The problems he
attacks and the analysis he uses are so various that they cannot
be described concisely and I have therefore selected five typical
problems to illustrate his methods. What seems to strike his
critics most is the ingenuity with which he selects as his unknown
some quantity which leads to equations such as he can
solve, and the artifices by which he finds numerical solutions of
his equations.

I select the following as characteristic examples.

(i)~\emph{Find four numbers, the sum of every arrangement three
at a time being given; say \emph{22}, \emph{24}, \emph{27}, and \emph{20}} [book~\textsc{i},
prob.~17].

Let $x$ be the sum of all four numbers; hence the numbers
are $x-22$, $x-24$, $x-27$, and $x-20$.
\begin{gather*}
 \therefore x = (x-22) + (x-24) + (x-27) + (x-20). \\
 \therefore x = 31. \\
 \therefore \text{the numbers are } 9, 7, 4, \text{ and } 11.
\end{gather*}

(ii)~\emph{Divide a number, such as \emph{13} which is the sum of two
squares \emph{4} and \emph{9}, into two other squares} [book~\textsc{ii}, prob.~10].

He says that since the given squares are $2^{2}$ and $3^{2}$ he will
take $(x + 2)^{2}$ and $(mx - 3)^{2}$ as the required squares, and will
assume $m = 2$.
\begin{gather*}
\therefore (x + 2)^{2} + (2x - 3)^{2} = 13.\\
\therefore x = 8/5.\\
\therefore \text{the required squares are } 324/25 \text{ and } 1/25.
\end{gather*}
\nblabel{xref:108}(iii)~\emph{Find two squares such that the sum of the product
and either is a square} [book~\textsc{ii}, prob.~29].

Let $x^{2}$ and $y^{2}$ be the numbers. Then $x^{2}y^{2} + y^{2}$ and $x^{2}y^{2} + x^{2}$
are squares. The first will be a square if $x^{2} + 1$ be a square,
which he assumes may be taken equal to $(x - 2)^{2}$, hence
$x = 3/4$. He has now to make $9 (y^{2}+ 1)/16$ a square, to do this
he assumes that $9y^{2} + 9 = (3y - 4)^{2}$, hence $y = 7/24$. Therefore
the squares required are $9/16$ and $49/576$.

It will be recollected that Diophantus had only one symbol
for an unknown quantity; and in this example he begins by
calling the unknowns $x^{2}$ and $1$, but as soon as he has found $x$
he then replaces the $1$ by the symbol for the unknown quantity,
and finds it in its turn.

(iv)~\emph{To find a \emph{[}rational\emph{]} right-angled triangle such that the
line bisecting an acute angle is rational} [book~\textsc{vi}, prob.~18].

\begin{center}
\labelincludegraphics{129a}
\end{center}

His solution is as follows. Let $ABC$ be the triangle of which
$C$ is the right-angle. Let the bisector $AD = 5x$, and
let $DC = 3x$, hence $AC = 4x$.
 Next let $BC$ be a multiple of $3$,
say $3,\ \therefore BD = 3-3x$, hence $AB = 4 - 4x$ (by Euc.~\textsc{vi}, 3).
Hence $(4-4x)^2 = 3^2 + (4x)^2$ (Euc.~\textsc{i},~47), $\therefore x = 7/32$. Multiplying
by 32 we get for the sides of the triangle 28, 96, and 100; and
for the bisector 35.

(v)~\emph{A man buys $x$ measures of wine, some at $8$ drachmae
a measure, the rest at $5$. He pays for them a square number of
drachmae, such that, if $60$ be added to it, the resulting number
is $x^2$. Find the number he bought at each price} [book~\textsc{v},
prob.~33].

The price paid was $x^2 - 60$, hence $8x > x^2-60$ and $5x < x^2-60$.
From this it follows that $x$ must be greater than 11 and less
than 12.

Again $x^2-60$ is to be a square; suppose it is equal to
$(x-m)^2$ then $x = (m^2 + 60)/2m$, we have therefore
\begin{gather*}
 11 < \frac{m^2+60}{2m} < 12; \\
 \therefore 19 < m < 21.
\end{gather*}

Diophantus therefore assumes that $m$ is equal to 20, which
gives him $x = 11\frac{1}{2}$; and makes the total cost, \emph{i.e.} $x^2-60$, equal
to $72\frac{1}{4}$ drachmae.

He has next to divide this cost into two parts which shall
give the cost of the 8 drachmae measures and the 5 drachmae
measures respectively. Let these parts be $y$ and $z$.
\begin{DPgather*}
\lintertext{Then}      \tfrac{1}{5}z + \tfrac{1}{8}(72\tfrac{1}{4} - z) = \tfrac{1}{2}.\\
\lintertext{Therefore} z = \frac{5 × 79}{12}\text{, and }y = \frac{8 × 59}{12}.
\end{DPgather*}
Therefore the number of 5 drachmae measures was $79/12$, and
of 8 drachmae measures was $59/12$.

\index{Numbers, theory of.!Treatment of, by Pythagoras!q2@by Diophantus|(}From the enunciation of this problem it would seem that
the wine was of a poor quality, and \index{Tannery, S. P., ref.\ to}Tannery ingeniously
suggested that the prices mentioned for such a wine are higher
than were usual until after the end of the second century. He
therefore rejected the view which was formerly held that
Diophantus lived in that century, but he did not seem to be
aware that \index{Demorgana@De~Morgan, A.!ref.\ to}De~Morgan had previously shewn that this opinion
was untenable. \index{Tannery, S. P., ref.\ to}Tannery inclined to think that Diophantus
lived half a century earlier than I have supposed.\index{Numbers, theory of.!Treatment of, by Pythagoras!q2@by Diophantus|)}

\nblabel{png131a}I mentioned that Diophantus wrote a third work entitled
\index{Porisms of Euclid!of Diophantus}\emph{Porisms}. The book is lost, but we have the enunciations of
\nblabel{png131b}some of the propositions, and though we cannot tell whether
they were rigorously proved by Diophantus they confirm our
opinion of his ability and sagacity. It has been suggested that
some of the theorems which he assumes in his arithmetic were
proved in the porisms. Among the more striking of these
results are the statements that the difference of the cubes of two
numbers can be always expressed as the sum of the cubes of two
other numbers; that no number of the form $4n-1$ can be
expressed as the sum of two squares; and that no number of the
form $8n-1$ (or possibly $24n + 7$) can be expressed as the sum
of three squares: to these we may perhaps add the proposition
that any number can be expressed as a square or as the sum of
two or three or four squares.

The writings of Diophantus exercised no perceptible influence
on Greek mathematics; but his \emph{Arithmetic}, when translated into
Arabic in the tenth century, influenced the Arabian school, and
so indirectly affected the progress of European mathematics. An
imperfect copy of the original work was discovered in 1462; it
was translated into Latin and published by \index{Xylander!ref.\ to}Xylander in 1575;
the translation excited general interest, and by that time the
European algebraists had, on the whole, advanced beyond the
point at which Diophantus had left off.\index{Diophantus|)}

\nblabel{png131c}\index{Iamblichus|(}\nbparagraph{Iamblichus.} \emph{Iamblichus}, circ.~350, to whom we owe a
valuable work on the \index{Pythagorean School, the!ref.\ to}Pythagorean discoveries and doctrines,
seems also to have studied the properties of numbers. He
enunciated the theorem that if a number which is equal to the
sum of three integers of the form $3n$, $3n-1$, $3n-2$ be taken,
and if the separate digits of this number be added, and if the
separate digits of the result be again added, and so on, then the
final result will be 6: for instance, the sum of 54, 53, and 52 is
159, the sum of the separate digits of 159 is 15, the sum of the
separate digits of 15 is 6. To any one confined to the usual
Greek numerical notation this must have been a difficult result
to prove: possibly it was reached empirically.\index{Algebra.!z2@Development of rhetorical and syncopated algebra in the fourth century after Christ|)}\index{Iamblichus|)}

The names of two commentators will practically conclude the
long roll of Alexandrian mathematicians.

\index{Theon of Alexandria}\nblabel{png132a}\nbparagraph{Theon.} The first of these is \emph{Theon of Alexandria}, who
flourished about 370. He was not a mathematician of special
note, but we are indebted to him for an edition of \index{Euclid's \emph{Elements}!ref.\ to}Euclid's
\emph{Elements} and a commentary on the \index{Almagest@\emph{Almagest}, the!ref.\ to}\emph{Almagest}; the latter\index{Halma, M., ref.\ to}\footnote
  {It was translated with comments by M.~Halma and published at Paris in 1821.}
gives a great deal of miscellaneous information about the numerical
methods used by the Greeks.

\index{Hypatia}\nbparagraph{Hypatia.} The other was \emph{Hypatia} the daughter of Theon.
She was more distinguished than her father, and was the last
Alexandrian mathematician of any general reputation: she wrote
a commentary on the \emph{Conics} of Apollonius and possibly some
other works, but none of her writings are now extant. \index{Christians (Eastern Church) opposed to Greek science}She was
murdered at the instigation of the Christians in 415.

\nblabel{png132b}The fate of \index{Hypatia!ref.\ to}Hypatia may serve to remind us that the Eastern
Christians, as soon as they became the dominant party in the
state, showed themselves bitterly hostile to all forms of learning.
That very singleness of purpose which had at first so materially
aided their progress developed into a one-sidedness which refused
to see any good outside their own body; and all who did not
actively assist them were persecuted. The final establishment of
Christianity in the East marks the end of the Greek scientific
schools, though nominally they continued to exist for two
hundred years more.


\section*{The Athenian School (in the fifth century).\nblabel{png132c}\protect\footnote
  {See \emph{Untersuchungen über die neu aufgefundenen Scholien des Proklus},
  by J.~H. Knoche, Herford, 1865.}
}\index{Knoche on Proclus}\index{Athens,!second School of|(}

The hostility of the Eastern church to Greek science is further
illustrated by the fall of the later Athenian school. This school
occupies but a small space in our history. Ever since Plato's
time a certain number of professional mathematicians had lived
at Athens; and about the year 420 this school again acquired
considerable reputation, largely in consequence of the numerous
students who after the murder of Hypatia migrated there
from Alexandria. Its most celebrated members were \index{Proclus}Proclus,
\index{Damascius}Damascius, and \index{Eutocius}Eutocius.

\nblabel{png133a}\nbparagraph{Proclus.} \emph{Proclus} was born at Constantinople in February
412 and died at Athens on April~17, 485. He wrote a commentary\index{Friedlein, G., ref.\ to}\footnote
  {It has been edited by G.~Friedlein, Leipzig, 1873.}
on the first book of \index{Euclid's \emph{Elements}!ref.\ to}Euclid's \emph{Elements}, which contains
a great deal of valuable information on the history of Greek
mathematics: he is verbose and dull, but luckily he has preserved
for us quotations from other and better authorities.
Proclus was succeeded as head of the school by \index{Marinus of Athens}\nbname{Marinus}, and
Marinus by \index{Isidorus of Athens}\nbname{Isidorus}.

\nbparagraph{Damascius.} \nbparagraph{Eutocius.} Two pupils of Isidorus, who in
their turn subsequently lectured at Athens, may be mentioned
in passing. One of these, \emph{Damascius} of Damascus, circ.~490,
is commonly said to have added to \index{Eucn@Euc.!n15@\textsc{xv}. ref.\ to}Euclid's \emph{Elements} a fifteenth
book on the inscription of one \index{Polyhedrons, regular}regular solid in another, but his
authorship of this has been questioned by some writers. The other,
\emph{Eutocius}, circ.~510, wrote commentaries on the first four books
of the \emph{Conics} of \index{Apollonius!ref.\ to}Apollonius and on various works of \index{Archimedes!ref.\ to}Archimedes.

\index{Christians (Eastern Church) opposed to Greek science}This later Athenian school was carried on under great
difficulties owing to the opposition of the Christians. Proclus,
for example, was repeatedly threatened with death because he
was ``a philosopher.'' His remark, ``after all my body does
not matter, it is the spirit that I shall take with me when
I die,'' which he made to some students who had offered to
defend him, has been often quoted. The Christians, after
several ineffectual attempts, at last got a decree from \index{Justinian, the Emperor}Justinian
in 529 that ``heathen learning'' should no longer be studied at
Athens. That date therefore marks the end of the Athenian school.

The church at Alexandria was less influential, and the city
was more remote from the centre of civil power. The schools
there were thus suffered to continue, though their existence was
of a precarious character. Under these conditions mathematics
continued to be read in Egypt for another hundred years, but
all interest in the study had gone.\index{Athens,!second School of|)}

\section*{Roman Mathematics\nblabel{png134a}\index{Hankel, ref.\ to}\protect\footnote
  {The subject is discussed by Cantor, chaps.~xxv,~xxvi, and xxvii; also
  by Hankel, pp.~294--304.}
}\index{Cantor, M., ref.\ to}

\index{Rome, mathematics at|(}\index{Roman mathematics|(}\nblabel{png134b}I ought not to conclude this part of the history without any
mention of Roman mathematics, for it was through Rome that
mathematics first passed into the curriculum of medieval Europe,
and in Rome all modern history has its origin. There is, however,
very little to say on the subject. The chief study of the
place was in fact the art of government, whether by law, by
persuasion, or by those material means on which all government
ultimately rests. There were, no doubt, professors who could
teach the results of Greek science, but there was no demand for
a school of mathematics. Italians who wished to learn more
than the elements of the science went to \index{Alexandria, university of}Alexandria or to places
which drew their inspiration from Alexandria.

The subject as taught in the mathematical schools at Rome
seems to have been confined in arithmetic to the art of calculation
(no doubt by the aid of the \index{Abacus,!ref.\ to}abacus) and perhaps some of
the easier parts of the work of \index{Nicomachus!ref.\ to}Nicomachus, and in geometry
to a few practical rules; though some of the arts founded on
a knowledge of mathematics (especially that of surveying) were
carried to a high pitch of excellence. It would seem also that
special attention was paid to the representation of numbers by
signs. \nblabel{xref:113}The manner of indicating numbers up to ten by the use
of fingers must have been in practice from quite early times, but
about the first century it had been developed by the Romans
into a \index{Bede on finger symbolism}\index{Finger symbolism}finger-symbolism by which numbers up to 10,000 or
perhaps more could be represented: this would seem to have
been taught in the Roman schools. It is described by Bede,
and therefore would seem to have been known as far west as
Britain; \index{Jerome on finger symbolism}Jerome also alludes to it; its use has still survived in
the Persian bazaars.

I am not acquainted with any Latin work on the principles
of mechanics, but there were numerous books on the practical
side of the subject which dealt elaborately with architectural
and engineering problems. We may judge what they were like
by the \index{Mathematici Veteres@\emph{Mathematici Veteres}, the}\emph{Mathematici Veteres}, which is a collection of various
short treatises on catapults, engines of war, \&c.: and by the
\index{Kes@\mytextgreek{Kesto'i}}\mytextgreek{Kesto'i}, written by \index{Africanus, Julius}Sextus Julius Africanus about the end of
the second century, part of which is included in the \emph{Mathematici
Veteres}, which contains, amongst other things, rules for finding
the breadth of a river when the opposite bank is occupied by an
enemy, how to signal with a semaphore, \&c.

In the sixth century \index{Boethius!ref.\ to}Boethius published a geometry containing
a few propositions from \index{Euclid's \emph{Elements}!ref.\ to}Euclid and an arithmetic founded on
that of \index{Nicomachus!ref.\ to}Nicomachus; and about the same time \index{Cassiodorus!ref.\ to}Cassiodorus
discussed the foundation of a liberal education which, after the
preliminary \index{Trivium, the}trivium of grammar, logic, and rhetoric, meant the
\index{Music, in the quadrivium}\index{Quadrivium}quadrivium of arithmetic, geometry, music, and astronomy.
These works were written at Rome in the closing years of
the Athenian and Alexandrian schools, and I therefore mention
them here, but as their only value lies in the fact that they
became recognized text-books in medieval education I postpone
their consideration to chapter \hyperref[chapter:VIII]{\textsc{viii}}.

Theoretical mathematics was in fact an exotic study at Rome;
not only was the genius of the people essentially practical, but,
alike during the building of their empire, while it lasted, and under
the Goths, all the conditions were unfavourable to abstract science.

\nblabel{png135a}On the other hand, Alexandria was exceptionally well placed
to be a centre of science. From the foundation of the city to
its capture by the Mohammedans it was disturbed neither by
foreign nor by civil war, save only for a few years when the
rule of the \index{Ptolemies, dynasty of}Ptolemies gave way to that of Rome: it was wealthy,
and its rulers took a pride in endowing the university: and
lastly, just as in commerce it became the meeting-place of the
east and the west, so it had the good fortune to be the dwelling-place
alike of Greeks and of various Semitic people; the one
race shewed a peculiar aptitude for geometry, the other for
sciences which rest on measurement. Here too, however, as
time went on the conditions gradually became more unfavourable,
the endless discussions on theological dogmas and the
increasing insecurity of the empire tending to divert men's
thoughts into other channels.\index{Roman mathematics|)}\index{Rome, mathematics at|)}

\section*{End of the Second Alexandrian School.\nblabel{png136a}}

The precarious existence and unfruitful history of the last
two centuries of the second Alexandrian School need no record.
In 632 \index{Mohammed, ref.\ to}Mohammed died, and within ten years his successors
had subdued Syria, Palestine, Mesopotamia, Persia, and Egypt.
The precise date on which Alexandria fell is doubtful, but the
most reliable Arab historians give December~10, 641---a date
which at any rate is correct within eighteen months.

\nblabel{png136b}With the fall of Alexandria the long history of Greek
mathematics came to a conclusion. It seems probable that
the greater part of the famous \index{Alexandria, university of}\index{Alexandrian,!library}university library and museum
had been destroyed by the \index{Christians (Eastern Church) opposed to Greek science}Christians a hundred or two
hundred years previously, and what remained was unvalued
and neglected. Some two or three years after the first capture
of Alexandria a serious revolt occurred in Egypt, which was
ultimately put down with great severity. I see no reason to
doubt the truth of the account that after the capture of the
city the Mohammedans destroyed such university buildings and
collections as were still left. It is said that, when the Arab
commander ordered the library to be burnt, the Greeks made
such energetic protests that he consented to refer the matter to
the \index{Omar, Caliph, ref.\ to}caliph Omar. The caliph returned the answer, ``As to the
books you have mentioned, if they contain what is agreeable
with the book of God, the book of God is sufficient without
them; and, if they contain what is contrary to the book of God,
there is no need for them; so give orders for their destruction.''
The account goes on to say that they were burnt in the public baths
of the city, and that it took six months to consume them all.

\nbChapter{VI}{The Byzantine School}{}{641--1453.}{}
\index{Byzantine School, chapter \textsc{vi}}\nblabel{png137a}

\First{It} will be convenient to consider the Byzantine school in
connection with the history of Greek mathematics. After the
capture of Alexandria by the Mohammedans the majority of the
philosophers, who previously had been teaching there, migrated
to Constantinople, which then became the centre of Greek learning
in the East and remained so for 800 years. \nblabel{png137b}But though
the history of the Byzantine school stretches over so many
years---a period about as long as that from the Norman Conquest
to the present day---it is utterly barren of any scientific
interest; and its chief merit is that it preserved for us the
works of the different Greek schools. The revelation of these
works to the West in the fifteenth century was one of the most
important sources of the stream of modern European thought,
and the history of the Byzantine school may be summed up by
saying that it played the part of a conduit-pipe in conveying to
us the results of an earlier and brighter age.

The time was one of constant war, and men's minds during
the short intervals of peace were mainly occupied with theological
subtleties and pedantic scholarship. I should not have
mentioned any of the following writers had they lived in the
Alexandrian period, but in default of any others they may be
noticed as illustrating the character of the school. I ought also,
perhaps, to call the attention of the reader explicitly to the fact
that I am here departing from chronological order, and that the
mathematicians mentioned in this chapter were contemporaries
of those discussed in the chapters devoted to the mathematics
of the middle ages. The Byzantine school was so isolated that
I deem this the best arrangement of the subject.

\index{Hero of Constantinople}\nblabel{png138a}\nbparagraph{Hero.} One of the earliest members of the Byzantine school
was \emph{Hero of Constantinople}, circ.~900, sometimes called the
younger to distinguish him from Hero of Alexandria. Hero
would seem to have written on geodesy and mechanics as applied
to engines of war.

During the tenth century two emperors, \index{Leo VI. of Constantinople}Leo~VI. and \index{Constantine VII., the Emperor}Constantine~VII.,
shewed considerable interest in astronomy and
mathematics, but the stimulus thus given to the study of these
subjects was only temporary.

\index{Psellus}\nbparagraph{Psellus.} In the eleventh century \emph{Michael Psellus}, born in
1020, wrote a pamphlet\footnote
  {It was printed at Bâle in 1536. Psellus also wrote a \emph{Compendium
  Mathematicum} which was printed at Leyden in 1647.}
on the \index{Quadrivium}quadrivium: it is now in the
National Library at Paris.

In the fourteenth century we find the names of three monks
who paid attention to mathematics.

\nblabel{xref:117}\index{Barlaam|(}\index{Planudes}\nbparagraph{Planudes.} \nbparagraph{Barlaam.} \nbparagraph{Argyrus.} The first of the three
was \emph{Maximus Planudes}.\index{Xylander!ref.\ to}\index{Gerhardt, ref.\ to}\footnote
  {His arithmetical commentary was published by Xylander, Bâle, 1575:
  his work on Hindoo arithmetic, edited by C.~J.~Gerhardt, was published at
  Halle, 1865.}
He wrote a commentary on the
first two books of the \emph{Arithmetic} of \index{Diophantus!ref.\ to}Diophantus; a work on
\index{Indian numerals}Hindoo arithmetic in which he used the \index{Arabic numerals}Arabic numerals;
and another on proportions which is now in the National
Library at Paris. The next was a Calabrian monk named
\emph{Barlaam}, who was born in 1290 and died in 1348. He
was the author of a work, \emph{Logistic}, on the Greek methods
of calculation from which we derive a good deal of information
as to the way in which the Greeks treated numerical
fractions.\footnote
  {Barlaam's \emph{Logistic}, edited by Dasypodius, was published at Strassburg,
  1572; another edition was issued at Paris in 1600.}
Barlaam seems to have been a man of great
intelligence. He was sent as an ambassador to the Pope at
Avignon, and acquitted himself creditably of a difficult mission;
while there he taught Greek to \index{Petrarch}Petrarch. He was famous at
Constantinople for the ridicule he threw on the preposterous
pretensions of the monks at \index{Athos, Mount}Mount Athos who taught that those
who joined them could, by steadily regarding their bodies,
see a mystic light which was the essence of God. Barlaam
advised them to substitute the light of reason for that of their
bodies---a piece of advice which nearly cost him his life.
The last of these monks was \index{Argyrus}\emph{Isaac Argyrus}, who died in 1372.
He wrote three astronomical tracts, the manuscripts of which
are in the libraries at the Vatican, Leyden, and Vienna: one
on geodesy, the manuscript of which is at the Escurial: one
on geometry, the manuscript of which is in the National Library
at Paris: one on the arithmetic of \index{Nicomachus!ref.\ to}Nicomachus, the manuscript
of which is in the National Library at Paris: and one on
trigonometry, the manuscript of which is in the Bodleian at
Oxford.\index{Barlaam|)}

\index{Rhabdas}\index{Nicholas Rhabdas of Smyrna}\nblabel{png139a}\nbparagraph{Rhabdas.} In the fourteenth or perhaps the fifteenth century
\emph{Nicholas Rhabdas} of \emph{Smyrna} wrote two papers\index{Tannery, S. P., ref.\ to}\footnote
  {They have been edited by S.~P.~Tannery, Paris, 1886.}
on arithmetic
which are now in the National Library at Paris. He gave an
account of the \index{Finger symbolism}finger-symbolism\footnote
  {See above, page~\pageref{xref:113}.}
which the Romans had introduced
into the East and was then current there.

\index{Pachymeres}\nbparagraph{Pachymeres.} Early in the fifteenth century \emph{Pachymeres}
wrote tracts on arithmetic, geometry, and four mechanical
machines.

\index{Moschopulus|(}\nbparagraph{Moschopulus.} A few years later \emph{Emmanuel Moschopulus},
who died in Italy circ.~1460, wrote a treatise on \index{Magic squares|(}magic squares.
A \emph{magic square}\index{Gunther@Günther, S.}\footnote
  {On the formation and history of magic squares, see \index{Ball, W. W. R., ref.\ to}my \emph{Mathematical
  Recreations}, London, ninth edition, 1920, chap.~vii. On the work of
  Moschopulus, see S.~Günther's \emph{Geschichte der mathematischen Wissenschaften},
  Leipzig, 1876, chap.~iv.}
consists of a number of integers arranged in
the form of a square so that the sum of the numbers in every
row, in every column, and in each diagonal is the same. If the
integers be the consecutive numbers from 1 to $n^{2}$, the square is
said to be of the $n$th order, and in this case the sum of the
numbers in any row, column, or diagonal is equal to $\frac{1}{2} n(n^{2} + 1)$.
Thus the first 16 integers, arranged in either of the forms given
below, form a magic square of the fourth order, the sum of
the numbers in every row, every column, and each diagonal
being 34.\index{Magic squares|)}

\begin{figure*}[!hbt]
\centering
\hspace*{\fill}
\begin{minipage}{0.3\textwidth}
\centering
\begin{MagicSquare}{4}
1 & {15} & {14} & 4 \\
{12} & 6 & 7 & 9 \\
8 & {10} & {11} & 5 \\
{13} & 3 & 2 & {16}
\end{MagicSquare}
\end{minipage}
\hfill
\begin{minipage}{0.3\textwidth}
\centering
\begin{MagicSquare}{4}
{15} & {10} & 3 & 6 \\
4 & 5 & {16} & 9 \\
{14} & {11} & 2 & 7 \\
1 & 8 & {13} & {12} \\
\end{MagicSquare}
\end{minipage}
\hspace*{\fill}
\end{figure*}

In the mystical philosophy then current certain metaphysical
ideas were often associated with particular numbers, and thus it
was natural that such arrangements of numbers should attract
attention and be deemed to possess magical properties. The
theory of the formation of magic squares is elegant, and several
distinguished mathematicians have written on it, but, though
interesting, I need hardly say it is not useful. Moschopulus
seems to have been the earliest European writer who attempted
to deal with the mathematical theory, but his rules apply only
to odd squares. The astrologers of the fifteenth and sixteenth
centuries were much impressed by such arrangements. In
particular the famous \index{Cornelius Agrippa, ref.\ to}\index{Agrippa, Cornelius, ref.\ to}Cornelius Agrippa (1486--1535) constructed
magic squares of the orders 3, 4, 5, 6, 7, 8, 9, which were associated
respectively with the \index{Planets, astrological}seven astrological ``planets,'' namely,
Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon.
He taught that a square of one cell, in which unity was inserted,
represented the unity and eternity of God; while the fact that
a square of the second order could not be constructed illustrated
the imperfection of the four elements, air, earth, fire, and water;
and later writers added that it was symbolic of original sin. A
magic square engraved on a silver plate was often prescribed as
a charm against the plague, and one (namely, that in the first
diagram on the last page) is drawn in the picture of melancholy
painted about the year 1500 by \index{Durer@Dürer!ref.\ to}Albrecht Dürer. Such charms
are still worn in the East.\index{Moschopulus|)}

\index{Constantinople, fall of}\nblabel{png141a}Constantinople was captured by the Turks in 1453, and the
last semblance of a Greek school of mathematics then disappeared.
Numerous Greeks took refuge in Italy. In the
West the memory of Greek science had vanished, and even the
names of all but a few Greek writers were unknown; thus the
books brought by these refugees came as a revelation to Europe,
and, as we shall see later, gave a considerable stimulus to the
study of science.


\nbChapter{VII}{Systems of Numeration and Primitive Arithmetic}{Systems of Numeration}{}
  {The subject of this chapter has been discussed by Cantor and by Hankel.
  See also the \emph{Philosophy of Arithmetic} by John Leslie, second edition, Edinburgh,
  1820. Besides these authorities the article on \emph{Arithmetic} by George Peacock
  in the \emph{Encyclopaedia Metropolitana, Pure Sciences,} London, 1845; E.~B.~Tylor's
  \emph{Primitive Culture,} London, 1873; \emph{Les signes numéraux et l'arithmétique
  chez les peuples de l'antiquité \dots }by T.~H.~Martin, Rome, 1864; and
  \emph{Die Zahlzeichen \dots }by G.~Friedlein, Erlangen, 1869, should be consulted.}
\index{Arithmetic.!z1@Primitive, chapter \textsc{vii}}\index{Peacock!ref.\ to}\index{Tylor, E. B., ref.\ to}\index{Leslie on arithmetic}\index{Martin, ref.\ to}\index{Cantor, M., ref.\ to}\index{Friedlein, G., ref.\ to}\index{Hankel, ref.\ to}\nblabel{png142a}

\index{Numerals, symbols for|(}\nblabel{png142b}\First{I have} in many places alluded to the Greek method of expressing
numbers in writing, and I have thought it best to defer to
this chapter the whole of what I wanted to say on the various
systems of numerical notation which were displaced by the
system introduced by the Arabs.

\nblabel{png142c}First, as to symbolism and language. The plan of indicating
numbers by the digits of one or both hands is so natural that we
find it in universal use among early races, and the members of
all tribes now extant are able to indicate by signs numbers at
least as high as ten: it is stated that in some languages the
names for the first ten numbers are derived from the \index{Finger symbolism}fingers used
to denote them. For larger numbers we soon, however, reach a
limit beyond which primitive man is unable to count, while as
far as language goes it is well known that many tribes have no
word for any number higher than ten, and some have no word
for any number beyond four, all higher numbers being expressed
by the words plenty or \index{Heap for unknown number}heap: in connection with this it is worth
remarking that (as stated above) the Egyptians used the \index{Unknown quantity, word or symbol for}symbol
for the word heap to denote an unknown quantity in algebra.

\index{Five, things counted by|(}\index{Hand used to denote five}The number five is generally represented by the open hand,
and it is said that in almost all languages the words five and
hand are derived from the same root. It is possible that in
early times men did not readily count beyond five, and things if
more numerous were counted by multiples of it. It may be that
the Roman symbol X for ten represents two ``V''s, placed apex
to apex, and, if so, this seems to point to a time when things were
counted by fives.\footnote
  {See also the \emph{Odyssey}, iv, 413--415, in which apparently reference is made
  to a similar custom.}
In connection with this it is worth noticing that
both in Java and among the Aztecs a week consisted of five days.\index{Five, things counted by|)}

The members of nearly all races of which we have now any
knowledge seem, however, to have used the digits of both hands
to represent numbers. They could thus count up to and including
ten, and therefore were led to take ten as their radix of
notation. In the English language, for example, all the words
for numbers higher than ten are expressed on the decimal
system: those for 11 and 12, which at first sight seem to be
exceptions, being derived from Anglo-Saxon words for one and
ten and two and ten respectively.

Some tribes seem to have gone further, and by making use of
their toes were accustomed to count by multiples of twenty.
The Aztecs, for example, are said to have done so. \index{Scores, things counted by}It may be
noticed that we still count some things (for instance, sheep) by
scores, the word score signifying a notch or scratch made on the
completion of the twenty; while the French also talk of quatrevingts,
as though at one time they counted things by multiples
of twenty. I am not, however, sure whether the latter argument
is worth anything, for I have an impression that I have
seen the word \index{Octante for eighty}\emph{octante} in old French books; and there is no
question\footnote
  {See, for example, \index{Kempten, de}\index{Dekempten@De Kempten, ref.\ to}V.~M.~de Kempten's \emph{Practique\ldots à ciffrer}, Antwerp,
  1556.}
that \index{Septante for seventy}\emph{septante} and \index{Nonante for ninety}\emph{nonante} were at one time common
words for seventy and ninety, and indeed they are still retained
in some dialects.

The only tribes of whom I have read who did not count in
terms either of five or of some multiple of five are the Bolans
of West Africa who are said to have counted by multiples of
seven, and the Maories who are said to have counted by
multiples of eleven.

Up to ten it is comparatively easy to count, but primitive
people find great difficulty in counting higher numbers;
apparently at first this difficulty was only overcome by the
method (still in use in South Africa) of getting two men, one
to count the units up to ten on his fingers, and the other to
count the number of groups of ten so formed. To us it is
obvious that it is equally effectual to make a mark of some
kind on the completion of each group of ten, but it is alleged
that the members of many tribes never succeeded in counting
numbers higher than ten unless by the aid of two men.

\nblabel{png144a}\index{Abacus,!description of|(}Most races who shewed any aptitude for civilization proceeded
further and invented a way of representing numbers by
means of pebbles or counters arranged in sets of ten; and this
in its turn developed into the abacus or swan-pan. This instrument
was in use among nations so widely separated as the
Etruscans, Greeks, Egyptians, Hindoos, Chinese, and Mexicans;
and was, it is believed, invented independently at several
different centres. It is still in common use in Russia, China,
and Japan.

\begin{center}
\labelincludegraphics[width=0.5\textwidth]{145-fig1}\\
\small\textsc{Figure~1.}
\end{center}

In its simplest form (see \figureref{145-fig1}{Figure~1}) the abacus %[**TN: Removed ', on the next page']
consists of a wooden board with a number of grooves cut in it,
or of a table covered with sand in which grooves are made with
the fingers. To represent a number, as many counters or pebbles
are put on the first groove as there are units, as many on the
second as there are tens, and so on. When by its aid a number
of objects are counted, for each object a pebble is put on the
first groove; and, as soon as there are ten pebbles there, they
are taken off and one pebble put on the second groove; and so
on. It was sometimes, as in the Aztec \emph{quipus,} made with a
number of parallel wires or strings stuck in a piece of wood on
which beads could be threaded; and in that form is called a
swan-pan. In the number represented in each of the instruments
drawn on the next page there are seven thousands, three
hundreds, no tens, and five units, that is, the number is 7305.
Some races counted from left to right, others from right to left,
but this is a mere matter of convention.

The Roman abaci seem to have been rather more elaborate.
They contained two marginal grooves or wires, one with four
beads to facilitate the addition of fractions whose denominators
were four, and one with twelve beads for fractions whose
denominators were twelve: but otherwise they do not differ in
principle from those described above. They were commonly
made to represent numbers up to 100,000,000. The Greek
abaci were similar to the Roman ones. The Greeks and Romans
used their abaci as boards on which they played a game something like backgammon.

\begin{center}
\labelincludegraphics[width=0.5\textwidth]{145-fig2}\\
\small\textsc{Figure~2.}
\end{center}

In the Russian \emph{tschotü} (\figureref{145-fig2}{Figure~2}) the instrument is improved
by having the wires set in a rectangular frame, and ten (or nine)
beads are permanently threaded on each of the wires, the wires
being considerably longer than is necessary to hold them. If
the frame be held horizontal, and all the beads be towards one
side, say the lower side of the frame, it is possible to represent
any number by pushing towards the other or upper side as
many beads on the first wire as there are units in the number,
as many beads on the second wire as there are tens in the
number, and so on. Calculations can be made somewhat more
rapidly if the five beads on each wire next to the upper side
be coloured differently to those next to the lower side, and they
can be still further facilitated if the first, second, \dots , ninth
counters in each column be respectively marked with symbols
for the numbers 1, 2, \dots , 9. Gerbert\footnote
  {See below, page~\pageref{xref:138}.}
is said to have introduced
the use of such marks, called \index{Apices}apices, towards the close
of the tenth century.

\begin{center}
\labelincludegraphics[width=0.5\textwidth]{145-fig3}\\
\small\textsc{Figure~3.}
\end{center}

\figureref{145-fig3}{Figure~3} represents the form of swan-pan or saroban in
common use in China and Japan. There the development is
carried one step further, and five beads on each wire are replaced
by a single bead of a different form or on a different division,
but apices are not used. I am told that an expert Japanese can,
by the aid of a swan-pan, add numbers as rapidly as they can
be read out to him. It will be noticed that the instrument
represented in \figureref{145-fig3}{Figure~3} is made so that two numbers can be
expressed at the same time on it.

The use of the abacus in addition and subtraction is evident.
It can be used also in multiplication and division; rules for these
processes, illustrated by examples, are given in various old works
on arithmetic.\footnote
  {For example in \index{Record!ref.\ to}R.~Record's \emph{Grounde of Artes}, edition of 1610, London,
  pp.~225--262.}

The abacus obviously presents a concrete way of representing
a number in the decimal system of notation, that is, by means
of the local value of the digits. Unfortunately the method of
writing numbers developed on different lines, and it was not
until about the thirteenth century of our era, when a symbol
zero used in conjunction with nine other symbols was introduced,
that a corresponding notation in writing was adopted in Europe.\index{Abacus,!description of|)}

\nblabel{png147a}Next, as to the means of representing numbers in writing.
In general we may say that in the earliest times a number
was (if represented by a sign and not a word) indicated by the
requisite number of strokes. \index{Finger symbolism}Thus in an inscription from
Tralles in Caria of the date 398~\textsc{b.c.}\ the phrase seventh year is
represented by \mytextgreek{eteos} $|\ |\ |\ |\ |\ |\ |$. These strokes may have been
mere marks; or perhaps they originally represented fingers,
since in the Egyptian hieroglyphics the symbols for the
numbers 1, 2, 3, are one, two, and three fingers respectively,
though in the later hieratic writing these symbols had become
reduced to straight lines. Additional symbols for 10 and 100
were soon introduced: and the oldest extant Egyptian and
Phoenician writings repeat the symbol for unity as many times
(up to 9) as was necessary, and then repeat the symbol for ten
as many times (up to 9) as was necessary, and so on. No
specimens of Greek numeration of a similar kind are in existence,
but there is every reason to believe the testimony of \index{Iamblichus!ref.\ to}Iamblichus
who asserts that this was the method by which the Greeks first
expressed numbers in writing.

\nblabel{png147b}This way of representing numbers remained in current use
throughout Roman history; and for greater brevity they or
the Etruscans added separate signs for 5, 50, \&c. \index{Roman mathematics!symbols for numbers}The Roman
symbols are generally merely the initial letters of the names of
the numbers; thus \textsc{c} stood for centum or 100, \textsc{m} for mille or
1000. The symbol \textsc{v} for 5 seems to have originally represented
an open palm with the thumb extended. The symbols \textsc{l} for 50
and \textsc{d} for 500 are said to represent the upper halves of the
symbols used in early times for \textsc{c} and \textsc{m}\@. The subtractive forms
like \textsc{iv} for \textsc{iiii} are probably of a later origin.

\index{Attic symbols for numbers|(}Similarly in Attica five was denoted by $\Pi$, the first letter of
\mytextgreek{p'ente}, or sometimes by $\Gamma$; ten by $\Delta$, the initial letter of \mytextgreek{d'eka}; a
hundred by $\Eta$ for \mytextgreek{<ekat'on}; a thousand by $\Chi$ for \mytextgreek{q'ilioi}; while
50 was represented by a $\Delta$ written inside a $\Pi$; and so on.
These Attic symbols continued to be used for inscriptions and
formal documents until a late date.\index{Attic symbols for numbers|)}

\index{Arithmetic.!z5@Theory of,!x2@and thenceforward treated empirically (Boethian arithmetic) by most of the Greek and European mathematicians to the end of the fourteenth century after Christ|(}\index{Arithmetic.!z4@Practical Greek}This, if a clumsy, is a perfectly intelligible system; but the
Greeks at some time in the third century before Christ abandoned
it for one which offers no special advantages in denoting a given
number, while it makes all the operations of arithmetic exceedingly
difficult. \nblabel{png148a}In this, which is known from the place where it
was introduced as the \index{Alexandrian,!symbols for numbers}Alexandrian system, the numbers from 1
to 9 are represented by the first nine letters of the alphabet;
the tens from 10 to 90 by the next nine letters; and the
hundreds from 100 to 900 by the next nine letters. To do this
the Greeks wanted 27 letters, and as their alphabet contained
only 24, they reinserted two letters (the digamma and koppa)
which had formerly been in it but had become obsolete, and
introduced at the end another symbol taken from the Phoenician
alphabet. \nblabel{png148b}Thus the ten letters $\alpha$ to $\iota$ stood respectively for the
numbers from 1 to 10; the next eight letters for the multiples
of 10 from 20 to 90; and the last nine letters for 100, 200, etc.,
up to 900. Intermediate numbers like 11 were represented as
the sum of 10 and 1, that is, by the symbol $\iota\alpha'$. This afforded
a notation for all numbers up to 999; and by a system of
suffixes and indices it was extended so as to represent numbers
up to 100,000,000.

\index{Inductive arithmetic|(}There is no doubt that at first the results were obtained by
the use of the \index{Abacus,!ref.\ to}abacus or some similar mechanical method, and
that the signs were only employed to record the result; the idea
of operating with the symbols themselves in order to obtain the
results is of a later growth, and is one with which the Greeks
never became familiar. \index{Multiplication, processes of|(}The non-progressive character of Greek
arithmetic may be partly due to their unlucky adoption of the
Alexandrian system which caused them for most practical purposes
to rely on the abacus, and to supplement it by a table of
multiplications which was learnt by heart. The results of the
multiplication or division of numbers other than those in the
multiplication table might have been obtained by the use of the
abacus, but in fact they were generally got by repeated additions
and subtractions. Thus, as late as 944, a certain mathematician
who in the course of his work wants to multiply 400 by
5 finds the result by addition. The same writer, when he wants
to divide 6152 by 15, tries all the multiples of 15 until he gets
to 6000, this gives him 400 and a remainder 152; he then
begins again with all the multiples of 15 until he gets to 150,
and this gives him 10 and a remainder 2. Hence the answer is
410 with a remainder 2.\index{Multiplication, processes of|)}\index{Inductive arithmetic|)}

\index{Arithmetic.!z4@Practical Greek}\index{Multiplication, processes of}A few mathematicians, however, such as \index{Hero of Alexandria!ref.\ to}Hero of Alexandria,
\index{Theon of Alexandria!ref.\ to}Theon, and \index{Eutocius!ref.\ to}Eutocius, multiplied and divided in what is essentially
the same way as we do. Thus to multiply 18 by 13 they
proceeded as follows:---
\begin{flalign*}
\iota\gamma + \iota\eta &= (\iota + \gamma)(\iota + \eta)            &    13×18&=(10+3) (10+8) \\
                        &=\iota(\iota + \eta) + \gamma (\iota + \eta)&         &=10 (10+8) +3 (10+8) \\
                        &=\rho + \pi + \lambda + \kappa\delta        &         &=100+80+30+24      \\
                        &=\sigma\lambda\delta                        &         &=234
\end{flalign*}
I suspect that the last step, in which they had to add four
numbers together, was obtained by the aid of the abacus.\index{Arithmetic.!z5@Theory of,!x2@and thenceforward treated empirically (Boethian arithmetic) by most of the Greek and European mathematicians to the end of the fourteenth century after Christ|)}

These, however, were men of exceptional genius, and we must
recollect that for all ordinary purposes the art of calculation was
performed only by the use of the abacus and the multiplication
table, while the term arithmetic was confined to the theories of
ratio, proportion, and of numbers.\index{Numerals, symbols for|)}

\nblabel{png149a}All the systems here described were more or less clumsy, and
they have been displaced among civilized races by the \index{Indian numerals}\index{Arabic numerals}Arabic
system in which there are ten digits or symbols, namely, nine
for the first nine numbers and another for zero. In this system
an integral number is denoted by a succession of digits, each
digit representing the product of that digit and a power of ten,
and the number being equal to the sum of these products.
Thus, by means of the local value attached to nine symbols and
a symbol for zero, any number in the decimal scale of notation
can be expressed. The history of the development of the science
of arithmetic with this notation will be considered below in
chapter~\hyperref[chapter:XI]{\textsc{xi}}.

\nbpart{SECOND PERIOD}{Mathematics of the Middle Ages and Renaissance}{middle}
\nblabel{png150a}

\emph{This period begins about the sixth century, and may be said
to end with the invention of analytical geometry and of the
infinitesimal calculus. The characteristic feature of this period
is the creation or development of modern arithmetic, algebra,
and trigonometry.}
\medskip

In this period I consider first, in chapter~\hyperref[chapter:VIII]{\textsc{viii}}, the rise of
learning in Western Europe, and the mathematics of the middle
ages. Next, in chapter~\hyperref[chapter:IX]{\textsc{ix}}, I discuss the nature and history of
Hindoo and Arabian mathematics, and in chapter~\hyperref[chapter:X]{\textsc{x}} their introduction
into Europe. Then, in chapter~\hyperref[chapter:XI]{\textsc{xi}}, I trace the subsequent
progress of arithmetic to the year 1637. Next, in chapter~\hyperref[chapter:XII]{\textsc{xii}},
I treat of the general history of mathematics during the
renaissance, from the invention of printing to the beginning of
the seventeenth century, say, from 1450 to 1637; this contains
an account of the commencement of the modern treatment of
arithmetic, algebra, and trigonometry. Lastly, in chapter~\hyperref[chapter:XIII]{\textsc{xiii}},
I consider the revival of interest in mechanics, experimental
methods, and pure geometry which marks the last few
years of this period, and serves as a connecting link between the
mathematics of the renaissance and the mathematics of modern
times.

\nbChapter{VIII}{The Rise of Learning in Western Europe}{The Rise of Learning in Europe}{Circ.~600--1200.}
  {The mathematics of this period has been discussed by Cantor, by
  S.~Günther, \emph{Geschichte des mathematischen Unterrichtes im deutschen
  Mittelalter}, Berlin, 1887; and by H.~Weissenborn, \emph{Gerbert, Beiträge zur
  Kenntniss der Mathematik des Mittelalters}, Berlin, 1888; and \emph{Zur Geschichte
  der Einführung der jetzigen Ziffers}, Berlin, 1892.}
\index{Geometry.!z4@Arab and medieval geometry, founded on Greek works, chapters \textsc{viii, ix, x}}\nblabel{png151a}\index{Cantor, M., ref.\ to}\index{Gunther@Günther, S.}\index{Weissenborn, ref.\ to}

\section*{Education in the sixth, seventh, and eighth centuries.\nblabel{png151b}}

\nblabel{png151c}\First{The} first few centuries of this second period of our history are
singularly barren of interest; and indeed it would be strange if
we found science or mathematics studied by those who lived in
a condition of perpetual war. \nblabel{png151d}\index{Monastic mathematics|(}Broadly speaking we may say
that from the sixth to the eighth centuries the only places of
study in western Europe were the \index{Benedictine monasteries}Benedictine monasteries.
We may find there some slight attempts at a study of literature;
but the science usually taught was confined to the use of the
\index{Abacus,!ref.\ to}abacus, the method of keeping accounts, and a knowledge of
the rule by which the date of Easter could be determined. Nor
was this unreasonable, for the monk had renounced the world,
and there was no reason why he should learn more science than
was required for the services of the Church and his monastery.
The traditions of Greek and Alexandrian learning gradually
died away. Possibly in Rome and a few favoured places copies
of the works of the great Greek mathematicians were obtainable
though with difficulty, but there were no students, the
books were unvalued, and in time became very scarce.

\index{Boethius|(}Three authors of the sixth century---Boethius, Cassiodorus,
and Isidorus---may be named whose writings serve as a connecting
link between the mathematics of classical and of
medieval times. As their works remained standard text-books
for some six or seven centuries it is necessary to mention them,
but it should be understood that this is the only reason for
doing so; they show no special mathematical ability. It will
be noticed that these authors were contemporaries of the later
Athenian and Alexandrian schools.

\nblabel{png152a}\nbparagraph{Boethius.} \emph{Anicius Manlius Severinus Boethius}, or as the
name is sometimes written \emph{Boetius}, born at Rome about 475
and died in 526, belonged to a family which for the two preceding
centuries had been esteemed one of the most illustrious
in Rome. It was formerly believed that he was educated at
Athens: this is somewhat doubtful, but at any rate he was
exceptionally well read in Greek literature and science.

Boethius would seem to have wished to devote his life to
literary pursuits; but recognizing ``that the world would be
happy only when kings became philosophers or philosophers
kings,'' he yielded to the pressure put on him and took an
active share in politics. He was celebrated for his extensive
charities, and, what in those days was very rare, the care that
he took to see that the recipients were worthy of them. He
was elected consul at an unusually early age, and took advantage
of his position to reform the coinage and to introduce the public
use of sun-dials, water-clocks, etc. He reached the height of
his prosperity in 522 when his two sons were inaugurated as
consuls. His integrity and attempts to protect the provincials
from the plunder of the public officials brought on him the
hatred of the Court. He was sentenced to death while absent
from Rome, seized at Ticinum, and in the baptistery of the
church there tortured by drawing a cord round his head till
the eyes were forced out of the sockets, and finally beaten
to death with clubs on October~23, 526. Such at least is the
account that has come down to us. At a later time his merits
were recognized, and tombs and statues erected in his honour by
the state.

Boethius was the last Roman of note who studied the
language and literature of Greece, and his works afforded to
medieval Europe some glimpse of the intellectual life of the
old world. His importance in the history of literature is thus
very great, but it arises merely from the accident of the time
at which he lived. After the introduction of \index{Aristotle!ref.\ to}Aristotle's works
in the thirteenth century his fame died away, and he has now
sunk into an obscurity which is as great as was once his
reputation. He is best known by his \emph{Consolatio}, which was
translated by \index{Alfred the Great, ref.\ to}Alfred the Great into Anglo-Saxon. For our
purpose it is sufficient to note that the teaching of early
medieval mathematics was mainly founded on his geometry
and arithmetic.

\nblabel{png153a}His \emph{Geometry}\footnote
  {His works on geometry and arithmetic were edited by \index{Friedlein, G., ref.\ to}G.~Friedlein,
  Leipzig, 1867.}
consists of the enunciations (only) of the first
book of \index{Euclid's \emph{Elements}!ref.\ to}Euclid, and of a few selected propositions in the third
and fourth books, but with numerous practical applications to
finding areas, etc. He adds an appendix with proofs of the
first three propositions to shew that the enunciations may be
relied on. His \emph{Arithmetic} is founded on that of \index{Nicomachus!ref.\ to}Nicomachus.\index{Boethius|)}

\index{Cassiodorus}\nblabel{png153b}\nbparagraph{Cassiodorus.} A few years later another Roman, \emph{Magnus
Aurelius Cassiodorus}, who was born about 490 and died in
566, published two works, \emph{De Institutione Divinarum Litterarum}
and \emph{De Artibus ac Disciplinis}, in which not only the
preliminary \index{Trivium, the}trivium of grammar, logic, and rhetoric were discussed,
but also the \index{Music, in the quadrivium|(}\index{Quadrivium}scientific quadrivium of arithmetic, geometry,
music, and astronomy. These were considered standard works
during the middle ages; the former was printed at Venice
in 1598.

\index{Isidorus of Seville|(}\nbparagraph{Isidorus.} \emph{Isidorus}, bishop of Seville, born in 570 and
died in 636, was the author of an encyclopaedic work in
twenty volumes called \emph{Origines}, of which the third volume is
given up to the quadrivium.  It was published at Leipzig in
1833.

\section*{The Cathedral and Conventual Schools.\nblabel{png154a}\protect\footnote
  {See \emph{The Schools of Charles the Great and the Restoration of Education
  in the Ninth Century} by J.~B.~Mullinger, London, 1877.}
}\index{Schools of Charles|(}\index{Mullinger, ref.\ to}\index{Conventual Schools|(}\index{Charf@Charles the Great}\index{Cathedral Schools, the|(}

\nblabel{png154b}When, in the latter half of the eighth century, Charles the
Great had established his empire, he determined to promote
learning so far as he was able. He began by commanding
that schools should be opened in connection with every
cathedral and monastery in his kingdom; an order which was
approved and materially assisted by the popes. It is interesting
to us to know that this was done at the instance and under the
direction of two Englishmen, \index{Alcuin}Alcuin and \index{Clement IV. of Rome, ref.\ to}\index{Clement, ref.\ to}Clement, who had
attached themselves to his court.\index{Isidorus of Seville|)}

\nblabel{png154c}\index{Alcuin}\nbparagraph{Alcuin.\footnote
  {See the life of Alcuin by \index{Lorentz on Alcuin}F.~Lorentz, Halle, 1829, translated by \index{Slee on Alcuin}J.~M.~Slee,
  London, 1837; \emph{Alcuin und sein Jahrhundert} by \index{Werner, ref.\ to}K.~Werner, Paderborn,
  1876; and \index{Cantor, M., ref.\ to}Cantor, vol.~i, pp.~712--721.}}
Of these the more prominent was \emph{Alcuin}, who
was born in Yorkshire in 735 and died at Tours in 804. He
was educated at York under archbishop \index{Egbert, Archbishop, ref.\ to}Egbert, his ``beloved
master,'' whom he succeeded as director of the school there.
Subsequently he became abbot of Canterbury, and was sent to
Rome by \index{Offa, ref.\ to}Offa to procure the \emph{pallium} for archbishop \index{Eanbald, Archbishop, ref.\ to}Eanbald.
On his journey back he met Charles at Parma; the emperor
took a great liking to him, and finally induced him to take up
his residence at the imperial court, and there teach rhetoric,
logic, mathematics, and divinity. \index{Alcuin}Alcuin remained for many
years one of the most intimate and influential friends of Charles
and was constantly employed as a confidential ambassador;
as such he spent the years 791 and 792 in England, and while
there reorganized the studies at his old school at York. In 801
he begged permission to retire from the court so as to be able
to spend the last years of his life in quiet: with difficulty he
obtained leave, and went to the abbey of St.~Martin at Tours,
of which he had been made head in 796. He established a
school in connection with the abbey which became very
celebrated, and he remained and taught there till his death on
May~19, 804.

Most of the extant writings of \index{Alcuin}Alcuin deal with theology
or history, but they include a collection of arithmetical propositions
suitable for the instruction of the young. The
majority of the propositions are easy problems, either determinate
or indeterminate, and are, I presume, founded on works
with which he had become acquainted when at Rome. The
following is one of the most difficult, and will give an idea of
the character of the work. If one hundred bushels of corn be
distributed among one hundred people in such a manner that
each man receives three bushels, each woman two, and each
child half a bushel: how many men, women, and children
were there? The general solution is $(20 - 3n)$ men, $5n$ women,
and $(80 - 2n)$ children, where $n$ may have any of the values
1, 2, 3, 4, 5, 6. \index{Alcuin}Alcuin only states the solution for which
$n = 3$; that is, he gives as the answer 11 men, 15 women, and
74 children.

This collection however was the work of a man of exceptional
genius, and probably we shall be correct in saying that
mathematics, if taught at all in a school, was generally confined
to the geometry of \index{Boethius!ref.\ to}Boethius, the use of the abacus and
multiplication table, and possibly the arithmetic of Boethius;
while except in one of these schools or in a \index{Benedictine monasteries}Benedictine cloister
it was hardly possible to get either instruction or opportunities
for study. It was of course natural that the works used should
come from Roman sources, for Britain and all the countries
included in the empire of \index{Charf@Charles the Great}Charles had at one time formed part
of the western half of the Roman empire, and their inhabitants
continued for a long time to regard Rome as the centre of
civilization, while the higher clergy kept up a tolerably constant
intercourse with Rome.

After the death of Charles many of his schools confined
themselves to teaching Latin, music, and theology, some
knowledge of which was essential to the worldly success of
the higher clergy. Hardly any science or mathematics was
taught, but the continued existence of the schools gave an
opportunity to any teacher whose learning or zeal exceeded
the narrow limits fixed by tradition; and though there were
but few who availed themselves of the opportunity, yet the
number of those desiring instruction was so large that it
would seem as if any one who could teach was sure to attract
a considerable audience.\index{Monastic mathematics|)}

A few schools, where the teachers were of repute, became
large and acquired a certain degree of permanence, but even in
them the teaching was still usually confined to the \index{Trivium, the}trivium
and \index{Quadrivium}quadrivium. The former comprised the three arts of
grammar, logic, and rhetoric, but practically meant the art
of reading and writing Latin; nominally the latter included
arithmetic and geometry with their applications, especially to
music and astronomy, but in fact it rarely meant more than
arithmetic sufficient to enable one to keep accounts, music for
the church services, geometry for the purpose of land-surveying,
and astronomy sufficient to enable one to calculate the feasts
and fasts of the church. The seven liberal arts are enumerated
in the line, \emph{Lingua, tropus, ratio; numerus, tonus, angulus,
astra}. Any student who got beyond the trivium was looked
on as a man of great erudition, \emph{Qui tria, qui septem, qui totum
scibile novit}, as a verse of the eleventh century runs. The
special questions which then and long afterwards attracted
the best thinkers were logic and certain portions of transcendental
theology and philosophy.\index{Music, in the quadrivium|)}

\nblabel{png156a}We may sum the matter up by saying that during the
ninth and tenth centuries the mathematics taught was still
usually confined to that comprised in the two works of
\index{Boethius!ref.\ to}Boethius together with the practical use of the abacus and the
multiplication table, though during the latter part of the time
a wider range of reading was undoubtedly accessible.

\index{Gerbert (Sylvester II.)}\nblabel{png156b}\nbparagraph{Gerbert.\footnote
  {\index{Weissenborn, ref.\ to}Weissenborn, in the works already mentioned, treats Gerbert very fully;
  see also \emph{La Vie et les \OE{}uvres de Gerbert}, by \index{Olleris on Gerbert}A.~Olleris, Clermont, 1867;
  \emph{Gerbert von Aurillac}, by \index{Werner, ref.\ to}K.~Werner, second edition, Vienna, 1881; and \emph{Gerberti
  \dots Opera mathematica}, edited by \index{Bubnov on Gerbert}N.~Bubnov, Berlin, 1899.}}
In the tenth century a man appeared who
would in any age have been remarkable and who gave a great
stimulus to learning. This was \emph{Gerbert}, an Aquitanian by
birth, who died in 1003 at about the age of fifty. His abilities
attracted attention to him even when a boy, and procured his
removal from the abbey school at Aurillac to the Spanish
march where he received a good education. He was in Rome
in 971, where his proficiency in music and astronomy excited
considerable interest: but his interests were not confined to
these subjects, and he had already mastered all the branches of
the trivium and quadrivium, as then taught, except logic; and
to learn this he moved to Rheims, which \index{Adalbero of Rheims}Archbishop Adalbero
had made the most famous school in Europe. Here he was at
once invited to teach, and so great was his fame that to him
\index{Capet, Hugh, ref.\ to}Hugh Capet entrusted the education of his son Robert who
was afterwards king of France.

Gerbert was especially famous for his construction of abaci
and of \index{Globes}terrestrial and celestial globes; he was accustomed to
use the latter to illustrate his lectures. These globes excited
great admiration; and he utilized this by offering to exchange
them for copies of classical Latin works, which seem already
to have become very scarce; the better to effect this he appointed
agents in the chief towns of Europe. To his efforts it
is believed we owe the preservation of several Latin works.
In 982 he received the abbey of Bobbio, and the rest of his life
was taken up with political affairs; he became Archbishop of
Rheims in 991, and of Ravenna in 998; in 999 he was elected
Pope, when he took the title of \index{Sylvester II.}Sylvester II.; as head of the
Church, he at once commenced an appeal to Christendom to arm
and defend the Holy Land, thus forestalling \index{Peter the Hermit, ref.\ to}Peter the Hermit by
a century, but he died on May~12, 1003, before he had time to
elaborate his plans. His library is, I believe, preserved in the
Vatican.

So remarkable a personality left a deep impress on his
generation, and all sorts of fables soon began to collect around
his memory. It seems certain that he made a clock which
was long preserved at Magdeburg, and an organ worked by
steam which was still at Rheims two centuries after his death.
All this only tended to confirm the suspicions of his contemporaries
that he had sold himself to the devil; and the details
of his interviews with that gentleman, the powers he purchased,
and his effort to escape from his bargain when he was dying,
may be read in the pages of \index{William of Malmesbury, ref.\ to}William of Malmesbury, \index{Vitalis, ref.\ to}\index{Orderic Vitalis, ref.\ to}Orderic
Vitalis, and \index{Platina, ref.\ to}Platina. To these anecdotes the first named
writer adds the story of the statue inscribed with the words
``strike here,'' which having amused our ancestors in the \index{Gesta Romanorum@\emph{Gesta Romanorum}}\emph{Gesta
Romanorum} has been recently told again in the \emph{Earthly
Paradise}.

Extensive though his influence was, it must not be supposed
that Gerbert's writings shew any great originality. His mathematical
works comprise a treatise on arithmetic entitled \emph{De
Numerorum Divisione}, and one on \emph{geometry}. \nblabel{xref:138}An improvement
in the \index{Abacus,!ref.\ to}abacus, attributed by some writers to \index{Boethius!ref.\ to}Boethius, but which
is more likely due to Gerbert, is the introduction in every
column of beads marked by different characters, called \index{Apices}\emph{apices},
for each of the numbers from 1 to 9, instead of nine exactly
similar counters or beads. \index{Numerals, symbols for}These apices lead to a representation
of numbers essentially the same as the Arabic numerals. There
was however no symbol for zero; the step from this concrete
system of denoting numbers by a decimal system on an abacus
to the system of denoting them by similar symbols in writing
seems to us to be a small one, but it would appear that Gerbert
did not make it. He found at Mantua a copy of the geometry
of Boethius, and introduced it into the medieval schools.
Gerbert's own work on geometry is of unequal ability; it includes
a few applications to land-surveying and the determination of
the heights of inaccessible objects, but much of it seems to be
copied from some Pythagorean text-book. In the course of it
he however solves one problem which was of remarkable
difficulty for that time. The question is to find the sides of a
right-angled triangle whose hypotenuse and area are given.
He says, in effect, that if these latter be denoted respectively
by $c$ and $h^2$, then the lengths of the two sides will be
\[
\tfrac{1}{2}\left\{\sqrt{c^2+4h^2} + \sqrt{c^2-4h^2}\right\} \text{ and } \tfrac{1}{2}\left\{\sqrt{c^2+4h^2}-\sqrt{c^2-4h^2}\right\}.
\]

\index{Bernelinus}\nblabel{png159a}\nbparagraph{Bernelinus.} One of Gerbert's pupils, \emph{Bernelinus}, published
a work on the \index{Abacus,!ref.\ to}abacus\footnote
  {It is reprinted in \index{Olleris on Gerbert}Olleris's edition of Gerbert's works, pp.~311--326.}
which is, there is very little doubt, a
reproduction of the teaching of Gerbert. It is valuable as
indicating that the Arabic system of writing numbers was still
unknown in Europe.\index{Cathedral Schools, the|)}\index{Conventual Schools|)}\index{Schools of Charles|)}

\section*{The Early Medieval Universities.\nblabel{png159b}\protect\footnote
  {See the \emph{Universities of Europe in the Middle Ages} by H.~Rashdall,
  Oxford, 1895; \emph{Die Universitäten des Mittelalters bis} 1400 by P.~H.~Denifle,
  1885; and vol.~i of the \emph{University of Cambridge} by J.~B.~Mullinger,
  Cambridge, 1873.}
}\index{Rashdall, ref.\ to}\index{Universities, medieval|(}\index{Medieval universities|(}\index{Denifle, P. H., ref.\ to}\index{Mullinger, ref.\ to}

\nblabel{png159c}At the end of the eleventh century or the beginning of the
twelfth a revival of learning took place at several of these
cathedral or monastic schools; and in some cases, at the same
time, teachers who were not members of the school settled in
its vicinity and, with the sanction of the authorities, gave
lectures which were in fact always on theology, logic, or civil
law. As the students at these centres grew in numbers, it
became desirable to act together whenever any interest common
to all was concerned. The association thus formed was a sort
of guild or trades union, or in the language of the time a \emph{universitas
magistrorum et scholarium}. This was the first stage
in the development of the earliest medieval universities. In
some cases, as at \index{Paris, university of}Paris, the governing body of the university
was formed by the teachers alone, in others, as at \index{Bologna, university of}Bologna, by
both teachers and students; but in all cases precise rules for
the conduct of business and the regulation of the internal
economy of the guild were formulated at an early stage in its
history. The municipalities and numerous societies which
existed in Italy supplied plenty of models for the construction
of such rules, but it is possible that some of the regulations
were derived from those in force in the Mohammedan schools
at \index{Cordova, School of}Cordova.

\nblabel{png160a}We are, almost inevitably, unable to fix the exact date of
the commencement of these voluntary associations, but they
existed at \index{Paris, university of}Paris, \index{Bologna, university of}Bologna, \index{Salerno, university of}Salerno, Oxford, and Cambridge
before the end of the twelfth century: these may be considered
the earliest universities in Europe. The instruction given at
Salerno and Bologna was mainly technical---at Salerno in medicine,
and at Bologna in law---and their claim to recognition as
universities, as long as they were merely technical schools, has
been disputed.

Although the organization of these early universities was
independent of the neighbouring church and monastic schools
they seem in general to have been, at any rate originally, associated
with such schools, and perhaps indebted to them for the
use of rooms, etc. The universities or guilds (self-governing
and formed by teachers and students), and the adjacent schools
(under the direct control of church or monastic authorities), continued
to exist side by side, but in course of time the latter
diminished in importance, and often ended by becoming subject
to the rule of the university authorities. Nearly all the medieval
universities grew up under the protection of a bishop (or abbot),
and were in some matters subject to his authority or to that of
his \index{Chancellor of a university}chancellor, from the latter of whom the head of the university
subsequently took his title. The universities, however,
were not ecclesiastical organizations, and, though the bulk of
their members were ordained, their direct connection with the
Church arose chiefly from the fact that clerks were then the
only class of the community who were left free by the state to
pursue intellectual studies.

A \emph{universitas magistrorum et scholarium}, if successful in
attracting students and acquiring permanency, always sought
special legal privileges, such as the right to fix the price of
provisions and the power to try legal actions in which its
members were concerned. These privileges generally led to a
recognition of its power to grant degrees which conferred a right
of teaching anywhere within the kingdom. The university was
frequently incorporated at or about the same time. Paris
received its charter in 1200, and probably was the earliest
university in Europe thus officially recognized. Legal privileges
were conferred on Oxford in 1214, and on Cambridge in 1231:
the development of Oxford and Cambridge followed closely the
precedent of Paris on which their organization was modelled.
In the course of the thirteenth century universities were founded
at (among other places) \index{Naples, university of}Naples, \index{Orleans, university of}Orleans, \index{Padua, university of}Padua, and \index{Prague, university of}Prague;
and in the course of the fourteenth century at \index{Pavia, university of}Pavia and \index{Vienna, university of}Vienna.
The title of university was generally accredited to any teaching
body as soon as it was recognized as a \index{Studium generale}\emph{studium generale}.

The most famous medieval universities aspired to a still
wider recognition, and the final step in their evolution was an
acknowledgment by the pope or emperor of their degrees as a
title to teach throughout Christendom---such universities were
closely related one with the other. \index{Paris, university of}Paris was thus recognized
in 1283, Oxford in 1296, and Cambridge in 1318.

\nblabel{png161a}The standard of education in mathematics has been largely
fixed by the universities, and most of the mathematicians of
subsequent times have been closely connected with one or more
of them; and therefore I may be pardoned for adding a few
words on the general course of studies\footnote
  {For fuller details as to their organization of studies, their system of
  instruction, and their constitution, see \index{Ball, W. W. R., ref.\ to}my \emph{History of the Study of Mathematics
  at Cambridge}, Cambridge, 1889.}
in a university in
medieval times.

\index{Universities, medieval!curriculum at|(}The students entered when quite young, sometimes not being
more than eleven or twelve years old when first coming into
residence. It is misleading to describe them as undergraduates,
for their age, their studies, the discipline to which they were
subjected, and their position in the university shew that they
should be regarded as schoolboys. The first four years of their
residence were supposed to be spent in the study of the \index{Trivium, the|(}trivium,
that is, Latin grammar, logic, and rhetoric. In quite early
times, a considerable number of the students did not progress
beyond the \index{Grammar, students in}study of Latin grammar---they formed an inferior
faculty and were eligible only for the degree of master of
grammar or master of rhetoric---but the more advanced students
(and in later times all students) spent these years in the study
of the trivium.\index{Trivium, the|)}

\index{Arts!Bachelor of}\index{Bachelor of Arts, degree of}The title of bachelor of arts was conferred at the end of this
course, and signified that the student was no longer a schoolboy
and therefore in pupilage. The average age of a commencing
bachelor may be taken as having been about seventeen or
eighteen. Thus at Cambridge in the presentation for a degree
the technical term still used for an undergraduate is \emph{juvenis},
while that for a bachelor is \emph{vir}. A bachelor could not take
pupils, could teach only under special restrictions, and probably
occupied a position closely analogous to that of an undergraduate
nowadays. Some few bachelors proceeded to the study of \index{Law, faculty of}civil
or canon law, but it was assumed in theory that they next
studied the \index{Quadrivium}quadrivium, the course for which took three years,
and which included about as much science as was to be found
in the pages of \index{Boethius!ref.\ to}Boethius and \index{Isidorus of Seville!ref.\ to}Isidorus.

\index{Master, degree of}\index{Arts!Master of|(}The degree of master of arts was given at the end of this
course. In the twelfth and thirteenth centuries it was merely
a license to teach: no one sought it who did not intend to use
it for that purpose and to reside in the university, and only
those who had a natural aptitude for such work were likely to
enter a profession so ill-paid as that of a teacher. The degree
was obtainable by any student who had gone through the recognized
course of study, and shewn that he was of good moral
character. Outsiders were also admitted, but not as a matter
of course. I may here add that towards the end of the fourteenth
century students began to find that a degree had a pecuniary
value, and most universities subsequently conferred it only on
condition that the new master should reside and teach for at
least a year. Somewhat later the universities took a further
step and began to refuse degrees to those who were not intellectually
qualified. This power was assumed on the precedent
of a case which arose in Paris in 1426, when the university
declined to confer a degree on a student---a Slavonian, one
\index{Nicholas, Paul, ref.\ to}Paul Nicholas---who had performed the necessary exercises in
a very indifferent manner: he took legal proceedings to compel
the university to grant the degree, but their right to withhold
it was established. Nicholas accordingly has the distinction
of being the first student who under modern conditions was
``plucked.''\index{Arts!Master of|)}

Although science and mathematics were recognised as the
standard subjects of study for a bachelor, it is probable that,
until the renaissance, the majority of the students devoted most
of their time to logic, philosophy, and theology. The subtleties
of scholastic philosophy were dreary and barren, but it is only
just to say that they provided a severe intellectual training.\index{Medieval universities|)}\index{Universities, medieval|)}\index{Universities, medieval!curriculum at|)}

We have now arrived at a time when the results of Arab
and Greek science became known in Europe. The history of
Greek mathematics has been already discussed; I must now
temporarily leave the subject of medieval mathematics, and
trace the development of the Arabian schools to the same date;
and I must then explain how the schoolmen became acquainted
with the Arab and Greek text-books, and how their introduction
affected the progress of European mathematics.

\nbChapter{IX}{The Mathematics of the Arabs}{}{}
  {The subject is discussed at length by Cantor, chaps.~xxxii--xxxv; by
  Hankel, pp.~172--293; by A.~von Kremer in \emph{Kulturgeschichte des Orientes
  unter den Chalifen}, Vienna, 1877; and by H.~Suter in his ``Die Mathematiker
  und Astronomen der Araber und ihre Werke,'' \emph{Zeitschrift für Mathematik
  und Physik, Abhandlungen zur Geschichte der Mathematik}, Leipzig, vol.~xlv,
  1900. See also \emph{Matériaux pour servir à l'histoire comparée des sciences
  mathématiques chez les Grecs et les Orientaux}, by L.~A.~Sédillot, Paris,
  1845--9; and the following articles by Fr.~Woepcke, \emph{Sur l'introduction
  de l'arithmétique Indienne en Occident}, Rome, 1859; \emph{Sur l'histoire des
  sciences mathématiques chez les Orientaux}, Paris, 1860; and \emph{Mémoire sur la
  propagation des chiffres Indiens}, Paris, 1863.}
\index{Indian mathematics, chapter \textsc{ix}}
\index{Geometry.!z4@Arab and medieval geometry, founded on Greek works, chapters \textsc{viii, ix, x}}
\index{Arabs!q1@Mathematics of, chapter \textsc{ix}}
\index{Algebra.!z3@Discussed rhetorically by the Hindoo and Arab mathematicians!q1@chapter \textsc{ix}}
\index{Suter on Dionysodorus}\index{Kremer on Arab science}\index{Sedillot@Sédillot, ref.\ to}
\index{Woepcke, ref.\ to}\index{Cantor, M., ref.\ to}\index{Hankel, ref.\ to}\nblabel{png164a}

\nblabel{png164b}\First{The} story of Arab mathematics is known to us in its general
outlines, but we are as yet unable to speak with certainty on
many of its details. It is, however, quite clear that while part
of the early knowledge of the Arabs was derived from Greek
sources, part was obtained from Hindoo works; and that it was
on those foundations that Arab science was built. I will begin
by considering in turn the extent of mathematical knowledge
derived from these sources.

\section*{Extent of Mathematics obtained from Greek Sources.\nblabel{png164c}}

According to their traditions, in themselves very probable,
the scientific knowledge of the Arabs was at first derived from
the Greek doctors who attended the caliphs at Bagdad. It is
said that when the Arab conquerors settled in towns they
became subject to diseases which had been unknown to them
in their life in the desert. The study of \index{Medicine, Greek practitioners}medicine was then
confined mainly to Greeks and Jews, and many of these,
encouraged by the caliphs, settled at Bagdad, Damascus, and
other cities; their knowledge of all branches of learning was
far more extensive and accurate than that of the Arabs, and
the teaching of the young, as has often happened in similar
cases, fell into their hands. The introduction of European
science was rendered the more easy as various small Greek
schools existed in the countries subject to the Arabs: there
had for many years been one at \index{Edessa, Greek School at}Edessa among the Nestorian
Christians, and there were others at \index{Antioch, Greek School at}Antioch, \index{Emesa, Greek School at}Emesa, and
even at \index{Damascus, Greek School at}Damascus, which had preserved the traditions and some
of the results of Greek learning.

The Arabs soon remarked that the Greeks rested their
medical science on the works of \index{Hippocrates of Cos}Hippocrates, \index{Aristotle!ref.\ to}Aristotle, and
\index{Galen, ref.\ to}Galen; and these books were translated into Arabic by order
of the \index{Haroun Al Raschid, ref.\ to}\index{Alras@Al Raschid, Caliph, ref.\ to}caliph Haroun Al Raschid about the year 800. The
translation excited so much interest that his successor \index{Almam@Al Mamun, Caliph, ref.\ to}Al
Mamun (813--833) sent a commission to Constantinople to
obtain copies of as many scientific works as was possible, while
an embassy for a similar purpose was also sent to India. At
the same time a large staff of Syrian clerks was engaged, whose
duty it was to translate the works so obtained into Arabic and
Syriac. \nblabel{png165a}To disarm fanaticism these clerks were at first termed
the caliph's doctors, but in 851 they were formed into a college,
and their most celebrated member, \index{Ishak ibn Honein}\index{Honein ibn Ishak}Honein ibn Ishak, was made
its first president by the caliph \index{Mutawakkil, Caliph, ref.\ to}Mutawakkil (847--861). Honein
and his son Ishak ibn Honein revised the translations before
they were finally issued. Neither of them knew much mathematics,
and several blunders were made in the works issued on
that subject, but another member of the college, \index{Tabit ibn Korra!ref.\ to}Tabit ibn
Korra, shortly published fresh editions which thereafter became
the standard texts.

In this way before the end of the ninth century the Arabs
obtained translations of the works of \index{Euclid!ref.\ to}Euclid, \index{Archimedes!ref.\ to}Archimedes,
\index{Apollonius!ref.\ to}Apollonius, \index{Ptolemy!ref.\ to}Ptolemy, and others; and in some cases these
editions are the only copies of the books now extant. It is
curious, as indicating how completely \index{Diophantus!ref.\ to}Diophantus had dropped
out of notice, that as far as we know the Arabs got no manuscript
of his great work till 150 years later, by which time they
were already acquainted with the idea of algebraic notation and
processes.

\section*{Extent of Mathematics obtained from Hindoo Sources.\nblabel{png166a}}\index{Hindoo mathematics|(}

The Arabs had considerable commerce with India, and a
knowledge of one or both of the two great original Hindoo
works on algebra had been thus obtained in the \index{Almans@Al Mansur, Caliph, ref.\ to}caliphate of
Al Mansur (754--775), though it was not until fifty or sixty
years later that they attracted much attention. The algebra
and arithmetic of the Arabs were largely founded on these
treatises, and I therefore devote this section to the consideration
of Hindoo mathematics.

The Hindoos, like the Chinese, have pretended that they
are the most ancient people on the face of the earth, and
that to them all sciences owe their creation. But it is probable
that these pretensions have no foundation; and in fact no
science or useful art (except a rather fantastic architecture and
sculpture) can be definitely traced back to the inhabitants of
the Indian peninsula prior to the \index{Aryan invasion of India}Aryan invasion. This
invasion seems to have taken place at some time in the
latter half of the fifth century or in the sixth century,
when a tribe of Aryans entered India by the north-west
frontier, and established themselves as rulers over a large
part of the country. Their descendants, wherever they
have kept their blood pure, may still be recognised by
their superiority over the races they originally conquered;
but as is the case with the modern Europeans, they found
the climate trying and gradually degenerated. For the
first two or three centuries they, however, retained their
intellectual vigour, and produced one or two writers of great
ability.

\index{Arya-Bhata|(}\nblabel{png167a}\nbparagraph{Arya-Bhata.} The earliest of these, of whom we have definite
information, is \emph{Arya-Bhata},\footnote
  {The subject of prehistoric \index{Indian numerals}Indian mathematics has been discussed by \index{Thibaut, G.}G.~Thibaut,
  \index{Schroeder}Von~Schroeder, and \index{Vogt}H.~Vogt. \nblabel{png167b}A Sanskrit text of the \emph{Aryabhathiya},
  edited by \index{Kern on Arya-Bhata}H.~Kern, was published at Leyden in 1874; there is also an
  article on it by the same editor in the \emph{Journal of the Asiatic Society}, London,
  1863, vol.~xx, pp.~371--387; a French translation by \index{Rodet, ref.\ to}L.~Rodet of that part
  which deals with algebra and trigonometry is given in the \emph{Journal Asiatique},
  1879, Paris, series 7, vol.~xiii, pp.~393--434.}
who was born at Patna in the year
476. He is frequently quoted by \index{Brahmagupta!ref.\ to}Brahmagupta, and in the
opinion of many commentators he created algebraic analysis,
though it has been suggested that he may have seen \index{Diophantus!ref.\ to}Diophantus's
\emph{Arithmetic}. The chief work of Arya-Bhata with which we are
acquainted is his \emph{Aryabhathiya}, which consists of mnemonic
verses embodying the enunciations of various rules and propositions.
There are no proofs, and the language is so obscure and
concise that it long defied all efforts to translate it.

The book is divided into four parts: of these three are
devoted to \index{Astronomy.!z15@Studied by Hindoos and Arabs}astronomy and the \index{Trigonometry.!z4@Hindoo works on|(}elements of spherical trigonometry;
the remaining part contains the enunciations of thirty-three
rules in arithmetic, algebra, and plane trigonometry. It
is probable that Arya-Bhata regarded himself as an astronomer,
and studied mathematics only so far as it was useful to him in
his astronomy.

In algebra Arya-Bhata gives the sum of the first, second, and
third powers of the first $n$ natural numbers; the general solution
of a quadratic equation; and the solution in integers of certain
\index{Equations, differential!indeterminate}indeterminate equations of the first degree. His solutions of
numerical equations have been supposed to imply that he was
acquainted with the \index{Decimal numeration}decimal system of enumeration.

In trigonometry he gives a table of natural \index{Sine|(}sines of the
angles in the first quadrant, proceeding by multiples of $3\frac{3}{4}°$,
defining a sine as the semi-chord of double the angle. Assuming %[**TN: hyphen added to 'semichord' for consistency]
that for the angle $3\frac{3}{4}°$ the sine is equal to the circular measure,
he takes for its value 225, \emph{i.e.} the number of minutes in the
angle. \index{Trigonometrical functions}He then enunciates a rule which is nearly unintelligible,
but probably is the equivalent of the statement
\[
\sin (n + 1)\alpha - \sin n\alpha = \sin n\alpha - \sin (n - 1) \alpha - \sin n\alpha \cosec \alpha,
\]
where $\alpha$ stands for $3\frac{3}{4}^\circ$; and working with this formula he
constructs a table of sines, and finally finds the value of $\sin 90^\circ$
to be $3438$. This result is correct if we take \index{P1@$\pi$,!z1@value of}$3.1416$ as the
value of $\pi$, and it is interesting to note that this is the number
which in another place he gives for $\pi$. The correct trigonometrical formula is
\[
\sin (n + 1) \alpha - \sin n\alpha = \sin n\alpha - \sin (n - 1) \alpha - 4 \sin n\alpha \sin^{2} \tfrac{1}{2}\alpha.
\]
Arya-Bhata, therefore, took $4\sin^{2} \frac{1}{2}\alpha$ as equal to $\mathrm{cosec}\, \alpha$, that is,
he supposed that $2 \sin \alpha = 1 + \sin 2\alpha$: using the approximate
values of $\sin \alpha$ and $\sin 2\alpha$ given in his table, this reduces to
$2(225) = 1 + 449$, and hence to that degree of approximation his
formula is correct. A considerable proportion of the geometrical
propositions which he gives is wrong.\index{Sine|)}\index{Arya-Bhata|)}\index{Trigonometry.!z4@Hindoo works on|)}

\index{Brahmagupta|(}\nblabel{png167c}\nbparagraph{Brahmagupta.} The next Hindoo writer of note is \emph{Brahmagupta},
who is said to have been born in 598, and probably was
alive about 660. He wrote a work in verse entitled \emph{Brahma-Sphuta-Siddhanta},
that is, the \emph{Siddhanta}, or system of \index{Astronomy.!z15@Studied by Hindoos and Arabs}Brahma
in astronomy. In this, two chapters are devoted to arithmetic,
algebra, and geometry.\footnote
  {These two chapters (chaps.~xii and xviii) were translated by \index{Colebrooke, ref.\ to}H.~T.~Colebrooke,
  and published at London in 1817.}

\index{Rhetorical algebra}The arithmetic is entirely rhetorical. Most of the problems
are worked out by the rule of three, and a large proportion of
them are on the subject of interest.

\nblabel{png168a}In his algebra, which is also rhetorical, he works out the
fundamental propositions connected with an arithmetical progression,
and solves a \index{Quadratic equations|(}quadratic equation (but gives only the
positive value to the radical). As an illustration of the problems
given I may quote the following, which was reproduced in
slightly different forms by various subsequent writers, but I
replace the numbers by letters. ``Two apes lived at the top of
a cliff of height $h$, whose base was distant $mh$ from a neighbouring
village. One descended the cliff and walked to the village,
the other flew up a height $x$ and then flew in a straight line to
the village. The distance traversed by each was the same.
Find $x$.'' Brahmagupta gave the correct answer, namely
$x = mh/(m+2)$. In the question as enunciated originally
$h = 100$, $m = 2$.

Brahmagupta finds solutions in integers of \index{Equations, differential!indeterminate}several indeterminate
equations of the first degree, using the same method as
that now practised. He states one indeterminate equation of
the second degree, namely, $nx^{2} + 1 = y^{2}$, and gives as its solution
$x = 2t/(t^2 - n)$ and $y = (t^{2} + n)/(t^2 - n)$. To obtain this general
form he proved that, if one solution either of that or of certain
allied equations could be guessed, the general solution could be
written down; but he did not explain how one solution could be
obtained. Curiously enough this equation was sent by \index{Fermat!ref.\ to}Fermat
as a challenge to \index{Wallis!ref.\ to}Wallis and \index{Brouncker, Lord!ref.\ to}Lord Brouncker in the seventeenth
century, and the latter found the same solutions as Brahmagupta
had previously done. Brahmagupta also stated that the equation
$y^2 = nx^2 - 1$ could not be satisfied by integral values of $x$ and $y$
unless $n$ could be expressed as the sum of the squares of two
integers. It is perhaps worth noticing that the early algebraists,
whether Greeks, Hindoos, Arabs, or Italians, drew no distinction
between the problems which led to determinate and those
which led to indeterminate equations. It was only after the introduction
of syncopated algebra that attempts were made to give
general solutions of equations, and the difficulty of giving such
solutions of indeterminate equations other than those of the first
degree has led to their practical exclusion from elementary algebra.\index{Quadratic equations|)}

In geometry Brahmagupta proved the Pythagorean property
of a right-angled triangle \index{Eucn@Euc.!a47@\textsc{i}, 47. ref.\ to}(Euc.~\textsc{i}, 47). He gave expressions for
the \index{Quadrilateral, area of}area of a triangle and of a quadrilateral inscribable in a
circle in terms of their sides; and shewed that the area of a
circle was equal to that of a rectangle whose sides were the
radius and semiperimeter. He was less successful in his
attempt to rectify a circle, and his result is equivalent to
taking $\surd 10$ for the \index{P1@$\pi$,!z1@value of}value of $\pi$. He also determined the
\index{Pyramid,!volume of}\index{Cone, sections of!surface of}\index{Cone, sections of!volume of}surface and volume of a pyramid and \index{Quadrature of circle!cone}cone; problems over
which \index{Arya-Bhata!ref.\ to}Arya-Bhata had blundered badly. The next part of
his geometry is almost unintelligible, but it seems to be an
attempt to find expressions for several magnitudes connected
with a quadrilateral inscribed in a circle in terms of its sides:
much of this is wrong.

It must not be supposed that in the original work all the
propositions which deal with any one subject are collected
together, and it is only for convenience that I have tried to
arrange them in that way. It is impossible to say whether the
whole of Brahmagupta's results given above are original. He
knew of Arya-Bhata's work, for he reproduces the table of \index{Sine}sines
there given; it is likely also that some progress in mathematics
had been made by Arya-Bhata's immediate successors, and that
Brahmagupta was acquainted with their works; but there seems
no reason to doubt that the bulk of Brahmagupta's algebra and
arithmetic is original, although perhaps influenced by \index{Diophantus!ref.\ to}Diophantus's
writings: the origin of the geometry is more doubtful,
probably some of it is derived from Hero's works, and maybe
some represents indigenous Hindoo work.\index{Brahmagupta|)}

\index{Bhaskara|(}\index{Bhaskara!ref.\ to}\nblabel{png170a}\nbparagraph{Bhaskara.} To make this account of Hindoo mathematics
complete I may depart from the chronological arrangement and
say that the only remaining Indian mathematician of exceptional
eminence of whose works we know anything was \emph{Bhaskara,} who
was born in 1114. He is said to have been the lineal successor
of Brahmagupta as head of an \index{Astronomy.!z15@Studied by Hindoos and Arabs}astronomical observatory at \index{Ujein}Ujein.
He wrote an astronomy, of which four chapters have been translated.
Of these one termed \index{Lilavati, the|(}\emph{Lilavati} is on arithmetic; a second
termed \index{Bija Ganita}\emph{Bija Ganita} is on algebra; the third and fourth are on
astronomy and the sphere;\index{Viga Ganita|(}\footnote
  {\nblabel{png170b}See the article \emph{Viga Ganita} in the \emph{Penny Cyclopaedia}, London, 1843;
  and the translations of the \emph{Lilavati} and the \emph{Bija Ganita} issued by \index{Colebrooke, ref.\ to}H.~T.~Colebrooke,
  London, 1817. The chapters on astronomy and the sphere were
  edited by \index{Wilkinson on Bhaskara}L.~Wilkinson, Calcutta, 1842.}
some of the other chapters also
involve mathematics. This work was, I believe, known to the
Arabs almost as soon as it was written, and influenced their
subsequent writings, though they failed to utilize or extend
most of the discoveries contained in it. The results thus became
indirectly known in the West before the end of the twelfth
century, but the text itself was not introduced into Europe till
within recent times.\index{Viga Ganita|)}

The treatise is in verse, but there are explanatory notes in
prose. It is not clear whether it is original or whether it is
merely an exposition of the results then known in India; but in
any case it is most probable that Bhaskara was acquainted with
the Arab works which had been written in the tenth and eleventh
centuries, and with the results of Greek mathematics as transmitted
through Arabian sources. The algebra is syncopated and
almost symbolic, which marks a great advance over that of
\index{Brahmagupta!ref.\ to}Brahmagupta and of the Arabs. The geometry is also superior
to that of Brahmagupta, but apparently this is due to the
knowledge of various Greek works obtained through the Arabs.

The first book or \emph{Lilavati} commences with a salutation to
the god of wisdom. The general arrangement of the work may
be gathered from the following table of contents. Systems of
weights and measures. Next decimal numeration, briefly described.
Then the eight operations of arithmetic, namely,
addition, subtraction, multiplication, division, square, cube,
square-root, and cube-root. Reduction of fractions to a common
denominator, fractions of fractions, mixed numbers, and the
eight rules applied to fractions. The ``rules of cipher,'' namely,
$a\pm0 = a,\; 0^{2} = 0,\; \sqrt{0} = 0,\; a÷{0} = \infty$. The solution of some
simple equations which are treated as questions of arithmetic.
\index{Assumption, rule of false}\index{False assumption, rule of}The rule of false assumption. Simultaneous equations of the
first degree with applications. Solution of a few quadratic
equations. Rule of three and compound rule of three, with
various cases. Interest, discount, and partnership. Time of
filling a cistern by several fountains. Barter. \index{Arithmetical!progressions}\index{Progressions,!arithmetical}Arithmetical
progressions, and sums of squares and cubes. \index{Geometrical progressions}\index{Progressions,!geometrical}Geometrical
progressions. Problems on triangles and quadrilaterals. \index{P1@$\pi$,!z1@value of}Approximate
value of $\pi$. \index{Trigonometry.!z4@Hindoo works on}\index{Trigonometrical functions}Some trigonometrical formulae. Contents
of solids. Indeterminate equations of the first degree. Lastly,
the book ends with a few questions on combinations.

This is the earliest known work which contains a systematic
exposition of the \index{Decimal numeration}decimal system of numeration. It is possible
that \index{Arya-Bhata!ref.\ to}Arya-Bhata was acquainted with it, and it is most likely
that \index{Brahmagupta!ref.\ to}Brahmagupta was so, but in Bhaskara's arithmetic we meet
with the \index{Numerals, symbols for}\index{Arabic numerals}\index{Indian numerals}Arabic or Indian numerals and a sign for zero as part
of a well-recognised notation. It is impossible at present to
definitely trace these numerals farther back than the eighth
century, but there is no reason to doubt the assertion that they
were in use at the beginning of the seventh century. Their
origin is a difficult and disputed question. I mention below\footnote
  {See below, page~\pageref{xref:184}.}
the view which on the whole seems most probable, and perhaps is
now generally accepted, and I reproduce there some of the forms
used in early times.

To sum the matter up briefly, it may be said that the
\emph{Lilavati} gives the rules now current for addition, subtraction,
multiplication, and division, as well as for the more common processes
in arithmetic; while the greater part of the work is taken
up with the discussion of the rule of three, which is divided
into direct and inverse, simple and compound, and is used to
solve numerous questions chiefly on interest and exchange---the
numerical questions being expressed in the decimal system of
notation with which we are familiar.

Bhaskara was celebrated as an \index{Astrology}astrologer no less than as a
mathematician. He learnt by this art that the event of his
daughter Lilavati marrying would be fatal to himself. He
therefore declined to allow her to leave his presence, but by
way of consolation he not only called the first book of his work
by her name, but propounded many of his problems in the form
of questions addressed to her. For example, ``Lovely and dear
Lilavati, whose eyes are like a fawn's, tell me what are the
numbers resulting from 135 multiplied by 12. If thou be
skilled in multiplication, whether by whole or by parts, whether
by division or by separation of digits, tell me, auspicious damsel,
what is the quotient of the product when divided by the same
multiplier.''

I may add here that the problems in the Indian works give
a great deal of interesting information about the social and
economic condition of the country in which they were written.
\index{Algebra.!z5@Introduction of syncopated algebra by Bhaskara}Thus Bhaskara discusses some questions on the price of slaves,
and incidentally remarks that a female slave was generally
supposed to be most valuable when 16 years old, and subsequently
to decrease in value in inverse proportion to the age;
for instance, if when 16 years old she were worth 32 nishkas,
her value when 20 would be represented by $(16×32)÷20$
nishkas. It would appear that, as a rough average, a female
slave of 16 was worth about 8 oxen which had worked for two
years. The interest charged for money in India varied from $3\frac{1}{2}$
to 5 per cent per month. Amongst other data thus given will
be found the prices of provisions and labour.

\nblabel{png173a}The chapter termed \index{Viga Ganita|(}\index{Bija Ganita|(}\emph{Bija Ganita} commences with a sentence
so ingeniously framed that it can be read as the enunciation of a
religious, or a philosophical, or a mathematical truth. Bhaskara
after alluding to his \emph{Lilavati}, or arithmetic, states that he intends
in this book to proceed to the general operations of analysis.
The idea of the notation is as follows. \index{Plus!symbols for}\index{Addition, processes of!symbols for}\index{Minus!z1@symbols for}\index{Negative sign}Abbreviations and
initials are used for symbols; \index{Subtraction,!symbols for}subtraction is indicated by a dot
placed above the coefficient of the quantity to be subtracted;
addition by juxtaposition merely; but no symbols are used for
multiplication, equality, or inequality, these being written at
length. A product is denoted by the first syllable of the word
subjoined to the factors, between which a dot is sometimes
placed. \index{Division,!symbols for}\index{Fractions, continued!symbols for}\index{Quotient!symbols for}In a quotient or fraction the divisor is written under
the dividend without a line of separation. The two sides of an
equation are written one under the other, confusion being prevented
by the recital in words of all the steps which accompany
the operation. \index{Unknown quantity, word or symbol for|(}Various symbols for the unknown quantity are
used, but most of them are the initials of names of colours, and
the word colour is often used as synonymous with unknown
quantity; its Sanskrit equivalent also signifies a letter, and
letters are sometimes used either from the alphabet or from the
initial syllables of subjects of the problem. \index{Radical, symbols for}In one or two cases
symbols are used for the given as well as for the unknown
quantities. \index{Exponents}The initials of the words square and solid denote
the second and third powers, and the initial syllable of \index{Roots, square, cube, \&c.}\index{Square root, symbols for}\index{Surds, symbols for}square
root marks a surd. Polynomials are arranged in powers, the
absolute quantity being always placed last and distinguished by
an initial syllable denoting known quantity. Most of the
equations have numerical coefficients, and the coefficient is
always written after the unknown quantity. Positive or
negative terms are indiscriminately allowed to come first; and
every power is repeated on both sides of an equation, with a
zero for the coefficient when the term is absent. After explaining
his notation, \index{Bhaskara!ref.\ to}Bhaskara goes on to give the rules for addition,
subtraction, multiplication, division, squaring, and extracting
the square root of algebraical expressions; he then gives the
rules of cipher as in the \emph{Lilavati}; solves a few equations; and
lastly concludes with some operations on surds. Many of the
problems are given in a poetical setting with allusions to fair
damsels and gallant warriors.\index{Lilavati, the|)}\index{Bija Ganita|)}\index{Viga Ganita|)}\index{Unknown quantity, word or symbol for|)}

\index{Trigonometry.!z4@Hindoo works on}Fragments of other chapters, involving algebra, trigonometry,
and geometrical applications, have been translated by \index{Colebrooke, ref.\ to}Colebrooke.
Amongst the trigonometrical formulae is one which is
equivalent to the equation $d(\sin \theta) = \cos \theta \,d\theta$.

I have departed from the chronological order in treating here
of Bhaskara, but I thought it better to mention him at the same
time as I was discussing his compatriots. It must be remembered,
however, that he flourished subsequently to all the Arab
mathematicians considered in the next section. The works with
which the Arabs first became acquainted were those of \index{Arya-Bhata!ref.\ to}Arya-Bhata
and \index{Brahmagupta!ref.\ to}Brahmagupta, and perhaps of their successors Sridhara
and Padmanabha; it is doubtful if they ever made much use of
the great treatise of Bhaskara.\index{Bhaskara|)}

\index{Arithmetic.!z7@Algoristic arithmetic invented by the Hindoos!x3@adopted by the Arabs}\index{Indian numerals|(}It is probable that the attention of the Arabs was called to
the works of the first two of these writers by the fact that the
Arabs adopted the Indian system of arithmetic, and were thus
led to look at the mathematical text-books of the Hindoos.
\index{Arabic numerals}The Arabs had always had considerable commerce with India,
and with the establishment of their empire the amount of trade
naturally increased; at that time, about the year 700, they
found the Hindoo \index{Mercantile arithmetic}merchants beginning to use the system of
numeration with which we are familiar, and adopted it at once.
This immediate acceptance of it was made the easier, as they
had no works of science or literature in which another system
was used, and it is doubtful whether they then possessed any
but the most primitive system of notation for expressing
numbers. The Arabs, like the Hindoos, seem also to have
made little or no use of the abacus, and therefore must have
found Greek and Roman methods of calculation extremely
laborious. \index{Numerals, symbols for}The earliest definite date assigned for the use in
Arabia of the \index{Decimal numeration}decimal system of numeration is 773. In that
year some Indian astronomical tables were brought to Bagdad,
and it is almost certain that in these Indian numerals (including
a zero) were employed.\index{Hindoo mathematics|)}\index{Indian numerals|)}

\section*{The Development of Mathematics in Arabia.\nblabel{png175a}\protect\footnote
  {A work by B.~Baldi on the lives of several of the Arab mathematicians
  was printed in Boncompagni's \emph{Bulletino di bibliografia.} 1872, vol.~v, pp.~427--534.}
}\index{Boncompagni, ref.\ to}\index{Baldi on Arab mathematics}

In the preceding sections of this chapter I have indicated
the two sources from which the Arabs derived their knowledge
of mathematics, and have sketched out roughly the amount of
knowledge obtained from each. We may sum the matter up
by saying that before the end of the eighth century the Arabs
were in possession of a good numerical notation and of
\index{Brahmagupta!ref.\ to}Brahmagupta's work on arithmetic and algebra; while before
the end of the ninth century they were acquainted with the
masterpieces of Greek mathematics in geometry, mechanics,
and astronomy. I have now to explain what use they made of
these materials.

\index{Alkarismi|(}\nblabel{png175b}\nbparagraph{Alkarismi.} The first and in some respects the most illustrious
of the Arabian mathematicians was \emph{Mohammed ibn Musa
Abu Djefar Al-Khw\={a}rizm\={\i}}. There is no common agreement as
to which of these names is the one by which he is to be known:
the last of them refers to the place where he was born, or in
connection with which he was best known, and I am told that
it is the one by which he would have been usually known
among his contemporaries. I shall therefore refer to him by
that name; and shall also generally adopt the corresponding
titles to designate the other Arabian mathematicians. Until
recently, this was almost always written in the corrupt form
\emph{Alkarismi}, and, though this way of spelling it is incorrect, it
has been sanctioned by so many writers that I shall make use
of it.

We know nothing of Alkarismi's life except that he was a
native of Khorassan and librarian of the \index{Almam@Al Mamun, Caliph, ref.\ to}caliph Al Mamun; and
that he accompanied a mission to Afghanistan, and possibly
came back through India. On his return, about 830, he wrote
an \index{Algebraa@Algebra,!origin of term}algebra,\footnote
  {It was published by \index{Rosen on Alkarismi}F.~Rosen, with an English translation, London,
  1831.}
which is founded on that of Brahmagupta, but in
which some of the proofs rest on the Greek method of representing
numbers by lines. He also wrote a treatise on arithmetic:
an anonymous tract termed \emph{Algoritmi De Numero
Indorum,} which is in the university library at Cambridge, is
believed to be a Latin translation of this treatise.\footnote
  {It was published by \index{Boncompagni, ref.\ to}B.~Boncompagni, Rome, 1857.}
Besides
these two works he compiled some astronomical tables, with
explanatory remarks; these included results taken from both
\index{Ptolemy!ref.\ to}Ptolemy and Brahmagupta.

The algebra of Alkarismi holds a most important place in the
history of mathematics, for we may say that the subsequent
Arab and the early medieval works on algebra were founded on
it, and also that through it the Arabic or Indian system of
decimal numeration was introduced into the West. \nblabel{png176a}The work
is termed \emph{Al-gebr we' l mukabala: al-gebr,} from which the word
algebra is derived, means \emph{the restoration,} and refers to the fact
that any the same magnitude may be added to or subtracted %[**TN: sic]
from both sides of an equation; \emph{al mukabala} means the process
of simplification, and is generally used in connection with the
combination of like terms into a single term. \index{Roots of equations,!origin of term}The \index{Unknown quantity, word or symbol for}unknown
quantity is termed either ``the thing'' or ``the root'' (that is,
of a plant), and from the latter phrase our use of the word root
as applied to the solution of an equation is derived. The
square of the unknown is called ``the power.'' All the known
quantities are numbers.

\nblabel{png177a}The work is divided into five parts. \index{Quadratic equations|(}In the first Alkarismi
gives rules for the solution of quadratic equations, divided
into five classes of the forms $ax^{2} = bx$, $ax^{2} = c$, $ax^{2} + bx= c$,
$ax^{2} + c= bx$, and $ax^{2} = bx + c$, where $a, b, c$ are positive numbers,
and in all the applications $a=1$. He considers only real and
positive roots, but he recognises the existence of two roots,
which as far as we know was never done by the Greeks. It is
somewhat curious that when both roots are positive he generally
takes only that root which is derived from the negative value of
the radical.

He next gives geometrical proofs of these rules in a
manner analogous to that of Euclid~\textsc{ii}, 4. For example, to
solve the equation $x^{2} + 10x = 39$, or any equation of the form
$x^{2} + px = q$, he gives two methods of which one is as follows.
Let $AB$ represent the value of $x$, and construct on it the
square $ABCD$ (see \figureref{177a}{figure} below). Produce $DA$ to $H$ and
$DC$ to $F$ so that $AH=CF=5$ (or $\frac{1}{2}p$); and complete the
figure as drawn below. Then the areas $AC$, $HB$, and $BF$
represent the magnitudes $x^{2}$, $5x$, and $5x$. Thus the left-hand
side of the equation is represented by the sum of the areas $AC$,
$HB$, and $BF$, that is, by the gnomon $HCG$. To both sides of
the equation add the square $KG$, the area of which is 25 (or
$\frac{1}{4}p^{2}$), and we shall get a new square whose area is by hypothesis
equal to 39+25, that is, to 64 (or $q + \frac{1}{4}p^{2}$) and whose side
therefore is 8. The side of this square $DH$, which is equal to
8, will exceed $AH$, which is equal to 5, by the value of the
unknown required, which, therefore, is 3.\index{Quadratic equations|)}

\begin{center}
\labelincludegraphics{177a}
\end{center}

In the third part of the book Alkarismi considers the
product of $(x\pm a)$ and $(x\pm b)$. In the fourth part he states
the rules for addition and subtraction of expressions which
involve the unknown, its square, or its square root; gives rules
for the calculation of square roots; and concludes with the
theorems that $a\sqrt{b} = \sqrt{a^{2}b}$ and $\sqrt{a}\sqrt{b}=\sqrt{ab}$. In the fifth
and last part he gives some problems, such, for example, as to
find two numbers whose sum is 10 and the difference of whose
squares is 40.

\nblabel{png178a}In all these early works there is no clear distinction between
arithmetic and algebra, and we find the account and explanation
of arithmetical processes mixed up with algebra and
treated as part of it. It was from this book then that
the Italians first obtained not only the ideas of algebra, but
also of an arithmetic founded on the \index{Decimal numeration}decimal system.
This arithmetic was long known as \index{Algorism}\emph{algorism}, or the art of
Alkarismi, which served to distinguish it from the arithmetic
of Boethius; this name remained in use till the eighteenth
century.\index{Alkarismi|)}

\index{Tabit ibn Korra|(}\nblabel{png178b}\nbparagraph{Tabit ibn Korra.} The work commenced by Alkarismi
was carried on by \emph{Tabit ibn Korra,} born at Harran in 836, and
died in 901, who was one of the most brilliant and accomplished
scholars produced by the Arabs. As I have already
stated, he issued translations of the chief works of \index{Euclid!ref.\ to}\index{Euclid's \emph{Elements}!ref.\ to}Euclid,
\index{Apollonius!ref.\ to}Apollonius, \index{Archimedes!ref.\ to}Archimedes, and \index{Ptolemy!ref.\ to}Ptolemy. He also wrote several
original works, all of which are lost with the exception of a
fragment on algebra, consisting of \index{Cubic equations}one chapter on cubic equations,
which are solved by the aid of geometry in somewhat
the same way as that given later.\footnote
  {See below, page~\pageref{xref:224}.}\index{Tabit ibn Korra|)}

Algebra continued to develop very rapidly, but it remained
entirely rhetorical. The problems with which the Arabs were
chiefly concerned were solution of equations, problems leading
to equations, or properties of numbers. The two most prominent
algebraists of a later date were Alkayami and Alkarki, both
of whom flourished at the beginning of the eleventh century.

\index{Alkayami}\nblabel{png179a}\nbparagraph{Alkayami.} The first of these, \emph{Omar Alkayami}, is noticeable
for his geometrical treatment of cubic equations by which
he obtained a root as the abscissa of a point of intersection
of a conic and a circle.\footnote
  {His treatise on algebra was published by \index{Woepcke, ref.\ to}Fr.~Woepcke, Paris, 1851.}
The equations he considers are of
the following forms, where $a$ and $c$ stand for positive integers,
(i)~$x^{3} + b^{2}x = b^{2}c$, whose root he says is the abscissa of a point
of intersection of $x^{2} = by$ and $y^{2} = x(c - x)$; (ii)~$x^{3} + ax^{2} = c^{3}$,
whose root he says is the abscissa of a point of intersection
of $xy = c^{2}$ and $y^{2} = c(x + a)$; (iii)~$x^{3}± ax^{2} + b^{2}x = b^{2}c$, whose
root he says is the abscissa of a point of intersection of
$y^{2}=(x\pm a)(c-x)$ and $x(b\pm y) = bc$. \index{Quartic equation}He gives one \index{Biquadratic equation}biquadratic,
namely, $(100-x^{2})(10-x)^{2} = 8100$, the root of which is determined
by the point of intersection of $(10 - x)y = 90$ and
$x^{2} + y^{2} = 100$. It is sometimes said that he stated that it was
impossible to solve the equation $x^{3} + y^{3} = z^{3}$ in positive integers,
or in other words that the sum of two cubes can never be a
cube; though whether he gave an accurate proof, or whether,
as is more likely, the proposition (if enunciated at all) was the
result of a wide induction, it is now impossible to say; but
the fact that such a theorem is attributed to him will serve to
illustrate the extraordinary progress the Arabs had made in
algebra.

\index{Alkarki|(}\nbparagraph{Alkarki.} The other mathematician of this time (circ.~1000)
whom I mentioned was \emph{Alkarki.}\footnote
  {His algebra was published by Fr.~Woepcke, 1853, and his arithmetic was
  translated into German by \index{Hochheim on Alkarki}Ad.~Hochheim, Halle, 1878.}
He gave expressions for the
sums of the first, second, and third powers of the first $n$ natural
numbers; solved various equations, including some of the forms
$ax^{2p}± bx^{p}± c=0$ and discussed surds, shewing, for example,
that $\sqrt{8}+\sqrt{18}=\sqrt{50}$.\index{Alkarki|)}

Even where the methods of Arab algebra are quite general
the applications are confined in all cases to numerical problems,
and the algebra is so arithmetical that it is difficult to treat the
subjects apart. From their books on arithmetic and from the
observations scattered through various works on algebra, we may
say that the methods used by the Arabs for the four fundamental
processes were analogous to, though more cumbrous
than, those now in use; but the problems to which the subject
was applied were similar to those given in modern books, and
were solved by similar methods, such as rule of three, \&c.
Some minor improvements in notation were introduced, such,
for instance, as the \index{Quotient!symbols for}introduction of a line to separate the numerator
from the denominator of a fraction; and hence a \index{Division,!symbols for}line
between two symbols came to be used as a symbol of division.\footnote
  {See below, page~\pageref{xref:241}.}
\index{Alhossein}Alhossein (980--1037) used a rule for testing the correctness of
the results of addition and multiplication by \index{Nines, casting out the}``casting out the
nines.'' Various forms of this rule have been given, but they
all depend on the proposition that, if each number in the question
be replaced by the remainder when it is divided by 9, and
if these remainders be added or multiplied as directed in the
question, then this result when divided by 9 will leave the same
remainder as the answer whose correctness it is desired to test
when divided by 9: if these remainders differ, there is an error.
The selection of 9 as a divisor was due to the fact that the
remainder when a number is divided by 9 can be obtained by
adding the digits of the number and dividing the sum
by 9.

I am not concerned with the views of Arab writers on
\index{Astronomy.!z15@Studied by Hindoos and Arabs}astronomy or the value of their observations, but I may remark
in passing that they accepted the theory as laid down by \index{Hipparchus!ref.\ to}Hipparchus
and \index{Ptolemy!ref.\ to}Ptolemy, and did not materially alter or advance it.
I may, however, add that Al Mamun caused the \index{Degree, length of}length of a
degree of latitude to be measured, and he, as well as the two
mathematicians to be next named, determined the obliquity of
the ecliptic.

\index{Albategni}\index{Albuzjani}\nblabel{png181a}\nbparagraph{Albategni.} \nbparagraph{Albuzjani.} Like the Greeks, the Arabs rarely,
if ever, \index{Trigonometry.!z3@Considered a part of astronomy, and treated as such by the Greeks and Arabs}employed trigonometry except in connection with
astronomy; but in effect they used the trigonometrical ratios
which are now current, and worked out the plane trigonometry
of a single angle. They are also acquainted with the elements
of \index{Spherical trigonometry}spherical trigonometry. \emph{Albategni}, born at Batan in Mesopotamia,
in 877, and died at Bagdad in 929, was among the
earliest of the many distinguished Arabian astronomers. He
wrote the \emph{Science of the Stars,}\footnote
  {It was edited by \index{Regiomontanus!ref.\ to}Regiomontanus, Nuremberg, 1537.}
which is worthy of note from
its containing a mention of the motion of the \index{Apogee, sun's}sun's apogee.
In this work angles are determined by ``the semi-chord of twice
the angle,'' that is, by the \index{Sine}sine of the angle (taking the radius
vector as unity). It is doubtful whether he was acquainted
with the previous introduction of sines by \index{Arya-Bhata!ref.\ to}Arya-Bhata and
\index{Brahmagupta!ref.\ to}Brahmagupta; \index{Hipparchus!ref.\ to}Hipparchus and \index{Ptolemy!ref.\ to}Ptolemy, it will be remembered,
had used the chord. Albategni was also acquainted with
the fundamental formula in spherical trigonometry, giving
the side of a triangle in terms of the other sides and the
angle included by them. Shortly after the death of \index{Albategni}Albategni,
\emph{Albuzjani}, who is also known as \emph{Abul-Wafa,} born in 940,
and died in 998, introduced certain \index{Trigonometrical functions}trigonometrical functions,
and constructed \index{Cotangents, table of}tables of \index{Tangent (trigonometrical)}tangents and \index{Cotangent}cotangents.
He was celebrated as a geometrician as well as an
astronomer.

\index{Alhazen|(}\index{Abd-al-gehl}\nblabel{png181b}\nbparagraph{Alhazen.} \nbparagraph{Abd-al-gehl.} The Arabs were at first content to
take the works of \index{Euclid's \emph{Elements}!ref.\ to}Euclid and \index{Apollonius!ref.\ to}Apollonius for their text-books
in geometry without attempting to comment on them, but
\emph{Alhazen}, born at Bassora in 987 and died at Cairo in 1038,
issued in 1036 a collection\footnote
  {It was translated by \index{Sedillot@Sédillot, ref.\ to}L.~A.~Sédillot, and published at Paris in 1836.}
of problems something like the \emph{Data}
of Euclid. Besides commentaries on the definitions of \index{Euclid!ref.\ to}Euclid
and on the \index{Almagest@\emph{Almagest}, the!ref.\ to}\emph{Almagest}, Alhazen also wrote a work on \index{Optics (geometrical).!Discussed by (among others) Euclid!z2@Alhazen}optics,\footnote
  {It was published at Bâle in 1572.}
which
includes the earliest scientific account of \index{Refraction!atmospheric}atmospheric refraction.
It also contains some ingenious geometry, amongst other things,
a geometrical solution of the problem to find at what point of a
concave mirror a ray from a given point must be incident so as
to be reflected to another given point. Another geometrician
of a slightly later date was \emph{Abd-al-gehl} (circ.~1100), who wrote on
conic sections, and was also the author of three small geometrical tracts.\index{Alhazen|)}

It was shortly after the last of the mathematicians mentioned
above that \index{Bhaskara!ref.\ to}Bhaskara, the third great Hindoo mathematician,
flourished; there is every reason to believe that he was
familiar with the works of the Arab school as described
above, and also that his writings were at once known in
Arabia.

\nblabel{png182a}The Arab schools continued to flourish until the fifteenth
century. But they produced no other mathematician of any
exceptional genius, nor was there any great advance on the
methods indicated above, and it is unnecessary for me to
crowd my pages with the names of a number of writers
who did not materially affect the progress of the science in
Europe.

From this rapid sketch it will be seen that the work of the
Arabs (including therein writers who wrote in Arabia and
lived under Eastern Mohammedan rule) in arithmetic, algebra,
and trigonometry was of a high order of excellence. They
appreciated geometry and the applications of geometry to
astronomy, but they did not extend the bounds of the science.
It may be also added that they made no special progress in
statics, or optics, or hydrostatics; though there is abundant
evidence that they had a thorough knowledge of practical
hydraulics.

The general impression left is that the Arabs were quick
to appreciate the work of others---notably of the Greek masters
and of the Hindoo mathematicians---but, like the ancient
Chinese and Egyptians, they did not systematically develop
a subject to any considerable extent. Their schools may be
taken to have lasted in all for about 650 years, and if the
work produced be compared with that of Greek or modern
European writers it is, as a whole, second-rate both in quantity
and quality.

\nbChapter{X}{The Introduction of Arab Works into Europe}{Introduction of Arab Works}{Circ.~1150--1450.}{}
\index{Geometry.!z4@Arab and medieval geometry, founded on Greek works, chapters \textsc{viii, ix, x}}
\index{Arabs!q3@introduced into Europe, chap. \textsc{x}}
\index{Algebra.!z3@Discussed rhetorically by the Hindoo and Arab mathematicians!q2@by the early Italian writers, chapter \textsc{x}}\nblabel{png184a}

\First{In} the \hyperref[chapter:VIII]{last chapter but one} I discussed the development of
European mathematics to a date which corresponds roughly
with the end of the ``dark ages''; and in the last chapter
I traced the history of the mathematics of the Indians and
Arabs to the same date. The mathematics of the two or
three centuries that follow and are treated in this chapter are
characterised by the introduction of the Arab mathematical
text-books and of Greek books derived from Arab sources, and
the assimilation of the new ideas thus presented.

\index{Spanish mathematics|(}It was, however, from Spain, and not from Arabia, that
a knowledge of eastern mathematics first came into western
Europe. \index{Moors, mathematics of|(}The Moors had established their rule in Spain in 747,
and by the tenth or eleventh century had attained a high
degree of civilisation. Though their political relations with the
caliphs at Bagdad were somewhat unfriendly, they gave a
ready welcome to the works of the great Arab mathematicians.
In this way the Arab translations of the writings of \index{Euclid!ref.\ to}\index{Euclid's \emph{Elements}!ref.\ to}Euclid,
\index{Archimedes!ref.\ to}Archimedes, \index{Apollonius!ref.\ to}Apollonius, \index{Ptolemy!ref.\ to}Ptolemy, and perhaps of other Greek
authors, together with the works of the Arabian algebraists,
were read and commented on at the three great Moorish schools of
\index{Granada, School of}Granada, \index{Cordova, School of}Cordova, and \index{Seville, School of}Seville. It seems probable that these
works indicate the full extent of Moorish learning, but, as
all knowledge was jealously guarded from Christians, it is
impossible to speak with certainty either on this point or
on that of the time when the Arab books were first introduced
into Spain.

\nblabel{png185a}\nbparagraph{The eleventh century.} \nblabel{png185b}The earliest Moorish writer of
distinction of whom I find mention is \index{Geber ibn Aphla}\nbname{Geber ibn Aphla}, who
was born at Seville and died towards the latter part of the
eleventh century at \index{Cordova, School of}Cordova. He wrote on \index{Astronomy.!z15@Studied by Hindoos and Arabs}astronomy and
trigonometry, and was acquainted with the theorem that the
sines of the angles of a spherical triangle are proportional to the
sines of the opposite sides.\footnote
  {Geber's works were translated into Latin by \index{Gerard!ref.\ to}Gerard, and published at
  Nuremberg in 1533.}

\index{Arzachel}\nbparagraph{Arzachel.\footnote
  {See a memoir by \index{Steinschneider on Arzachel}M.~Steinschneider in Boncompagni's \emph{Bulletino di
  Bibliografia,} 1887, vol xx.}}
Another Arab of about the same date was
\emph{Arzachel}, who was living at Toledo in 1080. He suggested
that the \index{Planetary motions}\index{Elliptic orbits of planets}planets moved in ellipses, but his contemporaries with
scientific intolerance declined to argue about a statement which
was contrary to \index{Ptolemy!ref.\ to}Ptolemy's conclusions in the \index{Almagest@\emph{Almagest}, the!ref.\ to}\emph{Almagest}.

\nblabel{png185c}\nbparagraph{The twelfth century.} During the course of the twelfth
century copies of the books used in Spain were obtained in
western Christendom. \nblabel{png185d}The first step towards procuring a
knowledge of Arab and Moorish science was taken by an
English monk, \nblabel{xref:165}\index{Adelhard of Bath}\nbname{Adelhard of Bath},\multifootnote{AB}
  {On the influence of Adelhard and Ben Ezra, see the ``Abhandlungen
  zur Geschichte der Mathematik'' in the \emph{Zeitschrift für Mathematik,} vol.~xxv,
  1880.}
who, under the disguise of
a Mohammedan student, attended some lectures at Cordova
about 1120 and obtained a copy of \index{Euclid's \emph{Elements}!ref.\ to}Euclid's \emph{Elements}. This
copy, translated into Latin, was the foundation of all the
editions known in Europe till 1533, when the Greek text
was recovered. How rapidly a knowledge of the work spread
we may judge when we recollect that before the end of the
thirteenth century \index{Bacon, Roger!ref.\ to}Roger Bacon was familiar with it, while
before the close of the fourteenth century the first five books
formed part of the regular curriculum at many universities.
The enunciations of Euclid seem to have been known before
Adelhard's time, and possibly as early as the year 1000, though
copies were rare. Adelhard also issued a text-book on the use
of the abacus.

\index{Ben Ezra}\nblabel{png186a}\nbparagraph{Ben Ezra.\multifootnote{AB}}
During the same century other translations of
the Arab text-books or commentaries on them were obtained.
Amongst those who were most influential in introducing
Moorish learning into Europe I may mention \emph{Abraham Ben
Ezra}. Ben Ezra was born at Toledo in 1097, and died at
Rome in 1167. He was one of the most distinguished \index{Jews, science of}Jewish
rabbis who had settled in Spain, where it must be recollected
that they were tolerated and even protected by the Moors
on account of their medical skill. Besides some astronomical
tables and an astrology, Ben Ezra wrote an arithmetic;\footnote
  {An analysis of it was published by \index{Terquem on Ben Ezra}O.~Terquem in Liouville's \emph{Journal}
  for 1841.}
in
this he explains the \index{Indian numerals}\index{Arabic numerals}\index{Decimal numeration}Arab system of numeration with nine
symbols and a zero, gives the fundamental processes of
arithmetic, and explains the rule of three.

\index{Gerard}\nbparagraph{Gerard.\footnote
  {See \index{Boncompagni, ref.\ to}Boncompagni's \emph{Della vita e delle opere di Gherardo Cremonese},
  Rome, 1851.}}
 Another European who was induced by the
reputation of the Arab schools to go to Toledo was \emph{Gerard},
who was born at Cremona in 1114 and died in 1187. He
translated the Arab edition of the \index{Almagest@\emph{Almagest}, the!ref.\ to}\emph{Almagest}, the works of
\index{Alhazen!ref.\ to}Alhazen, and the works of \index{Alfarabius, ref.\ to}Alfarabius, whose name is otherwise
unknown to us: it is believed that the Arabic numerals
were used in this translation, made in 1136, of \index{Ptolemy!ref.\ to}Ptolemy's work.
Gerard also wrote a short treatise on \index{Algorism}algorism which exists in
manuscript in the Bodleian Library at Oxford. He was
acquainted with one of the Arab editions of \index{Euclid's \emph{Elements}!ref.\ to}Euclid's \emph{Elements},
which he translated into Latin.

\index{John Hispalensis|(}\nbparagraph{John Hispalensis.} Among the contemporaries of Gerard
was \emph{John Hispalensis} of Seville, originally a rabbi, but converted
to Christianity and baptized under the name given above. He
made translations of several Arab and Moorish works, and also
wrote an algorism which contains the earliest examples of the
extraction of the square roots of numbers by the aid of the
decimal notation.\index{John Hispalensis|)}

\nblabel{png187a}\nbparagraph{The thirteenth century.} During the thirteenth century
there was a revival of learning throughout Europe, but the new
learning was, I believe, confined to a very limited class. The
early years of this century are memorable for the development
of several universities, and for the appearance of three remarkable
mathematicians---Leonardo of Pisa, \index{Jordanus!ref.\ to}Jordanus, and \index{Bacon, Roger!ref.\ to}Roger
Bacon, the Franciscan monk of Oxford. Henceforward it is
to Europeans that we have to look for the development of
mathematics, but until the invention of printing the knowledge
was confined to a very limited class.

\index{Leonardo of Pisa|(}\nblabel{png187b}\nbparagraph{Leonardo.\footnote
  {See the \emph{Leben und Schriften Leonardos da Pisa}, by \index{Giesing on Leonardo}J.~Giesing, Döbeln,
  1886; \index{Cantor, M., ref.\ to}Cantor, chaps.~xli,~xlii; and an article by \index{Lazzarini, V.}V.~Lazzarini in the
  \emph{Bollettino di Bibliografia e Storia}, Rome, 1904, vol.~vii. Most of Leonardo's
  writings were edited and published by \index{Boncompagni, ref.\ to}B.~Boncompagni, Rome, vol.~i, 1857,
  and vol.~ii, 1862.}}
\emph{Leonardo Fibonacci} (\emph{i.e.}~filius Bonaccii) generally
known as \emph{Leonardo of Pisa}, was born at Pisa about 1175.
His father Bonacci was a merchant, and was sent by his fellow-townsmen
to control the custom-house at Bugia in Barbary;
there Leonardo was educated, and he thus became acquainted
with the Arabic or decimal system of numeration, as also with
\index{Alkarismi!ref.\ to}Alkarismi's work on Algebra, which was described in the \hyperref[png175b]{last
chapter}. It would seem that Leonardo was entrusted with some
duties, in connection with the custom-house, which required him
to travel. \nblabel{png187c}He returned to Italy about 1200, and in 1202
published a work called \emph{Algebra et almuchabala} (the title being
taken from Alkarismi's work), but generally known as the \emph{Liber
Abaci}. He there explains the Arabic system of numeration, and
remarks on its great advantages over the Roman system. He
then gives an account of algebra, and points out the convenience
of using geometry to get rigid demonstrations of algebraical
formulae. He shews how to solve simple equations, solves a few
quadratic equations, and states some methods for the solution of
indeterminate equations; these rules are illustrated by problems
on numbers. \index{Rhetorical algebra}The algebra is rhetorical, but in one case letters
are employed as algebraical symbols. This work had a wide
circulation, and for at least two centuries remained a standard
authority from which numerous writers drew their inspiration.

The \emph{Liber Abaci} is especially interesting in the history of
arithmetic, since practically it introduced the use of the \index{Arabic numerals}Arabic
numerals into Christian Europe. The language of Leonardo
implies that they were previously unknown to his countrymen;
he says that having had to spend some years in Barbary he there
learnt the Arabic system, which he found much more convenient
than that used in Europe; he therefore published it ``in order
that the Latin\footnote
  {\index{Peacock!ref.\ to}Dean Peacock says that the earliest known application of the word
  Italians to describe the inhabitants of Italy occurs about the middle of the
  thirteenth century; by the end of that century it was in common use.}
race might no longer be deficient in that
knowledge.'' Now Leonardo had read very widely, and had
travelled in Greece, Sicily, and Italy; there is therefore every
presumption that the system was not then commonly employed
in Europe.

\nblabel{png188a}Though Leonardo introduced the use of \index{Numerals, symbols for}Arabic numerals
into \index{Mercantile arithmetic|(}commercial affairs, it is probable that a knowledge of them
as current in the East was previously not uncommon among
travellers and merchants, for the intercourse between Christians
and Mohammedans was sufficiently close for each to learn
something of the language and common practices of the other. We
can also hardly suppose that the Italian merchants were ignorant
of the method of keeping accounts used by some of their best
customers; and we must recollect, too, that there were numerous
Christians who had escaped or been ransomed after serving the
Mohammedans as slaves. It was, however, Leonardo who
brought the Arabic system into general use, and by the middle
of the thirteenth century a large proportion of the Italian
merchants employed it by the side of the old system.

\nblabel{png188b}The majority of mathematicians must have already known
of the system from the works of \index{Ben Ezra!ref.\ to}Ben Ezra, \index{Gerard!ref.\ to}Gerard, and \index{John Hispalensis!ref.\ to}John
Hispalensis. But shortly after the appearance of Leonardo's
book \index{Alfonso of Castile}\index{Alphonso of Castile}Alfonso of Castile (in 1252) published some \index{Alphonso's tables}\index{Alfonso's tables}astronomical
tables, founded on observations made in Arabia, which were
computed by Arabs, and which, it is generally believed, were
expressed in Arabic notation. Alfonso's tables had a wide
circulation among men of science, and probably were largely
instrumental in bringing these \index{Arabic numerals}\index{Numerals, symbols for}numerals into universal use
among mathematicians. By the end of the thirteenth century
it was generally assumed that all scientific men would be
acquainted with the system: thus \index{Bacon, Roger!ref.\ to}Roger Bacon writing in that
century recommends algorism (that is, the arithmetic founded
on the Arab notation) as a necessary study for theologians who
ought, he says, ``to abound in the power of numbering.'' We
may then consider that by the year 1300, or at the latest 1350,
these numerals were familiar both to mathematicians and to
Italian merchants.\index{Moors, mathematics of|)}\index{Spanish mathematics|)}\index{Mercantile arithmetic|)}

\nblabel{png189a}So great was Leonardo's reputation that the \index{Frederick II. of Germany!ref.\ to}Emperor
Frederick II. stopped at Pisa in 1225 in order to hold a sort
of mathematical tournament to test Leonardo's skill, of which
he had heard such marvellous accounts. The competitors were
informed beforehand of the questions to be asked, some or all
of which were composed by \index{John of Palermo}John of Palermo, who was one of
Frederick's suite. This is the first time that we meet with an
instance of those challenges to solve particular problems which
were so common in the sixteenth and seventeenth centuries.
The first question propounded was to find a number of which
the square, when either increased or decreased by 5, would
remain a square. Leonardo gave an answer, which is correct,
namely $41/12$. The next question was to find by the methods
used in the \index{Euclid's \emph{Elements}!ref.\ to}tenth book of Euclid a line whose length $x$
should satisfy the equation $x^3 + 2x^2 + 10x = 20$. \nblabel{xref:169}Leonardo
showed by geometry that the problem was impossible, but he
gave an approximate value of the root of this equation, namely,
$1\!\cdot\!22'\,7''\,42'''\,33''''\,4^{v}\,40^{vi}$, which is equal to $1.3688081075\ldots$,
and is correct to nine places of decimals.\footnote
  {See \index{Woepcke, ref.\ to}Fr.~Woepcke in Liouville's \emph{Journal} for 1854, p.~401.}
Another question
was as follows. Three men, \emph{A}, \emph{B}, \emph{C}, possess a sum of money $u$,
their shares being in the ratio $3:2:1$. \emph{A} takes away $x$, keeps
half of it, and deposits the remainder with \emph{D}; \emph{B} takes away $y$,
keeps two-thirds of it, and deposits the remainder with \emph{D}; \emph{C}
takes away all that is left, namely $z$, keeps five-sixths of it, and
deposits the remainder with \emph{D}. This deposit with \emph{D} is found
to belong to \emph{A}, \emph{B}, and \emph{C} in equal proportions. Find $u$, $x$, $y$,
and $z$. Leonardo showed that the problem was indeterminate,
and gave as one solution $u = 47$, $x = 33$, $y = 13$, $z = 1$. The other
competitors failed to solve any of these questions.

The chief work of Leonardo is the \emph{Liber Abaci} alluded to
above. This work contains a proof of the well-known result
\[
(a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (bc - ad)^2 = (ad + bc)^2 + (bd - ac)^2.
\]
He also wrote a geometry, termed \emph{Practica Geometriae}, which
was issued in 1220. This is a good compilation, and some
trigonometry is introduced; among other propositions and
examples he finds the area of a triangle in terms of its sides.
Subsequently he published a \emph{Liber Quadratorum} dealing with
problems similar to the first of the questions propounded at the
tournament.\footnote
  {\index{Woepcke, ref.\ to}Fr.~Woepcke in Liouville's \emph{Journal} for 1855, p.~54, has given an analysis
  of Leonardo's method of treating problems on square numbers.}
He also issued a tract dealing with determinate
algebraical problems: these are all solved by the \index{Assumption, rule of false}\index{False assumption, rule of}rule of false
assumption in the manner explained above.\index{Leonardo of Pisa|)}

\index{Frederick II. of Germany|(}\nblabel{png190a}\nbparagraph{Frederick II.} The Emperor \emph{Frederick II.}, who was born
in 1194, succeeded to the throne in 1210, and died in 1250,
was not only interested in science, but did as much as any
other single man of the thirteenth century to disseminate a
knowledge of the works of the Arab mathematicians in western
Europe. The \index{Naples, university of}university of Naples remains as a monument
of his munificence. I have already mentioned that the presence
of the \index{Jews, science of}Jews had been tolerated in Spain on account of their
medical skill and scientific knowledge, and as a matter of fact
the titles of physician and algebraist\footnote
  {For instance the reader may recollect that in \index{Don Quixote}\emph{Don Quixote} (part~ii,
  ch.~15), when Samson Carasco is thrown by the knight from his horse and
  has his ribs broken, an \index{Algebrista}\emph{algebrista} is summoned to bind up his wounds.}
were for a long time
nearly synonymous; thus the Jewish physicians were admirably
fitted both to get copies of the Arab works and to translate
them. Frederick II. made use of this fact to engage a staff of
learned Jews to translate the Arab works which he obtained,
though there is no doubt that he gave his patronage to them
the more readily because it was singularly offensive to the pope,
with whom he was then engaged in a quarrel. At any rate, by
the end of the thirteenth century copies of the works of \index{Euclid!ref.\ to}\index{Euclid's \emph{Elements}!ref.\ to}Euclid,
\index{Archimedes!ref.\ to}Archimedes, \index{Apollonius!ref.\ to}Apollonius, \index{Ptolemy!ref.\ to}Ptolemy, and of several Arab authors
were obtainable from this source, and by the end of the next
century were not uncommon. From this time, then, we may
say that the development of science in Europe was independent
of the aid of the Arabian schools.\index{Frederick II. of Germany|)}

\index{Algebra.!z5@Introduction of syncopated algebra by Bhaskara!x1@Jordanus|(}\index{Jordanus|(}\nblabel{png191a}\nbparagraph{Jordanus.\footnote
  {See \index{Cantor, M., ref.\ to}Cantor, chaps.~xliii, xliv, where references to the authorities on
  Jordanus are collected.}}
Among Leonardo's contemporaries was a German
mathematician, whose works were until the last few years almost
unknown. This was \emph{Jordanus Nemorarius}, sometimes called
\emph{Jordanus de Saxonia} or \emph{Teutonicus}. Of the details of his life
we know but little, save that he was elected general of the
Dominican order in 1222. The works enumerated in the footnote\footnote
  {\index{Curtze, M., ref.\ to}Prof.~Curtze, who has made a special study of the subject, considers that
  the following works are due to Jordanus. ``Geometria vel de Triangulis,''
  published by M.~Curtze in 1887 in vol.~vi of the \emph{Mitteilungen des Copernicus-Vereins
  zu Thorn}; \emph{De Isoperimetris}; \emph{Arithmetica Demonstrata}, published
  by \index{Stapulensis on Jordanus}\index{Faber Stapulensis on Jordanus}Faber Stapulensis at Paris in 1496, second edition, 1514; \emph{Algorithmus
  Demonstratus}, published by \index{Schoner@Schöner on Jordanus}J.~Schöner at Nuremberg in 1534; \emph{De Numeris
  Datis}, published by \index{Treutlein, ref.\ to}P.~Treutlein in 1879 and edited in 1891 with comments
  by M.~Curtze in vol.~xxxvi of the \emph{Zeitschrift für Mathematik und Physik};
  \emph{De Ponderibus}, published by \index{Apian on Jordanus}P.~Apian at Nuremberg in 1533, and reissued
  at Venice in 1565; and, lastly, two or three tracts on Ptolemaic astronomy.}
hereto are attributed to him, and if we assume that these
works have not been added to or improved by subsequent
annotators, we must esteem him one of the most eminent mathematicians
of the middle ages.

His knowledge of geometry is illustrated by his \emph{De Triangulis}
and \emph{De Isoperimetris}. The most important of these is the
\emph{De Triangulis}, which is divided into four books. The first
book, besides a few definitions, contains thirteen propositions on
triangles which are based on Euclid's \emph{Elements}. The second
book contains nineteen propositions, mainly on the ratios of
straight lines and the comparison of the areas of triangles; for
example, one problem is to find a point inside a triangle so that
the lines joining it to the angular points may divide the triangle
into three equal parts. The third book contains twelve propositions
mainly concerning arcs and chords of circles. The fourth
book contains twenty-eight propositions, partly on regular
polygons and partly on miscellaneous questions such as the
duplication and trisection problems.

The \emph{Algorithmus Demonstratus} contains practical rules for
the four fundamental processes, and Arabic numerals are
generally (but not always) used. It is divided into ten books
dealing with properties of numbers, primes, perfect numbers,
polygonal numbers, ratios, powers, and the progressions. It
would seem from it that Jordanus knew the general expression
for the square of any algebraic multinomial.

\nblabel{png192a}The \emph{De Numeris Datis} consists of four books containing
solutions of one hundred and fifteen problems. Some of these
lead to simple or quadratic equations involving more than one
unknown quantity. He shews a knowledge of proportion; but
many of the demonstrations of his general propositions are only
numerical illustrations of them.

In several of the propositions of the \emph{Algorithmus} and \emph{De
Numeris Datis} \index{Letters in diagrams!to indicate magnitudes}letters are employed to denote both known and
unknown quantities, and they are used in the demonstrations of
the rules of arithmetic as well as of algebra. As an example
of this I quote the following proposition,\footnote
  {From the \emph{De Numeris Datis}, book~i, prop.~3.}
the object of which is
to determine two quantities whose sum and product are known.

\begin{nbQuote}
\emph{Dato numero per duo diuiso si, quod ex ductu unius in alterum producitur,
datum fuerit, et utrumque eorum datum esse necesse est.}

Sit numerus datus \emph{abc} diuisus in \emph{ab} et \emph{c}, atque ex \emph{ab} in \emph{c} fiat \emph{d} datus,
itemque ex \emph{abc} in se fiat \emph{e}. Sumatur itaque quadruplum \emph{d}, qui fit \emph{f}, quo
dempto de \emph{e} remaneat \emph{g}, et ipse erit quadratum differentiae \emph{ab} ad \emph{c}.
Extrahatur ergo radix ex \emph{g}, et sit \emph{h}, eritque \emph{h} differentia \emph{ab} ad \emph{c}. cumque
sic \emph{h} datum, erit et \emph{c} et \emph{ab} datum.

Huius operatio facile constabit hoc modo. Verbi gratia sit x diuisus
in numeros duos, atque ex ductu unius eorum in alium fiat \textsc{xxi}; cuius
quadruplum et ipsum est \textsc{lxxxiiii}, tollatur de quadrato \textsc{x}, hoc est \textsc{c}, et
remanent \textsc{xvi}, cuius radix extrahatur, quae erit quatuor, et ipse est
differentia. Ipsa tollatur de \textsc{x} et reliquum, quod est \textsc{vi}, dimidietur,
eritque medietas \textsc{iii}, et ipse est minor portio et maior \textsc{vii}.
\end{nbQuote}

It will be noticed that Jordanus, like Diophantus and the
Hindoos, denotes \index{Addition, processes of!symbols for}\index{Plus!symbols for}addition by juxtaposition. Expressed in
modern notation his argument is as follows. Let the numbers
be $a + b$ (which I will denote by $\gamma$) and $c$. Then $\gamma + c$ is
given; hence $(\gamma + c)^{2}$ is known; denote it by $e$. Again $\gamma c$ is
given; denote it by $d$; hence $4\gamma c$, which is equal to $4d$, is
known; denote it by $f$. Then $(\gamma - c)^{2}$ is equal to $e-f$, which
is known; denote it by $g$. Therefore $\gamma - c = \surd g$, which is
known; denote it by $h$. Hence $\gamma + c$ and $\gamma - c$ are known,
and therefore $\gamma$ and $c$ can be at once found. It is curious
that he should have taken a sum like $a + b$ for one of his
unknowns. In his numerical illustration he takes the sum to
be 10 and the product 21.\index{Algebra.!z5@Introduction of syncopated algebra by Bhaskara!x1@Jordanus|)}

Save for one instance in Leonardo's writings, the above
works are the earliest instances known in European mathematics
of syncopated algebra in which letters are used for algebraical
symbols. It is probable that the \emph{Algorithmus} was not generally
known until it was printed in 1534, and it is doubtful how far
the works of Jordanus exercised any considerable influence on
the development of algebra. In fact it constantly happens in
the history of mathematics that improvements in notation or
method are made long before they are generally adopted or
their advantages realized. Thus the same thing may be discovered
over and over again, and it is not until the general
standard of knowledge requires some such improvement, or it is
enforced by some one whose zeal or attainments compel attention,
that it is adopted and becomes part of the science.
Jordanus in using letters or symbols to represent any quantities
which occur in analysis was far in advance of his contemporaries.
A similar notation was tentatively introduced by other and
later mathematicians, but it was not until it had been thus
independently discovered several times that it came into general
use.

It is not necessary to describe in detail the mechanics, optics,
or astronomy of Jordanus. The treatment of mechanics
throughout the middle ages was generally unintelligent.

No mathematicians of the same ability as Leonardo and
Jordanus appear in the history of the subject for over two
hundred years. Their individual achievements must not be
taken to imply the standard of knowledge then current, but
their works were accessible to students in the following two
centuries, though there were not many who seem to have
derived much benefit therefrom, or who attempted to extend the
bounds of arithmetic and algebra as there expounded.\index{Jordanus|)}

During the thirteenth century the most famous centres of
learning in western Europe were Paris and Oxford, and I must
now refer to the more eminent members of those schools.

\index{Holywood}\nblabel{png194a}\nbparagraph{Holywood.\footnote
  {See Cantor, chap.~xlv.}}
I will begin by mentioning \emph{John de Holywood},
whose name is often written in the latinized form of \emph{Sacrobosco}.
Holywood was born in Yorkshire and educated at Oxford; but
after taking his master's degree he moved to Paris, and taught
there till his death in 1244 or 1246. His lectures on \index{Algorism}algorism
and algebra are the earliest of which I can find mention. His
work on arithmetic was for many years a standard authority; it
contains rules, but no proofs; it was printed at Paris in 1496.
He also wrote a treatise on the sphere, which was made public
in 1256: this had a wide and long-continued circulation, and
indicates how rapidly a knowledge of mathematics was spreading.
Besides these, two pamphlets by him, entitled respectively \emph{De
Computo Ecclesiastico} and \emph{De Astrolabio}, are still extant.

\index{Bacon, Roger|(}\nblabel{png194b}\nbparagraph{Roger Bacon.\footnote
  {See \emph{Roger Bacon, sa vie, ses ouvrages} \dots\;by \index{Charles, E., on Roger Bacon}E.~Charles, Paris, 1861;
  and the memoir by \index{Brewer on Roger Bacon}J.~S.~Brewer, prefixed to the \emph{Opera Inedita}, Rolls Series,
  London, 1859: a somewhat depreciatory criticism of the former of these
  works is given in \emph{Roger Bacon, eine Monographie}, by \index{Schneider on Roger Bacon}L.~Schneider, Augsburg,
  1873.}}
 Another contemporary of Leonardo and
Jordanus was Roger Bacon, who for physical science did work
somewhat analogous to what they did for arithmetic and
algebra. \emph{Roger Bacon} was born near Ilchester in 1214, and
died at Oxford on June~11, 1294. He was the son of royalists,
most of whose property had been confiscated at the end of the
civil wars: at an early age he was entered as a student at
Oxford, and is said to have taken orders in 1233. In 1234
he removed to Paris, then the intellectual capital of western
Europe, where he lived for some years devoting himself especially
to languages and physics; and there he spent on books
and experiments all that remained of his family property and
his savings. He returned to Oxford soon after 1240, and there
for the following ten or twelve years he laboured incessantly,
being chiefly occupied in teaching science. His lecture room
was crowded, but everything that he earned was spent in buying
manuscripts and instruments. He tells us that altogether at
Paris and Oxford he spent over £2000 in this way---a sum
which represents at least £20,000 nowadays.

Bacon strove hard to replace logic in the university curriculum
by mathematical and linguistic studies, but the influences
of the age were too strong for him. His glowing eulogy on
``divine mathematics'' which should form the foundation of a
liberal education, and which ``alone can purge the intellect
and fit the student for the acquirement of all knowledge,'' fell
on deaf ears. We can judge how small was the amount of
geometry which was implied in the quadrivium, when he tells us
that in geometry few students at Oxford read beyond \index{Eucn@Euc.!a05@\textsc{i}, 5. ref.\ to}\index{Euclid's \emph{Elements}!ref.\ to}Euc.~\textsc{i}, 5;
though we might perhaps have inferred as much from the
character of the work of \index{Boethius!ref.\ to}Boethius.

At last worn out, neglected, and ruined, Bacon was persuaded
by his friend \index{Grosseteste, Bishop, ref.\ to}Grosseteste, the great Bishop of Lincoln,
to renounce the world and take the Franciscan vows. The
society to which he now found himself confined was singularly
uncongenial to him, and he beguiled the time by writing on
scientific questions and perhaps lecturing. The superior of the
order heard of this, and in 1257 forbade him to lecture or
publish anything under penalty of the most severe punishments,
and at the same time directed him to take up his residence at
Paris, where he could be more closely watched.

\index{Clement IV. of Rome, ref.\ to}Clement IV., when in England, had heard of Bacon's abilities,
and in 1266 when he became Pope he invited Bacon to write.
The Franciscan order reluctantly permitted him to do so, but
they refused him any assistance. With difficulty Bacon obtained
sufficient money to get paper and the loan of books, and in the
short space of fifteen months he produced in 1267 his \emph{Opus
Majus} with two supplements which summarized what was then
known in physical science, and laid down the principles on which
it, as well as philosophy and literature, should be studied. \index{Experiments, necessity of}He
stated as the fundamental principle that the study of natural
science must rest solely on experiment; and in the fourth part
he explained in detail how astronomy and physical sciences rest
ultimately on mathematics, and progress only when their fundamental
principles are expressed in a mathematical form. Mathematics,
he says, should be regarded as the alphabet of all
philosophy.

The results that he arrived at in this and his other works
are nearly in accordance with modern ideas, but were too far
in advance of that age to be capable of appreciation or perhaps
even of comprehension, and it was left for later generations to
rediscover his works, and give him that credit which he never
experienced in his lifetime. In astronomy he laid down the
principles for a reform of the calendar, explained the phenomena
of shooting stars, and stated that the \index{Ptolemy!ref.\ to}Ptolemaic system
was unscientific in so far as it rested on the assumption that
\index{Planetary motions}circular motion was the natural motion of a planet, while the
complexity of the explanations required made it improbable
that the theory was true. In \index{Optics (geometrical).!Discussed by (among others) Euclid!z3@Roger Bacon}optics he enunciated the laws of
reflexion and in a general way of \index{Refraction}refraction of light, and used
them to give a rough \index{Rainbow, explanation of}explanation of the rainbow and of magnifying
glasses. Most of his experiments in chemistry were directed
to the transmutation of metals, and led to no useful results. He
gave the composition of \index{Gunpowder, invention of|(}gunpowder, but there is no doubt that it
was not his own invention, though it is the earliest European
mention of it. On the other hand, some of his statements
appear to be guesses which are more or less ingenious, while
some of them are certainly erroneous.\index{Gunpowder, invention of|)}

In the years immediately following the publication of his
\emph{Opus Majus} he wrote numerous works which developed in
detail the principles there laid down. Most of these have now
been published, but I do not know of the existence of any
complete edition. They deal only with applied mathematics
and physics.

Clement took no notice of the great work for which he had
asked, except to obtain leave for Bacon to return to England.
On the death of Clement, the general of the Franciscan order
was elected Pope and took the title of \index{Nicholas IV. of Rome, ref.\ to}Nicholas IV\@. Bacon's
investigations had never been approved of by his superiors,
and he was now ordered to return to Paris, where we are told
he was immediately accused of magic; he was condemned in
1280 to imprisonment for life, but was released about a year
before his death.\index{Bacon, Roger|)}

\index{Campanus!ref.\ to}\index{Campanus}\nblabel{png197a}\nbparagraph{Campanus.} The only other mathematician of this century
whom I need mention is \emph{Giovanni Campano}, or in the latinized
form \emph{Campanus}, a canon of Paris. A copy of \index{Adelhard of Bath!ref.\ to}Adelhard's translation
of \index{Euclid's \emph{Elements}!ref.\ to}Euclid's \emph{Elements} fell into the hands of Campanus, who
added a commentary thereon in which he discussed the properties
of a regular re-entrant pentagon.\footnote
  {This edition of Euclid was printed by \index{Ratdolt on Campanus}Ratdolt at Venice in 1482, and
  was formerly believed to be due to Campanus. On this work see \index{Heiberg, ref.\ to}J.~L.~Heiberg
  in the \emph{Zeitschrift für Mathematik}, vol.~xxxv, 1890.}
He also, besides some minor
works, wrote the \emph{Theory of the Planets}, which was a free
translation of the \index{Almagest@\emph{Almagest}, the!ref.\ to}\emph{Almagest}.

\index{Universities, medieval!curriculum at|(}\nblabel{png197b}\nbparagraph{The fourteenth century.} The history of the fourteenth
century, like that of the one preceding it, is mostly concerned
with the assimilation of Arab mathematical text-books and of
Greek books derived from Arab sources.

\index{Bradwardine|(}\nblabel{png197c}\nbparagraph{Bradwardine.\footnote
  {See Cantor, vol.~ii, p.~102 \emph{et seq}.}}
A mathematician of this time, who was
perhaps sufficiently influential to justify a mention here, is
\emph{Thomas Bradwardine}, Archbishop of Canterbury. Bradwardine
was born at Chichester about 1290. He was educated at
Merton College, Oxford, and subsequently lectured in that
university. From 1335 to the time of his death he was chiefly
occupied with the politics of the church and state; he took a
prominent part in the invasion of France, the capture of Calais,
and the victory of Cressy. He died at Lambeth in 1349. His
mathematical works, which were probably written when he was
at Oxford, are the \emph{Tractatus de Proportionibus}, printed at Paris
in 1495; the \emph{Arithmetica Speculativa}, printed at Paris in 1502;
the \emph{Geometria Speculativa}, printed at Paris in 1511; and the
\emph{De Quadratura Circuli}, printed at Paris in 1495. They probably
give a fair idea of the nature of the mathematics then read
at an English university.\index{Bradwardine|)}

\index{Oresmus}\nblabel{png198a}\nbparagraph{Oresmus.\footnote
  {See \emph{Die mathematischen Schriften des Nicole Oresme}, by \index{Curtze, M., ref.\ to}M.~Curtze,
  Thorn, 1870.}}
\emph{Nicholas Oresmus} was another writer of the
fourteenth century. He was born at Caen in 1323, became the
confidential adviser of \index{Charles V. of France, ref.\ to}Charles V., by whom he was made tutor
to \index{Charles VI. of France, ref.\ to}Charles VI., and subsequently was appointed bishop of
Lisieux, at which city he died on July~11, 1382. He wrote the
\index{Algorism}\emph{Algorismus Proportionum}, in which the idea of \index{Indices}fractional indices
is introduced. He also issued a treatise dealing with questions
of coinage and commercial exchange; from the mathematical
point of view it is noticeable for the use of vulgar fractions and
the introduction of symbols for them.

By the middle of this century Euclidean geometry (as
expounded by Campanus) and algorism were fairly familiar to
all professed mathematicians, and the \index{Ptolemy!ref.\ to}Ptolemaic astronomy was
also generally known. About this time the \index{Almanacks}almanacks began to
add to the explanation of the Arabic symbols the rules of
addition, subtraction, multiplication, and division, ``de algorismo.''
The more important \index{Calendars}calendars and other treatises also inserted
a statement of the rules of proportion, illustrated by various
practical questions.

\nblabel{png199a}In the latter half of this century there was a general revolt
of the universities against the intellectual tyranny of the schoolmen.
This was largely due to \index{Petrarch}Petrarch, who in his own generation
was celebrated as a humanist rather than as a poet, and
who exerted all his power to destroy scholasticism and encourage
scholarship. The result of these influences on the study of
mathematics may be seen in the changes then introduced in
the study of the \index{Quadrivium}quadrivium. The stimulus came from the
university of \index{Paris, university of}Paris, where a statute to that effect was passed
in 1366, and a year or two later similar regulations were
made at other universities; unfortunately no text-books are
mentioned. We can, however, form a reasonable estimate of
the range of mathematical reading required, by looking at
the statutes of the universities of \index{Prague, university of}Prague, of \index{Vienna, university of}Vienna, and of
\index{Leipzig, university of}Leipzig.

By the statutes of Prague, dated 1384, candidates for the
bachelor's degree were required to have read \index{Holywood!ref.\ to}Holywood's treatise
on the sphere, and candidates for the master's degree to be
acquainted with the \index{Euclid's \emph{Elements}!ref.\ to}first six books of Euclid, optics, hydrostatics,
the theory of the lever, and astronomy. Lectures were actually
delivered on arithmetic, the art of reckoning with the fingers,
and the algorism of integers; on almanacks, which probably
meant elementary \index{Astrology}astrology; and on the \index{Almagest@\emph{Almagest}, the!ref.\ to}\emph{Almagest}, that is, on
Ptolemaic astronomy. There is, however, some reason for
thinking that mathematics received far more attention here than
was then usual at other universities.

At Vienna, in 1389, a candidate for a master's degree was
required to have read five books of Euclid, common perspective,
proportional parts, the measurement of superficies, and the
\emph{Theory of the Planets}. The book last named is the treatise by
\index{Campanus!ref.\ to}Campanus which was founded on that by \index{Ptolemy!ref.\ to}Ptolemy. This was a
fairly respectable mathematical standard, but I would remind
the reader that there was no such thing as ``plucking'' in a
medieval university. The student had to keep an act or give
a lecture on certain subjects, but whether he did it well or
badly he got his degree, and it is probable that it was only the
few students whose interests were mathematical who really
mastered the subjects mentioned above.

\nblabel{png200a}\nbparagraph{The fifteenth century.} A few facts gleaned from the history
of the fifteenth century tend to shew that the regulations about
the study of the \index{Quadrivium}quadrivium were not seriously enforced. The
lecture lists for the years 1437 and 1438 of the university of
\index{Leipzig, university of}Leipzig (founded in 1409, the statutes of which are almost
identical with those of \index{Prague, university of}Prague as quoted above) are extant, and
shew that the only lectures given there on mathematics in those
years were confined to astrology. The records of \index{Bologna, university of}Bologna,
\index{Padua, university of}Padua, and \index{Pisa, university of}Pisa seem to imply that there also astrology was
the only scientific subject taught in the fifteenth century, and
even as late as 1598 the professor of mathematics at Pisa was
required to lecture on the \emph{Quadripartitum}, an astrological work
purporting (probably falsely) to have been written by \index{Ptolemy!ref.\ to}Ptolemy.
The only mathematical subjects mentioned in the registers of
the university of \index{Oxford, university of}Oxford as read there between the years 1449
and 1463 were Ptolemy's astronomy, or some commentary on it,
and the first two books of \index{Euclid's \emph{Elements}!ref.\ to}Euclid. Whether most students got
as far as this is doubtful. It would seem, from an edition of
Euclid's \emph{Elements} published at \index{Paris, university of}Paris in 1536, that after 1452
candidates for the master's degree at that university had to take
an oath that they had attended lectures on the first six books of
that work.

\index{Beldomandi}\nblabel{png200b}\nbparagraph{Beldomandi.} The only writer of this time that I need
mention here is \emph{Prodocimo Beldomandi} of Padua, born about
1380, who wrote an algoristic arithmetic, published in 1410,
which contains the summation of a geometrical series; and
some geometrical works.\footnote
  {For further details see Boncompagni's \emph{Bulletino di bibliografia,}
  vols.~xii, xviii.}

By the middle of the fifteenth century printing had been
introduced, and the facilities it gave for disseminating knowledge
were so great as to revolutionize the progress of science. We
have now arrived at a time when the results of Arab and Greek
science were known in Europe; and this perhaps, then, is as
good a date as can be fixed for the close of this period, and the
commencement of that of the renaissance. The mathematical
history of the renaissance begins with the career of Regiomontanus;
but before proceeding with the general history it will be
convenient to collect together the chief facts connected with the
development of arithmetic during the middle ages and the
renaissance. To this the \hyperref[chapter:XI]{next chapter} is devoted.\index{Universities, medieval!curriculum at|)}

\nbChapter{XI}{The Development of Arithmetic}{}{Circ.~1300--1637.}
  {See the article on Arithmetic by G.~Peacock in the \emph{Encyclopaedia
  Metropolitana}, vol.~i, London, 1845; \emph{Arithmetical Books} by A.~De~Morgan,
  London, 1847; and an article by P.~Treutlein of Karlsruhe, in the \emph{Zeitschrift
  für Mathematik}, 1877, vol.~xxii, supplement, pp.~1--100.}
\index{Arithmetic.!z7@Algoristic arithmetic invented by the Hindoos!x5@development of European arithmetic, 1300--1637, chapter \textsc{xi}}\index{Demorgana@De~Morgan, A.!ref.\ to}\index{Peacock!ref.\ to}\index{Treutlein, ref.\ to}\nblabel{png202a}

\nblabel{png202b}\nblabel{png202c}\First{We} have seen in the \hyperref[chapter:X]{last chapter} that by the end of the
thirteenth century the Arabic arithmetic had been fairly introduced
into Europe and was practised by the side of the older
arithmetic which was founded on the work of \index{Boethius!ref.\ to}Boethius. It will
be convenient to depart from the chronological arrangement and
briefly to sum up the subsequent history of arithmetic, but I
hope, by references in the \hyperref[chapter:XII]{next chapter} to the inventions and
improvements in arithmetic here described, that I shall be able
to keep the order of events and discoveries clear.

\index{Arithmetic.!z5@Theory of,!x2@and thenceforward treated empirically (Boethian arithmetic) by most of the Greek and European mathematicians to the end of the fourteenth century after Christ|(}The older arithmetic consisted of two parts: practical arithmetic
or the art of calculation which was taught by means of
the abacus and possibly the multiplication table; and theoretical
arithmetic, by which was meant the ratios and properties of
numbers taught according to \index{Boethius!ref.\ to}Boethius---a knowledge of the
latter being confined to professed mathematicians. \index{Inductive arithmetic|(}\index{Mercantile arithmetic|(}The theoretical
part of this system continued to be taught till the middle
of the fifteenth century, and the practical part of it was used by
the smaller tradesmen in England,\footnote
  {See, for instance, \index{Chaucer, ref.\ to}Chaucer, \emph{The Miller's Tale}, \textsc{v}, 22--25; \index{Shakespeare, ref.\ to}Shakespeare,
  \emph{The Winter's Tale}, Act~\textsc{iv}, Sc.~2; \emph{Othello}, Act~\textsc{i}, Sc.~1. There are similar
  references in French and German literature; notably by Montaigne and
  Molière. I believe that the \index{Exchequer, Court of}Exchequer division of the High Court of Justice
  derives its name from the table before which the judges and officers of the
  court originally sat: this was covered with black cloth divided into squares
  or chequers by white lines, and apparently was used as an \index{Abacus,!ref.\ to}abacus.}
Germany, and France till
the beginning of the seventeenth century.\index{Arithmetic.!z5@Theory of,!x2@and thenceforward treated empirically (Boethian arithmetic) by most of the Greek and European mathematicians to the end of the fourteenth century after Christ|)}\index{Inductive arithmetic|)}

\nblabel{png203a}The new Arabian arithmetic was called \index{Algorism}\emph{algorism} or the art of
\index{Alkarismi!ref.\ to}Alkarismi, to distinguish it from the old or Boethian arithmetic.
The text-books on algorism commenced with the Arabic system
of notation, and began by giving rules for addition, subtraction,
multiplication, and division; the principles of proportion were
then applied to various practical problems, and the books usually
concluded with general rules for many of the common problems of
commerce. Algorism was in fact a mercantile arithmetic, though
at first it also included all that was then known as algebra.

\index{Algebraa@Algebra,!definitions of}Thus algebra has its origin in arithmetic; and to most people
the term \emph{universal arithmetic}, by which it was sometimes designated,
conveys a more accurate impression of its objects and
methods than the more elaborate definitions of modern
mathematicians---certainly better than the definition of \index{Hamilton, Sir Wm.!ref.\ to}Sir William
Hamilton as the science of pure time, or that of De~Morgan as
the calculus of succession. No doubt logically there is a marked
distinction between arithmetic and algebra, for the former is the
theory of discrete magnitude, while the latter is that of continuous
magnitude; but a scientific distinction such as this is of
comparatively recent origin, and the idea of continuity was not
introduced into mathematics before the time of \index{Kepler!ref.\ to}Kepler.

Of course the fundamental rules of this algorism were not at
first strictly proved---that is the work of advanced thought---but
until the middle of the seventeenth century there was some
discussion of the principles involved; since then very few arithmeticians
have attempted to justify or prove the processes used,
or to do more than enunciate rules and illustrate their use by
numerical examples.

\index{Arabic numerals|(}\index{Numerals, symbols for|(}\nblabel{png204a}I have alluded frequently to the Arabic system of numerical
notation. \index{Arabic numerals!origin of}I may therefore conveniently begin by a few notes on
the history of the symbols now current.

\nblabel{xref:184}Their origin is obscure and has been much disputed.\footnote
  {See A.~L'Esprit, \emph{Histoire des chiffres}, Paris, 1893; \index{Pihan on numerals}A.~P.~Pihan, \emph{Signes
  de numération}, Paris, 1860; \index{Woepcke, ref.\ to}Fr.~Woepcke, \emph{La propagation des chiffres
  Indiens}, Paris, 1863; \index{Burnell on numerals}A.~C.~Burnell, \emph{South Indian Palaeography}, Mangalore,
  1874; \index{Taylor, Is., on numerals}Is.~Taylor, \emph{The Alphabet}, London, 1883; and \index{Cantor, M., ref.\ to}Cantor.}
\index{Decimal numeration|(}\index{Indian numerals|(}On
the whole it seems probable that the symbols for the numbers 4,
5, 6, 7, and~9 (and possibly 8 too) are derived from the initial
letters of the corresponding words in the Indo-Bactrian alphabet
in use in the north of India perhaps 150 years before Christ;
that the symbols for the numbers 2~and~3 are derived respectively
from two and three parallel penstrokes written cursively; and
similarly that the symbol for the number~1 represents a single
penstroke. Numerals of this type were in use in India before
the end of the second century of our era. \index{Zero, symbol for|(}The origin of the
symbol for zero is unknown; it is not impossible that it was
originally a dot inserted to indicate a blank space, or it may
represent a closed hand, but these are mere conjectures; there
is reason to believe that it was introduced in India towards the
close of the fifth century of our era, but the earliest writing now
extant in which it occurs is assigned to the eighth century.

\medskip\noindent
{\small
\begin{tabular}{@{}m{0.33\textwidth}@{}m{0.05\textwidth}@{}m{0.62\textwidth}@{}}
Devanagari (Indian) numerals, \textit{circ.}~950.& \centering$\Bigr\}$
  & \raisebox{-2ex}{\labelincludegraphics[width=\linewidth]{205-deva}}   \\
Gobar Arabic numerals, \textit{circ.}~1100 (?).& \centering$\Bigr\}$
  & \labelincludegraphics[width=\linewidth]{205-gobar}  \\
From a missal, \textit{circ.}~1385, of German origin.& \centering$\Biggr\}$
  & \labelincludegraphics[width=\linewidth]{205-missal} \\
European (probably Italian) numerals, \textit{circ.}~1400.& \centering$\Biggr\}$
  & \labelincludegraphics[width=\linewidth]{205-italian}\\
From the \textit{Mirrour of the World}, printed by Caxton in 1480.& \centering$\Biggr\}$
  & \labelincludegraphics[width=\linewidth]{205-caxton} \\
\nbstretchyspace From a Scotch calendar for 1482, probably of French origin.& \centering$\Biggr\}$
  & \labelincludegraphics[width=\linewidth]{205-scotch}
\end{tabular}}

\medskip
\index{Indian numerals!origin of|(}The numerals used in India in the eighth century and for a
long time afterwards are termed \index{Devanagari numerals}Devanagari numerals, and their
forms are shewn in the first line of the table given above. %[**TN: Replaced 'on the next page' with 'above']
These forms were slightly modified by the eastern Arabs,
and the resulting symbols were again slightly modified by the
western Arabs or Moors. It is perhaps probable that at first
the Spanish Arabs discarded the use of the symbol for zero, and
only reinserted it when they found how inconvenient the omission
proved. The symbols ultimately adopted by the Arabs are
termed \index{Gobar numerals}Gobar numerals, and an idea of the forms most commonly
used may be gathered from those printed in the second line of
the table given above. From Spain or Barbary the Gobar %[**TN: Replaced 'on next page' with 'above']
numerals passed into western Europe, and they occur on a
Sicilian coin as early as 1138. The further evolution of
\index{Arabic numerals!origin of}the forms of the symbols to those with which we are familiar is
indicated below by facsimiles\index{Zero, symbol for|)}\footnote
  {The first, second, and fourth examples are taken from \index{Taylor, Is., on numerals}Is.~Taylor's
  \emph{Alphabet}, London, 1883, vol.~ii, p.~266; the others are taken from \index{Leslie on arithmetic}Leslie's
  \emph{Philosophy of Arithmetic}, 2nd ed., Edinburgh, 1820, pp.~114, 115.}
of the numerals used at different
times. All the sets of numerals here represented are written
from left to right and in the order 1,~2, 3, 4, 5, 6, 7, 8, 9,~10.
From 1500 onwards the symbols employed are practically the
same as those now in use.\footnote
  {See, for example, \index{Tonstall}Tonstall's \emph{De Arte Supputandi}, London, 1522; or
  \index{Record!ref.\ to}Record's \emph{Grounde of Artes}, London, 1540, and \emph{Whetstone of Witte}, London,
  1557.}

\begin{center}
\labelincludegraphics[width=0.6\linewidth]{205-arabic}
\end{center}

The further evolution in the East of the \index{Gobar numerals}Gobar numerals
proceeded almost independently of European influence. There
are minute differences in the forms used by various writers, and
in some cases alternative forms; without, however, entering into
these details we may say that the numerals they commonly
employed finally took the form shewn above, but the symbol
there given for~4 is at the present time generally written
cursively.\index{Indian numerals!origin of|)}

\index{Arithmetic.!z7@Algoristic arithmetic invented by the Hindoos!x4@and used since the fourteenth century in Europe|(}\nblabel{png206a}Leaving now the history of the symbols I proceed to discuss
their introduction into general use and the development of
algoristic arithmetic. I have already explained how men of
science, and particularly astronomers, had become acquainted
with the Arabic system by the middle of the thirteenth century.
The trade of Europe during the thirteenth and fourteenth
centuries was mostly in Italian hands, and the obvious advantages
of the algoristic system led to its general adoption
in Italy for mercantile purposes. This change was not effected,
however, without considerable opposition; thus, an edict was
issued at Florence in 1299 forbidding bankers to use Arabic
numerals, and in 1348 the authorities of the university of \index{Padua, university of}Padua
directed that a list should be kept of books for sale with the
prices marked ``non per cifras sed per literas claras.''

The rapid spread of the use of Arabic numerals and arithmetic
through the rest of Europe seems to have been as largely due to
the makers of \index{Almanacks|(}almanacks and \index{Calendars|(}calendars as to merchants and
men of science. These calendars had a wide circulation in
medieval times. Some of them were composed with special
reference to ecclesiastical purposes, and contained the dates of
the different festivals and fasts of the church for a period of
some seven or eight years in advance, as well as notes on church
ritual. Nearly every monastery and church of any pretensions
possessed one of these. Others were written specially for the
use of astrologers and physicians, and some of them contained
notes on various scientific subjects, especially medicine and astronomy.
Such almanacks were not then uncommon, but, since it
was only rarely that they found their way into any corporate
library, specimens are now rather scarce. It was the fashion to
use the Arabic symbols in ecclesiastical works; while their
occurrence in all astronomical tables and their Oriental origin
(which savoured of magic) secured their use in calendars intended
for scientific purposes. Thus the symbols were generally employed
in both kinds of almanacks, and there are but few specimens
of calendars issued after the year 1300 in which an explanation
of the Arabic numerals is not included. Towards the middle of
the fourteenth century the rules of arithmetic \emph{de algorismo} were
also sometimes added, and by the year 1400 we may consider
that the Arabic symbols were generally known throughout
Europe, and were used in most scientific and astronomical
works.\index{Indian numerals|)}\index{Decimal numeration|)}\index{Calendars|)}\index{Almanacks|)}

Outside Italy most merchants continued, however, to keep
their accounts in Roman numerals till about 1550, and
monasteries and colleges till about 1650; though in both
cases it is probable that in and after the fifteenth century the
processes of arithmetic were performed in the algoristic manner.
Arabic numerals are used in the pagination of some books issued
at Venice in 1471 and 1482. No instance of a date or number
being written in Arabic numerals is known to occur in any
English parish register or the court rolls of any English
manor before the sixteenth century; but in the rent-roll of
the St Andrews Chapter, Scotland, the Arabic numerals
were used in 1490. The Arabic numerals were used in
Constantinople by \index{Planudes!ref.\ to}Planudes\footnote
  {See above, p.~\pageref{xref:117}.}
in the fourteenth century.\index{Numerals, symbols for|)}\index{Arabic numerals|)}\index{Arithmetic.!z7@Algoristic arithmetic invented by the Hindoos!x4@and used since the fourteenth century in Europe|)}

The history of modern mercantile arithmetic in Europe
begins then with its use by Italian merchants, and it is
especially to the Florentine traders and writers that we owe
its early development and improvement. It was they who
invented the system of \index{Book-keeping}\index{Double entry, book-keeping by}book-keeping by double entry. In this
system every transaction is entered on the credit side in one
ledger, and on the debtor side in another; thus, if cloth be sold
to~\emph{A}, \emph{A}'s~account is debited with the price, and the stock-book,
containing the transactions in cloth, is credited with the amount
sold. It was they, too, who arranged the problems to which
arithmetic could be applied in different classes, such as rule of
three, interest, profit and loss, \&c. They also reduced the
fundamental operations of arithmetic ``to seven, in reverence,''
says \index{Pacioli!ref.\ to}Pacioli, ``of the seven gifts of the Holy Spirit: namely,
numeration, addition, subtraction, multiplication, division,
raising to powers, and extraction of roots.'' \index{Brahmagupta!ref.\ to}Brahmagupta
had enumerated twenty processes, besides eight subsidiary ones,
and had stated that ``a distinct and several knowledge of these''
was ``essential to all who wished to be calculators''; and,
whatever may be thought of \index{Pacioli!ref.\ to}Pacioli's reason for the alteration,
the consequent simplification of the elementary processes was
satisfactory. It may be added that arithmetical schools were
founded in various parts of Germany, especially in and after the
fourteenth century, and did much towards familiarizing traders
in northern and western Europe with commercial algoristic
arithmetic.

\nblabel{png208a}The operations of algoristic arithmetic were at first very
cumbersome. The chief improvements subsequently introduced
into the early Italian \index{Algorism}algorism were (i)~the simplification of the
four fundamental processes; (ii)~the introduction of signs for
addition, subtraction, equality, and (though not so important)
for multiplication and division; (iii)~the invention of
logarithms; and (iv)~the use of decimals. I will consider
these in succession.

\nblabel{png208b}(i)~In \index{Addition, processes of}addition and \index{Subtraction,!processes of}subtraction the Arabs usually worked
from left to right. The modern plan of working from right to
left is said to have been introduced by an Englishman named
\index{Garth, ref.\ to}Garth, of whose life I can find no account. The old plan continued
in partial use till about 1600; even now it would be
more convenient in approximations where it is necessary to keep
only a certain number of places of decimals.

\index{Multiplication, processes of|(}The Indians and Arabs had several systems of multiplication.
These were all somewhat laborious, and were made the more so
as multiplication tables, if not unknown, were at any rate used
but rarely. The operation was regarded as one of considerable
difficulty, and the test of the accuracy of the result by \index{Nines, casting out the}``casting
out the nines'' was invented as a check on the correctness of the
work. Various other systems of multiplication were
subsequently employed in Italy, of which several examples are
given by \index{Pacioli!ref.\ to}Pacioli and \index{Tartaglia!ref.\ to}Tartaglia; and the use of the multiplication
table---at least as far as $5 × 5$---became common. From
this limited table the resulting product of the multiplication of
all numbers up to $10 × 10$ can be deduced by what was termed
the \index{Regula ignavi}\emph{regula ignavi}. This is a statement of the identity
$(5+a)(5+b) = (5-a)(5-b)+10(a+b)$. The rule was usually
enunciated in the following form. Let the number five be
represented by the open hand; the number six by the hand with
one finger closed; the number seven by the hand with two
fingers closed; the number eight by the hand with three fingers
closed; and the number nine by the hand with four fingers
closed. To multiply one number by another let the multiplier be
represented by one hand, and the number multiplied by the
other, according to the above convention. Then the required
answer is the product of the number of fingers (counting the
thumb as a finger) open in the one hand by the number of
fingers open in the other together with ten times the total
number of fingers closed. The system of multiplication now
in use seems to have been first introduced at Florence.

\begin{figure*}[!hbt]
\nblabel{fig:napier}
\centering
\usesanstrue
\hspace*{1em}
\begin{minipage}{16em}
\centering
\begin{NapierRods}{10}[9]
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 0\\
{/2} & { /4} & { /6} & { /8} & {1/0} & {1/2} & {1/4} & {1/6} & {1/8} & {/0}\\
{/3} & { /6} & { /9} & {1/2} & {1/5} & {1/8} & {2/1} & {2/4} & {2/7} & {/0}\\
{/4} & { /8} & {1/2} & {1/6} & {2/0} & {2/4} & {2/8} & {3/2} & {3/6} & {/0}\\
{/5} & {1/0} & {1/5} & {2/0} & {2/5} & {3/0} & {3/5} & {4/0} & {4/5} & {/0}\\
{/6} & {1/2} & {1/8} & {2/4} & {3/0} & {3/6} & {4/2} & {4/8} & {5/4} & {/0}\\
{/7} & {1/4} & {2/1} & {2/8} & {3/5} & {4/2} & {4/9} & {5/6} & {6/3} & {/0}\\
{/8} & {1/6} & {2/4} & {3/2} & {4/0} & {4/8} & {5/6} & {6/4} & {7/2} & {/0}\\
{/9} & {1/8} & {2/7} & {3/6} & {4/5} & {5/4} & {6/3} & {7/2} & {8/1} & {/0}
\end{NapierRods}
\footnotesize Figure~1.
\end{minipage}
\hfill
\begin{minipage}{4em}
\centering
\begin{NapierRods}{1}[9]
7 \\
{1/4}\\
{2/1}\\
{2/8}\\
{3/5}\\
{4/2}\\
{4/9}\\
{5/6}\\
{6/3}
\end{NapierRods}
\footnotesize Figure~2.
\end{minipage}
\hfill
\begin{minipage}{7em}
\centering\makeatletter
\begin{NapierRods}{4}[9]
2 & 9 & 8 & 5\\
{ /4} & {1/8} & {1/6} & {1/0}\\
{ /6} & {2/7} & {2/4} & {1/5}\\
{ /8} & {3/6} & {3/2} & {2/0}\\
{1/0} & {4/5} & {4/0} & {2/5}\\
{1/2} & {5/4} & {4/8} & {3/0}\\
{1/4} & {6/3} & {5/6} & {3/5}\\
{1/6} & {7/2} & {6/4} & {4/0}\\
{1/8} & {8/1} & {7/2} & {4/5}
\end{NapierRods}
\footnotesize Figure~3.
\end{minipage}
\hspace*{1em}
\end{figure*}


The difficulty which all but professed mathematicians
experienced in the multiplication of large numbers led to the
invention of several mechanical ways of effecting the process.
Of these the most celebrated is that of \index{Rods, Napier's|(}\index{Napier's rods|(}Napier's rods invented in
1617. In principle it is the same as a method which had been
long in use both in India and Persia, and which has been
described in the diaries of several travellers, and notably in
the \index{Chardin, Sir John, ref.\ to}\emph{Travels of Sir John Chardin in Persia}, London, 1686.
To use the method a number of rectangular slips of bone, wood,
metal, or cardboard are prepared, and each of them divided by
cross lines into nine little squares, a slip being generally about
three inches long and a third of an inch across. In the top
square one of the digits is engraved, and the results of multiplying
it by 2,~3, 4, 5, 6, 7, 8, and~9 are respectively entered in the
eight lower squares; where the result is a number of two digits,
the ten-digit is written above and to the left of the unit-digit
and separated from it by a diagonal line. The slips are usually
arranged in a box. \figureref{napier}{Figure~1} above represents nine %[**TN: Replaced 'on the next page' with 'above']
such slips side by side; \figureref{napier}{figure~2} shews the seventh slip, which
is supposed to be taken out of the box and put by itself.
Suppose we wish to multiply 2985 by 317. The process as
effected by the use of these slips is as follows. The slips headed
2, 9, 8, and 5 are taken out of the box and put side by side as
shewn in \figureref{napier}{figure~3} above. The result of multiplying 2985 by 7
may be written thus---

\begin{center}
\begin{tabular}{@{} c @{} c @{} c @{} c @{} c @{}}
  & 2 & 9 & 8 & 5\\
  &   &   &   & 7\\\cline{2-5}
  &   &   & 3 & 5\\
  &   & 5 & 6 &  \\
  & 6 & 3 &   &  \\
1 & 4 &   &   &  \\ \cline{1-5}
2 & 0 & 8 & 9 & 5\\\hline\hline
\end{tabular}
\end{center}

Now if the reader will look at the seventh line in \figureref{napier}{figure~3},
he will see that the upper and lower rows of figures are respectively
1653 and 4365; moreover, these are arranged by the
diagonals so that roughly the 4 is under the 6, the 3 under the
5, and the 6 under the 3; thus

\begin{center}
\begin{tabular}{ c c c c c}
1 & 6 & 5 & 3 &   \\
  & 4 & 3 & 6 & 5
\end{tabular}
\end{center}

The addition of these two numbers gives the required result.
Hence the result of multiplying by 7, 1, and 3 can be successively
determined in this way, and the required answer (namely,
the product of 2985 and 317) is then obtained by addition.

The whole process was written as follows:
\begin{center}
\begin{tabular}{@{}l@{}l@{}}
\phantom{82}2985 & \\ \hline
\phantom{8}20895  &\phantom{000}/7 \\
\phantom{8}2985   &\phantom{00}/1 \\
8955              &\phantom{0}/3 \\ \hline
946245 & \\ \hline \hline
\end{tabular}
\end{center}

The modification introduced by Napier in his \index{Rabdologia@\emph{Rabdologia}, the}\emph{Rabdologia},
published in 1617, consisted merely in replacing each slip by a
prism with square ends, which he called ``a rod,'' each lateral
face being divided and marked in the same way as one of the
slips above described. These rods not only economized space,
but were easier to handle, and were arranged in such a way as
to facilitate the operations required.\index{Napier's rods|)}\index{Rods, Napier's|)}

\begin{center}
\nblabel{fig:division}
\setlength{\doublerulesep}{1.5pt}
\small\sffamily
\begin{tabular}{|*{17}{c|}}
                                        \cline{1-5}\cline{7-11}\cline{13-17}
1&7&9&7&8&   &1&7&9&7&8&   &1&7&9&7&8\\
 & & & & &   &1&2& & & &   &1&2& & & \\            \cline{7-11}\cline{13-17}
 & & & & &   & &5&9&7&8&   & &5&9&7&8\\
 & & & & &   & &2&1& & &   & &2&1& & \\            \cline{8-11}\cline{14-17}
 & & & & &   & &3&8&7&8&   & &3&8&7&8\\
 & & & & &   & & & &6& &   & & & &6& \\ \hhline{|*{5}{~|}~|~|*{4}{-|}~|-|*{4}{=|}}
 & & & & &   & &3&8&1&8&   & &3&8&1&8\\
 & & & & &   & & & & & &   & &3&2& & \\                        \cline{14-17}
 & & & & &   & & & & & &   & & &6&1&8\\
 & & & & &   & & & & & &   & & &5&6& \\                        \cline{15-17}
 & & & & &   & & & & & &   & & & &5&8\\
 & & & & &   & & & & & &   & & & &1&6\\ \hhline{|*{5}{~|}~|*{5}{~|}~|-|-|-|=|=|}
 & & & & &   & & & & & &   & & & &4&2\\ \hhline{|*{5}{~|}~|*{5}{~|}~|*{5}{=|}}
 & & & & &   & & & & & &   & & &4&7&2\\
 & & & & &   & &4&7&2& &   & &4&7&2& \\
4&7&2& & &&\rlap{$\backslash$}4&\rlap{$\backslash$}7&\rlap{$\backslash$}2&&& &4&7&2&&\\ \cline{1-5}\cline{7-11}\cline{13-17}
 & &0& & &   & & &0&3& &   & & &0&3&8\\ \hhline{|*{5}{=|}~|*{5}{=|}~|*{5}{=|}}
\multicolumn{5}{c}{\normalfont\rule{0pt}{3ex}Figure 1.} & \multicolumn{1}{c}{} &
\multicolumn{5}{c}{\normalfont Figure 2.} & \multicolumn{1}{c}{} &
\multicolumn{5}{c}{\normalfont Figure 3.}
\end{tabular}
\end{center}

\index{Division,!processes of|(}If multiplication was considered difficult, division was at first
regarded as a feat which could be performed only by skilled
mathematicians. The method commonly employed by the
Arabs and Persians for the division of one number by another
will be sufficiently illustrated by a concrete instance. Suppose
we require to divide 17978 by 472. A sheet of paper is divided
into as many vertical columns as there are figures in the number
to be divided. The number to be divided is written at the top
and the divisor at the bottom; the first digit of each number
being placed at the left-hand side of the paper. Then, taking
the left-hand column, 4 will go into~1 no times, hence the first
figure in the dividend is~0, which is written under the last figure
of the divisor. This is represented in \figureref{division}{figure~1} above. %[**TN: Replaced 'on the next page' with 'above']
Next (see \figureref{division}{figure~2}) rewrite the 472 immediately above its
former position, but shifted one place to the right, and cancel
the old figures. Then 4 will go into~17 four times; but, as
on trial it is found that 4 is too big for the first digit of the
dividend, 3 is selected; 3 is therefore written below the last
digit of the divisor and next to the digit of the dividend last
found. The process of multiplying the divisor by 3 and subtracting
from the number to be divided is indicated in \figureref{division}{figure~2},
and shews that the remainder is 3818. A similar process is
then repeated, that is, 472 is divided into 3818, shewing that
the quotient is 38 and the remainder 42. This is represented in
\figureref{division}{figure~3}, which shews the whole operation.\index{Multiplication, processes of|)}

The method described above never found much favour in
Italy. The present system was in use there as early as the
beginning of the fourteenth century, but the method generally
employed was that known as the \index{Galley system of division|(}\emph{galley} or \index{Scratch system of division|(}\emph{scratch} system. The
following example from \index{Tartaglia!ref.\ to|(}Tartaglia, in which it is required
to divide 1330 by 84, will serve to illustrate this method: the
arithmetic given by Tartaglia is shewn below, where numbers in %[**TN: "above" changed to "below"]
thin type are supposed to be scratched out in the course of the
work.\index{Tartaglia!ref.\ to|)}
\begin{center}
\begin{tabular}{c *{5}{@{\;}c}}
 &\textbf{0}&\textbf{7}& & &\\
 &4&9& & &\\
0&5&9&\textbf{0}& &\\
1&3&3&0&(&\textbf{15}\\
 &8&4&4& &\\
 & &8& & &
 \end{tabular}
\end{center}

The process is as follows. First write the 84 beneath the
1330, as indicated below, then 84 will go into 133 once, hence
the first figure in the quotient is 1. Now $1 × 8 = 8$, which subtracted
from 13 leaves 5. Write this above the 13, and cancel
the 13 and the 8, and we have as the result of the first step
\begin{center}
\begin{tabular}{c *{5}{@{\;}c}}
 &\textbf{5}& & & & \\
1&3&\textbf{3}&\textbf{0}&(&\textbf{1}\\
 &8&\textbf{4}& & &
 \end{tabular}
\end{center}
Next, $1 × 4 = 4$, which subtracted from 53 leaves 49. Insert
the 49, and cancel the 53 and the 4, and we have as the next
step
\begin{center}
\begin{tabular}{c *{5}{@{\;}c}}
 &\textbf{4}& & & &\\
 &5&\textbf{9}& & &\\
1&3&3&\textbf{0}&(&\textbf{1}\\
 &8&4& &
 \end{tabular}
\end{center}
which shews a remainder 490.

We have now to divide 490 by 84. Hence the next figure
in the quotient will be 5, and re-writing the divisor we have

\begin{center}
\begin{tabular}{c *{5}{@{\;}c}}
 &\textbf{4}& & & &\\
 &5&\textbf{9}& & &\\
1&3&3&\textbf{0}&(&\textbf{15}\\
 &8&4&\textbf{4}& &\\
 & &\textbf{8}& & &
 \end{tabular}
\end{center}
Then $5 × 8 = 40$, and this subtracted from 49 leaves 9. Insert
the 9, and cancel the 49 and the 8, and we have the following
result

\begin{center}
\begin{tabular}{c *{5}{@{\;}c}}
 &4&\textbf{9}& & &\\
 &5&9& & &\\
1&3&3&\textbf{0}&(&\textbf{15}\\
 &8&4&\textbf{4}& &\\
 & &8& & &
 \end{tabular}
\end{center}

Next $5 × 4 = 20$, and this subtracted from 90 leaves 70. Insert
the 70, and cancel the 90 and the 4, and the final result, shewing
a remainder 70, is

\begin{center}
\begin{tabular}{c *{5}{@{\;}c}}
 & &\textbf{7}& & &\\
 &4&9& & &\\
 &5&9&\textbf{0}& &\\
1&3&3&0&(&\textbf{15}\\
 &8&4&4& &\\
 & &8& & &
 \end{tabular}
\end{center}
The three extra zeros inserted in Tartaglia's work are unnecessary,
but they do not affect the result, as it is evident that a
figure in the dividend may be shifted one or more places up in
the same vertical column if it be convenient to do so.\index{Mercantile arithmetic|)}\index{Division,!processes of|)}\index{Scratch system of division|)}\index{Galley system of division|)}

The medieval writers were acquainted with the method now
in use, but considered the scratch method more simple. In
some cases the latter is very clumsy, as may be illustrated by the
following example taken from \index{Pacioli!ref.\ to}Pacioli. The object is to divide
23400 by 100. The result is obtained thus

\begin{center}
\begin{tabular}{c *{6}{@{\;}c}}
 & &0& & & &\\
 &0&4&0& & &\\
0&3&4&0&0& &\\
2&3&4&0&0&(&\textbf{234}\\
1&0&0&0&0& &\\
 &1&0&0& & &\\
 & &1& & & &
 \end{tabular}
\end{center}

The galley method was used in India, and the Italians may
have derived it thence. In Italy it became obsolete somewhere
about 1600; but it continued in partial use for at least another
century in other countries. I should add that \index{Napier of Merchiston!ref.\ to}Napier's rods can
be, and sometimes were used to obtain the result of dividing
one number by another.

\nblabel{png214a}(ii)~\index{Addition, processes of!symbols for}\index{Plus!symbols for}\index{Subtraction,!symbols for|(}\index{Minus!z1@symbols for|(}\index{Negative sign|(}The signs $+$ and $-$ to indicate addition and subtraction\footnote
  {See below, pp.~\pageref{xref:206},~\pageref{xref:207},~\pageref{xref:214},~\pageref{xref:216}.}
occur in \index{Widman!ref.\ to}Widman's arithmetic published in 1489, but were first
brought into general notice, at any rate as symbols of operation,
by \index{Stifel!ref.\ to}Stifel in 1544. They occur, however, in a work by
\index{Hoecke, G. V.}G.~V.~Hoecke, published at Antwerp in 1514. I believe I am
correct in saying that \index{Vieta!ref.\ to}Vieta in 1591 was the first well-known
writer who used these signs consistently throughout his work,
and that it was not until the beginning of the seventeenth
century that they became recognized as well-known symbols.
\index{Equality, symbols for}The sign $=$ to denote equality\footnote
  {See below, p.~\pageref{xref:214}.}
was introduced by \index{Record!ref.\ to}Record in
1557.\index{Subtraction,!symbols for|)}\index{Minus!z1@symbols for|)}\index{Negative sign|)}

\nblabel{png215a}(iii)~The invention of \index{Logarithms|(}logarithms,\footnote
  {See the article on \emph{Logarithms} in the \emph{Encyclopaedia Britannica}, ninth
  edition; see also below, pp.~\pageref{png255a}, \pageref{png256a}.}
without which many of
the numerical calculations which have constantly to be made
would be practically impossible, was due to \index{Napier of Merchiston!ref.\ to}Napier of Merchiston.
The first public announcement of the discovery was
made in his \emph{Mirifici Logarithmorum Canonis Descriptio}, published
in 1614, and of which an English translation was issued
in the following year; but he had privately communicated a
summary of his results to \index{Tycho Brahe}Tycho Brahe as early as 1594. In
this work Napier explains the nature of logarithms by a comparison
between corresponding terms of an arithmetical and
geometrical progression. He illustrates their use, and gives
tables of the logarithms of the sines and tangents of all angles
in the first quadrant, for differences of every minute, calculated
to seven places of decimals. His definition of the logarithm of a
quantity $n$ was what we should now express by $10^7\log _e(10^7/n)$.
This work is the more interesting to us as it is the first valuable
contribution to the progress of mathematics which was made by
any British writer. The method by which the logarithms were
calculated was explained in the \emph{Constructio}, a posthumous work
issued in 1619: it seems to have been very laborious, and
depended either on direct involution and evolution, or on the
formation of geometrical means. The method by finding the
approximate value of a convergent series was introduced by
\index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton, \index{Cotes!ref.\ to}Cotes, and \index{Euler!ref.\ to}Euler. Napier had determined to change
the base to one which was a power of 10, but died before he
could effect it.

The rapid recognition throughout Europe of the advantages
of using logarithms in practical calculations was mainly due to
\index{Briggs!ref.\ to}Briggs, who was one of the earliest to recognize the value of
Napier's invention. Briggs at once realized that the base to
which Napier's logarithms were calculated was inconvenient;
he accordingly visited Napier in 1616, and urged the change
to a decimal base, which was recognized by Napier as an improvement.
On his return Briggs immediately set to work to
calculate tables to a decimal base, and in 1617 he brought out
a table of logarithms of the numbers from 1 to 1000 calculated
to fourteen places of decimals.

It would seem that \index{Burgi@Bürgi, J.}J.~Bürgi, independently of \index{Napier of Merchiston!ref.\ to}Napier, had
constructed before 1611 a table of antilogarithms of a series of
natural numbers: this was published in 1620. In the same
year a table of the logarithms, to seven places of decimals, of
the sines and tangents of angles in the first quadrant was
brought out by \index{Gunter, E.}Edmund Gunter, one of the Gresham lecturers.
Four years later the latter mathematician introduced a ``line of
numbers,'' which provided a mechanical method for finding the
product of two numbers: this was the precursor of the \index{Slide-rule}slide-rule,
first described by \index{Oughtred!ref.\ to}Oughtred in 1632. In 1624, Briggs published
tables of the logarithms of some additional numbers and of %[**TN: 'tales of the logarithms' in the original]
various trigonometrical functions. His logarithms of the natural
numbers are equal to those to the base 10 when multiplied by
$10^8$, and of the sines of angles to those to the base 10 when
multiplied by $10^{12}$. The calculation of the logarithms of
70,000 numbers which had been omitted by Briggs from his
tables of 1624 was performed by \index{Vlacq}Adrian Vlacq and published
in 1628: with this addition the table gave the logarithms of
all numbers from 1 to 101,000.

The \emph{Arithmetica Logarithmica} of Briggs and Vlacq are substantially
the same as the existing tables: parts have at different
times been recalculated, but no tables of an equal range and
fulness entirely founded on fresh computations have been published
since. These tables were supplemented by Briggs's
\emph{Trigonometrica Britannica}, which contains tables not only of
the logarithms of the trigonometrical functions, but also of their
natural values: it was published posthumously in 1633. A
table of logarithms to the base~$e$ of the numbers from 1 to 1000
and of the sines, tangents, and secants of angles in the first
quadrant was published by \index{Speidell on logarithms}John Speidell at London as early
as 1619, but of course these were not so useful in practical
calculations as those to the base~10. By 1630 tables of
logarithms were in general use.\index{Logarithms|)}

\index{Decimal measures}\nblabel{png217a}(iv)~The introduction of the \index{Decimal fractions|(}decimal notation for fractions
is due to \index{Pitiscus!ref.\ to}Pitiscus, in whose Tables, 1608 and 1612, it appears;
it was employed in 1616 in the English translation of Napier's
\emph{Descriptio}, and occurs in the logarithmic Tables published by
\index{Briggs!ref.\ to}Briggs in 1617, after which date its use may be taken to be established.
The idea was not new, for \index{Stevinus!ref.\ to}Stevinus had in 1585 used a
somewhat similar notation, writing a number such as $25·379$ either
in the form $25,~3'~7''~9'''$, or in the form
25~\circledigit{0}~3~\circledigit{1}~7~\circledigit{2}~9~\circledigit{3}.
This latter notation was also used by \index{Napier of Merchiston!ref.\ to}Napier in 1617 in his essay
on Rods, and by Rudolff. These writers employed it only as
a concise way of stating results, and made no use of it as an
operative form; probably Briggs did more than any other writer
to establish its use as an operative form. The subject is one of
much interest, and a considerable body of literature has grown
up about it. Some of the facts are in dispute, and the above
statement must only be taken to represent my general conclusions.
The reader interested in the subject may consult the Napier
Tercentenary Volume issued by the Edinburgh Royal Society
in 1915. Before the sixteenth century fractions were commonly
written in the sexagesimal notation.\footnote
  {For examples, see above, pp.~\pageref{xref:097},~\pageref{xref:101},~\pageref{xref:169}.}

\index{Fractions, continued!treatment of}In Napier's work of 1619 the \index{Decimal point|(}point is written in the form now
adopted in England. \index{Witt}Witt in 1613 and Napier in 1617 used
a solidus to separate the integral from the fractional part.
Briggs underlined the decimal figures, and would have printed a
number such as $25·379$ in the form
$\inlinetabular{@{}c@{}c@{}}{25&379\\\cline{2-2}}$.
Subsequent writers added another line, and would have written it as
$\inlinetabular{@{}c@{}|@{}c@{}}{25\,&\,379\\\cline{2-2}}$;
nor was it till the beginning of the eighteenth century that the
current notation was generally employed. Even now the
notation varies slightly in different countries: thus the fraction
$\frac{1}{4}$ would in the decimal notation be written in England as $0·25$,
in America as $0.25$, and in Germany and France as $0\mathord{,}25$. A
knowledge of the decimal notation became general among
practical men with the introduction of the French decimal
standards.\index{Decimal fractions|)}\index{Decimal point|)}

\nbChapter{XII}{The Mathematics of the Renaissance}{}{Circ.\ 1450--1637.}
  {Where no other references are given, see parts~xii, xiii, xiv, and the early
  chapters of part~xv of Cantor's \emph{Vorlesungen}; on the Italian mathematicians
  of this period see also G.~Libri, \emph{Histoire des sciences mathématiques en Italie},
  4~vols., Paris, 1838--1841.}
\index{Trigonometry.!z7@Treated by most of the mathematicians of the renaissance, chapters \textsc{xii} and \textsc{xiii}}\index{Renaissance, the mathematics of, chapters \textsc{xii, xiii}}\index{Geometry.!z5@Geometry of the renaissance; characterized by a free use of algebra and trigonometry, chapters \textsc{xii, xiii}}\index{Libri, ref.\ to}\index{Cantor, M., ref.\ to}\nblabel{png219a}

\nblabel{png219b}\First{The} \hyperref[chapter:XI]{last chapter} is a digression from the chronological arrangement
to which, as far as possible, I have throughout adhered,
but I trust by references in this chapter to keep the order of
events and discoveries clear. I return now to the general
history of mathematics in western Europe. Mathematicians
had barely assimilated the knowledge obtained from the Arabs,
including their translations of Greek writers, when the refugees
who escaped from Constantinople after the fall of the eastern
empire brought the original works and the traditions of Greek
science into Italy. Thus by the middle of the fifteenth century
the chief results of Greek and Arabian mathematics were
accessible to European students.

\index{Printing, invention of}\nblabel{png219c}The invention of printing about that time rendered the
dissemination of discoveries comparatively easy. It is almost a
truism to remark that until printing was introduced a writer
appealed to a very limited class of readers, but we are perhaps
apt to forget that when a medieval writer ``published'' a work
the results were known to only a few of his contemporaries.
This had not been the case in classical times, for then and
until the fourth century of our era Alexandria was the recognized
centre for the reception and dissemination of new works
and discoveries. In medieval Europe, on the other hand, there
was no common centre through which men of science could
communicate with one another, and to this cause the slow
and fitful development of medieval mathematics may be partly
ascribed.

\index{Printing, invention of}The introduction of printing marks the beginning of the
modern world in science as in politics; for it was contemporaneous
with the assimilation by the indigenous European
school (which was born from scholasticism, and whose history
was traced in chapter~\hyperref[chapter:VIII]{\textsc{viii}}) of the results of the Indian and
Arabian schools (whose history and influence were traced in
chapters~\hyperref[chapter:IX]{\textsc{ix}} and \hyperref[chapter:X]{\textsc{x}}), and of the Greek schools (whose history
was traced in chapters~\hyperref[chapter:II]{\textsc{ii}} to \hyperref[chapter:V]{\textsc{v}}).

The last two centuries of this period of our history, which
may be described as the renaissance, were distinguished by
great mental activity in all branches of learning. The creation
of a \index{Universities of renaissance}fresh group of universities (including those in Scotland),
of a somewhat less complex type than the medieval universities
above described, testify to the general desire for knowledge.
\index{America, discovery of}The discovery of America in 1492 and the discussions that
preceded the \index{Reformation, the}Reformation flooded Europe with new ideas which,
by the invention of printing, were widely disseminated; but the
advance in mathematics was at least as well marked as that in
literature and that in politics.

During the first part of this time the attention of mathematicians
was to a large extent concentrated on syncopated
algebra and trigonometry; the treatment of these subjects is
discussed in the first section of this chapter, but the relative
importance of the mathematicians of this period is not very
easy to determine. The middle years of the renaissance were
distinguished by the development of symbolic algebra: this is
treated in the second section of this chapter. The close of the
sixteenth century saw the creation of the science of dynamics:
this forms the subject of the first section of chapter~\hyperref[chapter:XIII]{\textsc{xiii}}\@.
About the same time and in the early years of the seventeenth
century considerable attention was paid to pure geometry: this
forms the subject of the second section of chapter~\hyperref[chapter:XIII]{\textsc{xiii}}.

\section*{The development of syncopated algebra and trigonometry.\nblabel{png221a}}

\index{Regiomontanus|(}\nblabel{png221b}\nbparagraph{Regiomontanus.\footnote
  {His life was written by \index{Gassendi, ref.\ to}P.~Gassendi, The Hague, second edition, 1655.
  His letters, which afford much valuable information on the mathematics of his
  time, were collected and edited by \index{Von Murr, ref.\ to}\index{Murr on Regiomontanus}C.~G.~von~Murr, Nuremberg, 1786. An
  account of his works will be found in \index{Copernicus!ref.\ to}\emph{Regiomontanus, ein geistiger Vorläufer
  des Copernicus}, by \index{Ziegler on Regiomontanus}A.~Ziegler, Dresden, 1874; see also \index{Cantor, M., ref.\ to}Cantor, chap.~lv.}}
Amongst the many distinguished writers
of this time \emph{Johann Regiomontanus} was the earliest and one
of the most able. He was born at Königsberg on June~6,
1436, and died at Rome on July~6, 1476. His real name
was \emph{Johannes Müller}, but, following the custom of that time,
he issued his publications under a Latin pseudonym which in
his case was taken from his birthplace. To his friends, his
neighbours, and his tradespeople he may have been Johannes
Müller, but the literary and scientific world knew him as
Regiomontanus, just as they knew Zepernik as Copernicus,
and Schwarzerd as \index{Melanchthon, ref.\ to}Melanchthon. It seems as pedantic as it is
confusing to refer to an author by his actual name when he
is universally recognized under another: I shall therefore in all
cases as far as possible use that title only, whether latinized or
not, by which a writer is generally known.

Regiomontanus studied mathematics at the university of
Vienna, then one of the chief centres of mathematical studies
in Europe, under \index{Purbach!ref.\ to}Purbach who was professor there. His
first work, done in conjunction with Purbach, consisted of an
analysis of the \index{Almagest@\emph{Almagest}, the!ref.\ to}\emph{Almagest}. \index{Trigonometrical functions|(}In this the trigonometrical functions
\index{Sine}\emph{sine} and \index{Cosine}\emph{cosine} were used and a table of natural sines was
introduced. Purbach died before the book was finished: it
was finally published at Venice, but not till 1496. As soon as
this was completed Regiomontanus wrote a work on astrology,
which contains some astronomical tables and a table of natural
tangents: this was published in 1490.

Leaving Vienna in 1462, Regiomontanus travelled for some
time in Italy and Germany; and at last in 1471 settled for
a few years at Nuremberg, where he established an observatory,
opened a printing-press, and probably lectured. Three tracts on
astronomy by him were written here. A mechanical eagle, which
flapped its wings and saluted the \index{Maximilian I. of Germany}Emperor Maximilian I. on his
entry into the city, bears witness to his mechanical ingenuity,
and was reckoned among the marvels of the age. Thence
Regiomontanus moved to Rome on an invitation from \index{Sixtus IV. of Rome, ref.\ to}Sixtus IV.
who wished him to reform the calendar. He was assassinated,
shortly after his arrival, at the age of 40.

Regiomontanus was among the first to take advantage of
the recovery of the original texts of the Greek mathematical
works in order to make himself acquainted with the methods of
reasoning and results there used; the earliest notice in modern
Europe of the \index{Algebra.!z5@Introduction of syncopated algebra by Bhaskara!x2@Regiomontanus|(}algebra of \index{Diophantus!ref.\ to}Diophantus is a remark of his that he
had seen a copy of it at the Vatican. He was also well read in
the works of the Arab mathematicians.

\nblabel{png222a}The fruit of his study was shewn in his \emph{De Triangulis}
written in 1464. This is the earliest modern systematic
exposition of trigonometry, plane and spherical, though the
only trigonometrical functions introduced are those of the sine
and cosine. It is divided into five books. The first four are
given up to plane trigonometry, and in particular to determining
triangles from three given conditions. The fifth book
is devoted to spherical trigonometry. The work was printed
at Nuremberg in 1533, nearly a century after the death of
Regiomontanus.\index{Trigonometrical functions|)}

As an example of the mathematics of this time I quote one
of his propositions at length. It is required to determine a
triangle when the difference of two sides, the perpendicular on
the base, and the difference between the segments into which
the base is thus divided are given [book~\textsc{ii}, prop.~23]. The
following is the solution given by Regiomontanus.

\begin{nbQuote}\nblabel{xref:203}
Sit talis triangulus $ABG$, cujus duo latera $AB$ et $AG$ differentia
habeant nota $HG$, ductaque perpendiculari $AD$ duorum casuum $BD$ et
$DG$, differentia sit $EG$: hae duae differentiae sint datae, et ipsa perpendicularis
$AD$ data. Dico quod omnia latera trianguli nota concludentur.
Per artem rei et census hoc problema absolvemus. Detur ergo differentia
laterum ut 3, differentia casuum 12, et perpendicularis 10. Pono pro
basi unam rem, et pro aggregato laterum 4 res, nae proportio basis ad
congeriem laterum est ut $HG$ ad $GE$, scilicet unius ad 4. Erit ergo $BD$
$\frac{1}{2}$
rei minus 6, sed $AB$ erit 2 res demptis $\frac{3}{2}$. Duco $AB$ in se, producuntur
4 census et $2\frac{1}{4}$ demptis 6 rebus. Item $BD$ in se facit $\frac{1}{4}$ census et 36
minus 6 rebus: huic addo quadratum de 10 qui est 100. Colliguntur $\frac{1}{4}$
census et 136 minus 6 rebus aequales videlicet 4 censibus et $2\frac{1}{4}$ demptis
6 rebus. Restaurando itaque defectus et auferendo utrobique aequalia,
quemadmodum ars ipsa praecipit, habemus census aliquot aequales
numero, unde cognitio rei patebit, et inde tria latera trianguli more suo
innotescet.

\begin{center}
\labelincludegraphics{223a}
\end{center}
\end{nbQuote}

To explain the language of the proof I should add that
Regiomontanus calls the \index{Unknown quantity, word or symbol for}\index{Res used for unknown quantity}unknown quantity \emph{res}, and its square
\index{Census}\emph{census} or \index{Zensus}\emph{zensus}; but though he uses these technical terms he
writes the words in full. He commences by saying that he will
solve the problem by means of a quadratic equation (per artem
rei et census); and that he will suppose the difference of the
sides of the triangle to be 3, the difference of the segments of
the base to be 12, and the altitude of the triangle to be 10.
He then takes for his unknown quantity (unam rem or $x$) the
base of the triangle, and therefore the sum of the sides will be
$4x$. Therefore $BD$ will be equal to $\frac{1}{2}x - 6$ ($\frac{1}{2}$ rei minus 6), and
$AB$ will be equal to $2x - \frac{3}{2}$ (2 res demptis $\frac{3}{2}$); hence $AB^2$ ($AB$
in se) will be $4x^2 + 2\frac{1}{4} - 6x$ (4 census et $2\frac{1}{4}$ demptis 6 rebus),
and $BD^2$ will be $\frac{1}{4}x^2 + 36 - 6x$. To $BD^2$ he adds $AD^2$ (quadratum
de 10) which is 100, and states that the sum of the two
is equal to $AB^2$. This he says will give the value of $x^2$ (census),
whence a knowledge of $x$ (cognitio rei) can be obtained, and the
triangle determined.

To express this in the language of modern algebra we have
\begin{align*}
AG^2 - DG^2 & = AB^2 - DB^2, \\
\therefore AG^2 - AB^2 & = DG^2 - DB^2,
\end{align*}
but by the given numerical conditions
\begin{align*}
AG - AB & = 3 = \frac{1}{4}(DG - DB), \\
\therefore AG + AB & = 4(DG + DB) = 4x.
\end{align*}
Therefore \hspace{2cm} $AB = 2x - \frac{3}{2}$, and $BD = \frac{1}{2}x - 6$.\\
Hence \hspace{3cm} $(2x - \frac{3}{2})^2 = (\frac{1}{2}x-6)^2 + 100$.\\
From which $x$ can be found, and all the elements of the triangle
determined.

It is worth noticing that Regiomontanus merely aimed at
giving a general method, and the numbers are not chosen with
any special reference to the particular problem. Thus in his
diagram he does not attempt to make $GE$ anything like four
times as long as $GH$, and, since $x$ is ultimately found to be
equal to $\frac{1}{3}\sqrt{321}$, the point $D$ really falls outside the base. The
order of the letters $ABG$, used to denote the triangle, is of
course derived from the Greek alphabet.

\index{Rhetorical algebra}Some of the solutions which he gives are unnecessarily
complicated, but it must be remembered that algebra and
trigonometry were still only in the rhetorical stage of development,
and when every step of the argument is expressed in
words at full length it is by no means easy to realize all that is
contained in a formula.

It will be observed from the above example that Regiomontanus
did not hesitate to apply algebra to the solution of
geometrical problems. Another illustration of this is to be found
in his discussion of a question which appears in \index{Brahmagupta!ref.\ to}Brahmagupta's
\emph{Siddhanta}. The problem was to construct a quadrilateral,
having its sides of given lengths, which should be inscribable in
a circle. The solution\footnote
  {It was published by \index{Murr on Regiomontanus}\index{Von Murr, ref.\ to}C.~G.~von~Murr at Nuremberg in 1786.}
given by Regiomontanus was effected
by means of algebra and trigonometry.\index{Algebra.!z5@Introduction of syncopated algebra by Bhaskara!x2@Regiomontanus|)}

The \emph{Algorithmus Demonstratus} of \index{Jordanus!ref.\ to}Jordanus, described above,
which was first printed in 1534, was formerly attributed to
Regiomontanus.

Regiomontanus was one of the most prominent mathematicians
of his generation, and I have dealt with his works in
some detail as typical of the most advanced mathematics of
the time. Of his contemporaries I shall do little more than
mention the names of a few of those who are best known;
none were quite of the first rank, and I should sacrifice the
proportion of the parts of the subject were I to devote much
space to them.\index{Regiomontanus|)}

\index{Purbach}\nblabel{png225a}\nbparagraph{Purbach.\footnote
  {Purbach's life was written by \index{Gassendi, ref.\ to}P.~Gassendi, The Hague, second edition,
  1655.}}
I may begin by mentioning \emph{George Purbach},
first the tutor and then the friend of Regiomontanus, born
near Linz on May~30, 1423, and died at Vienna on April~8,
1461, who wrote a work on planetary motions which was
published in 1460; an arithmetic, published in 1511; a table
of eclipses, published in 1514; and a table of natural sines,
published in 1541.

\index{Cusa, Cardinal de}\nbparagraph{Cusa.\footnote
  {Cusa's life was written by \index{Scharpff on Cusa}F.~A.~Scharpff, Tübingen, 1871; and his
  collected works, edited by \index{Petri on Cusa}H.~Petri, were published at Bâle in 1565.}}
Next I may mention \emph{Nicolas de Cusa}, who was
born in 1401 and died in 1464. Although the son of a poor
fisherman and without influence, he rose rapidly in the church,
and in spite of being ``a reformer before the reformation''
became a cardinal. His mathematical writings deal with the
reform of the \index{Calendars}calendar and the quadrature of the circle; in
the latter problem his construction is equivalent to taking
$\frac{3}{4}(\surd 3 + \surd 6)$ as the value of $\pi$. He argued in favour of the
diurnal rotation of the earth.

\index{Chuquet|(}\nbparagraph{Chuquet.} I may also here notice a treatise on arithmetic,
known as \emph{Le Triparty},\footnote
  {See an article by \index{Marre on Chuquet}A.~Marre in \index{Boncompagni, ref.\ to}Boncompagni's \emph{Bulletino di bibliografia}
  for 1880, vol.~xiii, pp.~555--659.}
by \emph{Nicolas Chuquet}, a bachelor of
medicine in the university of Paris, which was written in 1484.
This work indicates that the extent of mathematics then taught
was somewhat greater than was generally believed a few years
ago. It contains the earliest known use of the \index{Radical, symbols for}radical sign
with indices to mark the \index{Roots, square, cube, \&c.}\index{Square root, symbols for}root taken, 2 for a square-root, 3 for
a cube-root, and so on; and also a definite statement of the
rule of signs. The words plus and minus are denoted by the
contractions $\overline{p}$, $\overline{m}$. The work is in French.\index{Chuquet|)}

\nblabel{xref:206}\nblabel{png226a}\nbparagraph{Introduction\footnote
  {Recently new light has been thrown on the history of the subject by the
  researches of \index{Glaisher}J.~W.~L.~Glaisher, \emph{Messenger of Mathematics}, Cambridge, vol.~li,
  pp.~1 \emph{et seq}. The account in the text is based on the earlier investigations
  of \index{Treutlein, ref.\ to}P.~Treutlein, A.~de~Morgan, and Boncompagni.}
 of signs $+$ and $-$.}\index{Demorgana@De~Morgan, A.!ref.\ to} In England and Germany
algorists were less fettered by precedent and tradition than in
Italy, and introduced some improvements in notation which
were hardly likely to occur to an Italian. \index{Plus!syn@origin of symbol +|(}\index{Minus!z2@origin of symbol|(}Of these the most
prominent were the introduction, if not the invention, of the
\index{Addition, processes of!symbols for|(}\index{Plus!symbols for|(}\index{Subtraction,!symbols for|(}\index{Minus!z1@symbols for|(}\index{Negative sign|(}current symbols for addition, subtraction, and equality.

The earliest instances of the regular use of the signs $+$ and $-$
of which we have any knowledge occur in the fifteenth century.
\index{Widman}\emph{Johannes Widman} of Eger, born about 1460, matriculated at
Leipzig in 1480, and probably by profession a physician, wrote
a \emph{Mercantile Arithmetic}, published at Leipzig in 1489 (and
modelled on a work by \index{Wagner}Wagner printed some six or seven years
earlier): in this book these signs are used merely as marks
signifying excess or deficiency; the corresponding use of the
word surplus or overplus\footnote
  {See \emph{passim} Levit.~xxv, verse~27, and 1~Maccab.~x, verse~41.}
was once common and is still
retained in \index{Mercantile arithmetic}commerce.

It is noticeable that the signs generally occur only in practical
mercantile questions: hence it has been conjectured that they
were originally warehouse marks. Some kinds of goods were
sold in a sort of wooden chest called a \emph{lagel}, which when full
was apparently expected to weigh roughly either three or four
\emph{centners}; if one of these cases were a little lighter, say 5 lbs.,
than four centners, Widman describes it as weighing $4c - 5$ lbs.:
if it were 5 lbs.\ heavier than the normal weight it is described
as weighing $4c \nblongplus 5$ lbs. The symbols are used as if they
would be familiar to his readers; and there are some slight
reasons for thinking that these marks were chalked on the
chests as they came into the warehouses. We infer that the
more usual case was for a chest to weigh a little less than
its reputed weight, and, as the sign $-$ placed between two
numbers was a common symbol to signify some connection
between them, that seems to have been taken as the standard
case, while the vertical bar was originally a small mark super-added
on the sign $-$ to distinguish the two symbols. It will
be observed that the vertical line in the symbol for excess,
printed above, is somewhat shorter than the horizontal line.
This is also the case with \index{Stifel!ref.\ to}Stifel and most of the early writers
who used the symbol: some presses continued to print it in
this, its earliest form, till the end of the seventeenth century.
\index{Xylander!ref.\ to}Xylander, on the other hand, in 1575 has the vertical bar
much longer than the horizontal line, and the symbol is something
like $\nbshortplus$.

\nblabel{xref:207}Another conjecture is that the symbol for \emph{plus} is derived
from the Latin abbreviation \& for \emph{et}; while that for \emph{minus} is
obtained from the bar which is often used in ancient manuscripts
to indicate an omission, or which is written over the contracted
form of a word to signify that certain letters have been left out.
This view has been often supported on a priori grounds, but it
has recently found powerful advocates in \index{Zangmeister, ref.\ to}Professors Zangmeister
and \index{Lepage@Le Paige}Le~Paige who also consider that the introduction of these
symbols for plus and minus may be referred to the fourteenth
century.

These explanations of the origin of our symbols for \emph{plus} and
\emph{minus} are the most plausible that have been yet advanced, but
the question is difficult and cannot be said to be solved. Another
suggested derivation is that $+$ is a contraction of
\raisebox{-0.2ex}{\labelincludegraphics[height=2.1ex]{oldgermanp1}} the initial
letter in Old German of plus, while $-$ is the limiting form of
\emph{m} (for minus) when written rapidly. \index{Demorgana@De~Morgan, A.!ref.\ to}De~Morgan\footnote
  {See his \emph{Arithmetical Books}, London, 1847, p.~19.}
proposed
yet another derivation: the Hindoos sometimes used a dot to
indicate subtraction, and this dot might, he thought, have been
elongated into a bar, and thus give the sign for \emph{minus}; while
the origin of the sign for \emph{plus} was derived from it by a super-added
bar as explained above; but I take it that at a later
time he abandoned this theory for what has been called the
warehouse explanation.\index{Plus!syn@origin of symbol +|)}\index{Minus!z2@origin of symbol|)}

I should perhaps here add that till the close of the sixteenth
century the sign $+$ connecting two quantities like $a$ and $b$ was
also used in the sense that if $a$ were taken as the answer to some
question one of the given conditions would be too little by $b$.
This was a relation which constantly occurred in solutions of
questions by the \index{Assumption, rule of false}\index{False assumption, rule of}rule of false assumption.

Lastly, I would repeat again that these signs in Widman are
only abbreviations and not symbols of operation; he attached
little or no importance to them, and no doubt would have
been amazed if he had been told that their introduction was preparing
the way for a revolution of the processes used in algebra.\index{Addition, processes of!symbols for|)}\index{Plus!symbols for|)}\index{Subtraction,!symbols for|)}\index{Minus!z1@symbols for|)}\index{Negative sign|)}

The \emph{Algorithmus} of \index{Jordanus!ref.\ to}Jordanus was not published till 1534;
Widman's work was hardly known outside Germany; and it
is to Pacioli that we owe the introduction into general use
of syncopated algebra; that is, the use of abbreviations for
certain of the more common algebraical quantities and operations,
but where in using them the rules of syntax are observed.

\index{Pacioli|(}\nblabel{png228a}\nbparagraph{Pacioli.\footnote
  {See \index{Staigmüller, ref.\ to}H.~Staigmüller in the \emph{Zeitschrift für Mathematik}, 1889, vol.~xxxiv;
  also \index{Libri, ref.\ to}Libri, vol.~iii, pp.~133--145; and \index{Cantor, M., ref.\ to}Cantor, chap.~lvii.}}
 \emph{Lucas Pacioli}, sometimes known as \emph{Lucas di~Burgo},
and sometimes, but more rarely, as \emph{Lucas Paciolus}, was born at
Burgo in Tuscany about the middle of the fifteenth century.
We know little of his life except that he was a Franciscan
friar; that he lectured on mathematics at Rome, Pisa, Venice,
and Milan; and that at the last-named city he was the first
occupant of a chair of mathematics founded by \index{Sforza, ref.\ to}Sforza: he died
at Florence about the year 1510.

\nblabel{png229a}His chief work was printed at Venice in 1494 and is termed
\emph{Summa de arithmetica, geometria, proporzioni e proporzionalita}.
It is divided into two parts, the first dealing with arithmetic
and algebra, the second with geometry. This was the earliest
printed book on arithmetic and algebra. It is mainly based on
the writings of \index{Leonardo of Pisa!ref.\ to}Leonardo of Pisa, and its importance in the
history of mathematics is largely due to its wide circulation.

In the arithmetic Pacioli gives rules for the four simple
processes, and a method for extracting square roots. He deals
pretty fully with all questions connected with \index{Mercantile arithmetic}mercantile
arithmetic, in which he works out numerous examples, and
in particular discusses at great length bills of exchange and
the theory of \index{Book-keeping}\index{Double entry, book-keeping by}book-keeping by double entry. This part was
the first systematic exposition of algoristic arithmetic, and has
been already alluded to in chapter~\hyperref[chapter:XI]{\textsc{xi}}. It and the similar
work by \index{Tartaglia!ref.\ to}Tartaglia are the two standard authorities on the
subject.

Many of his problems are solved by \index{Assumption, rule of false}\index{False assumption, rule of}``the method of false
assumption,'' which consists in assuming any number for the
unknown quantity, and if on trial the given conditions be
not satisfied, altering it by a simple proportion as in rule of
three. As an example of this take the problem to find the
original capital of a merchant who spent a quarter of it in
Pisa and a fifth of it in Venice, who received on these transactions
180 ducats, and who has in hand 224 ducats. Suppose
that we assume that he had originally 100 ducats. Then if
he spent $25+20$ ducats at Pisa and Venice, he would have
had 55 ducats left. But by the enunciation he then had
$224-180$, that is, 44 ducats. Hence the ratio of his original
capital to 100 ducats is as 44 to 55. Thus his original capital
was 80 ducats.

The following example will serve as an illustration of the
kind of arithmetical problems discussed.

\begin{nbQuote}
I buy for 1440 ducats at Venice 2400 sugar loaves, whose nett weight
is 7200 lire; I pay as a fee to the agent 2 per cent.; to the weighers and
porters on the whole, 2 ducats; I afterwards spend in boxes, cords,
canvas, and in fees to the ordinary packers in the whole, 8 ducats; for
the tax or octroi duty on the first amount, 1 ducat per cent.; afterwards
for duty and tax at the office of exports, 3 ducats per cent.; for writing
directions on the boxes and booking their passage, 1 ducat; for the bark
to Rimini, 13 ducats; in compliments to the captains and in drink for
the crews of armed barks on several occasions, 2 ducats; in expenses for
provisions for myself and servant for one month, 6 ducats; for expenses
for several short journeys over land here and there, for barbers, for
washing of linen, and of boots for myself and servant, 1 ducat; upon my
arrival at Rimini I pay to the captain of the port for port dues in the
money of that city, 3 lire; for porters, disembarkation on land, and
carriage to the magazine, 5 lire; as a tax upon entrance, 4 soldi a load
which are in number 32 (such being the custom); for a booth at the fair,
4 soldi per load; I further find that the measures used at the fair are
different to those used at Venice, and that 140 lire of weight are there
equivalent to 100 at Venice, and that 4 lire of their silver coinage are
equal to a ducat of gold. I ask, therefore, at how much I must sell a
hundred lire Rimini in order that I may gain 10 per cent.\ upon my
whole adventure, and what is the sum which I must receive in Venetian
money?
\end{nbQuote}

\index{Algebra.!z3@Discussed rhetorically by the Hindoo and Arab mathematicians!q3@and Pacioli}\nblabel{xref:210}\index{Rhetorical algebra}In the algebra he discusses in some detail simple and
\index{Quadratic equations}quadratic equations, and problems on numbers which lead to
such equations. He mentions the Arabic classification of cubic
equations, but adds that their solution appears to be as impossible
as the quadrature of the circle. The following is the
rule he gives\footnote
  {Edition of 1494, p.~145.}
 for solving a quadratic equation of the form
$x^2 + x = a$: it is rhetorical and not syncopated, and will serve
to illustrate the inconvenience of that method.

\begin{quote}
\nbstretchyspace
``Si res et census numero coaequantur, a rebus\\
\phantom{``}dimidio sumpto censum producere debes,\\
\phantom{``}addereque numero, cujus a radice totiens\\
\phantom{``}tolle semis rerum, census latusque redibit.''
\end{quote}

He confines his attention to the positive roots of equations.

Though much of the matter described above is taken from
\index{Leonardo of Pisa!ref.\ to|(}Leonardo's \emph{Liber Abaci}, yet the notation in which it is expressed
is superior to that of Leonardo. Pacioli follows Leonardo and
the Arabs in calling the \index{Unknown quantity, word or symbol for}\index{Res used for unknown quantity}unknown quantity the \emph{thing}, in Italian
\index{Cosa}\emph{cosa}---hence algebra was sometimes known as the \index{Cossic art}cossic art---or
in Latin \emph{res}, and sometimes denotes it by \emph{co} or \emph{R} or \emph{Rj}. He
calls the square of it \index{Census}\emph{census} or \index{Zensus}\emph{zensus}, and sometimes denotes
it by \emph{ce} or \emph{Z}; similarly the cube of it, or \index{Cuba}\emph{cuba}, is sometimes
represented by \emph{cu} or \emph{C}; the fourth power, or \index{Censo di censo}\emph{censo di censo},
is written either at length or as \emph{ce di ce} or as \emph{ce ce}. It may
be noticed that all his equations are numerical, so that he did
not rise to the conception of representing known quantities by
letters as \index{Jordanus!ref.\ to}Jordanus had done and as is the case in modern
algebra; but \index{Libri, ref.\ to}Libri gives two instances in which in a proportion
he represents a number by a letter. \index{Addition, processes of!symbols for}\index{Plus!symbols for}He indicates addition by
$p$ or $\overline{p}$, the initial letter of the word \emph{plus}, but he generally evades
the introduction of a \index{Subtraction,!symbols for}\index{Minus!z1@symbols for}\index{Negative sign}symbol for \emph{minus} by writing his quantities
on that side of the equation which makes them positive, though
in a few places he denotes it by $\overline{m}$ for \emph{minus} or by \emph{de} for \index{Demptus for \emph{minus}}\emph{demptus}.
Similarly, \index{Equality, symbols for}equality is sometimes indicated by \emph{ae} for \emph{aequalis}.
This is a commencement of syncopated algebra.

There is nothing striking in the results he arrives at in the
second or geometrical part of the work; nor in two other tracts
on geometry which he wrote and which were printed at Venice
in 1508 and 1509. It may be noticed, however, that, like
\index{Regiomontanus!ref.\ to}Regiomontanus, he applied algebra to aid him in investigating
the geometrical properties of figures.\index{Leonardo of Pisa!ref.\ to|)}

The following problem will illustrate the kind of geometrical
questions he attacked. The radius of the inscribed circle
of a triangle is 4 inches, and the segments into which one side
is divided by the point of contact are 6 inches and 8 inches
respectively. Determine the other sides. To solve this it is
sufficient to remark that $rs = \Delta = \sqrt{s(s-a)(s-b)(s-c)}$ which
gives $4s = \sqrt{s × (s-14) × 6 × 8}$, hence $s=21$; therefore the
required sides are $21 - 6$ and $21 - 8$, that is, 15 and 13. But
Pacioli makes no use of these formulae (with which he was
acquainted), but gives an elaborate geometrical construction, and
then uses algebra to find the lengths of various segments of the
lines he wants. The work is too long for me to reproduce here,
but the following analysis of it will afford sufficient materials
for its reproduction. Let $ABC$ be the triangle, $D$, $E$, $F$ the
points of contact of the sides, and $O$ the centre of the given
circle. Let $H$ be the point of intersection of $OB$ and $DF$, and
$K$ that of $OC$ and $DE$. Let $L$ and $M$ be the feet of the
perpendiculars drawn from $E$ and $F$ on $BC$. Draw $EP$
parallel to $AB$ and cutting $BC$ in $P$. Then \index{Pacioli!ref.\ to}Pacioli determines
in succession the magnitudes of the following lines: (i)~$OB$,
(ii)~$OC$, (iii)~$FD$, (iv)~$FH$, (v)~$ED$, (vi)~$EK$. He then
forms a quadratic equation, from the solution of which he
obtains the values of $MB$ and $MD$. Similarly he finds the
values of $LC$ and $LD$. He now finds in succession the values
of $EL$, $FM$, $EP$, and $LP$; and then by similar triangles obtains
the value of $AB$, which is 13. This proof was, even sixty years
later, quoted by \index{Cardan!ref.\ to}Cardan as ``incomparably simple and excellent,
and the very crown of mathematics.'' I cite it as an illustration
of the involved and inelegant methods then current. The
problems enunciated are very similar to those in the \emph{De
Triangulis} of \index{Regiomontanus!ref.\ to}Regiomontanus.\index{Pacioli|)}

\index{Vinci, Leonardo da|(}\index{Leonardo da Vinci|(}\nblabel{png232a}\nbparagraph{Leonardo da Vinci.} The fame of \emph{Leonardo da Vinci} as
an artist has overshadowed his claim to consideration as a
mathematician, but he may be said to have prepared the way
for a more accurate conception of mechanics and physics, while
his reputation and influence drew some attention to the subject;
he was an intimate friend of Pacioli. Leonardo was the
illegitimate son of a lawyer of Vinci in Tuscany, was born in
1452, and died in France in 1519 while on a visit to \index{Francis I. of France, ref.\ to}Francis I\@.
Several manuscripts by him were seized by the French revolutionary
armies at the end of the last century, and \index{Venturi on Leonardo da Vinci}Venturi, at
the request of the Institute, reported on those concerned with
physical or mathematical subjects.\footnote
  {\emph{Essai sur les ouvrages physico-mathématiques de Léonard de Vinci}, by
  J.\nobreakdash-B. Venturi, Paris, 1797.}

{\nbstretchyspace Leaving out of account Leonardo's numerous and important
artistic works, his mathematical writings are concerned chiefly
with mechanics, hydraulics, and optics---his conclusions being
usually based on experiments.} His treatment of hydraulics
and optics involves but little mathematics. The mechanics
contain numerous and serious errors; the best portions are
those dealing with the equilibrium of a lever under any forces,
the laws of friction, the stability of a body as affected by the
position of its centre of gravity, the strength of beams, and the
orbit of a particle under a central force; he also treated a few
easy problems by virtual moments. A knowledge of the \index{Forces!triangle of}\index{Triangle of forces}triangle
of forces is occasionally attributed to him, but it is probable
that his views on the subject were somewhat indefinite.
Broadly speaking, we may say that his mathematical work
is unfinished, and consists largely of suggestions which he
did not discuss in detail and could not (or at any rate did
not) verify.\index{Leonardo da Vinci|)}\index{Vinci, Leonardo da|)}

\index{Durer@Dürer}\nblabel{png233a}\nbparagraph{Dürer.} \emph{Albrecht Dürer}\footnote
  {See \emph{Dürer als Mathematiker}, by \index{Staigmüller, ref.\ to}H.~Staigmüller, Stuttgart, 1891.}
was another artist of the same
time who was also known as a mathematician. He was born at
Nuremberg on May~21, 1471, and died there on April~6, 1528.
His chief mathematical work was issued in 1525, and contains a
discussion of perspective, some geometry, and certain graphical
solutions; Latin translations of it were issued in 1532, 1555,
and 1605.

\index{Copernicus}\nbparagraph{Copernicus.} An account of \emph{Nicolaus Copernicus}, born at
Thorn on Feb.~19, 1473, and died at Frauenberg on May~7,
1543, and his \index{Astronomy.!z16@Modern theory of, created by Copernicus}conjecture that the \index{Planetary motions}earth and planets all revolved
round the sun, belong to astronomy rather than to mathematics.
I may, however, add that Copernicus wrote on trigonometry, his
results being published as a text-book at Wittenberg in 1542;
it is clear though it contains nothing new. It is evident from
this and his astronomy that he was well read in the literature
of mathematics, and was himself a mathematician of considerable power.
I describe his statement as to the motion of the
earth as a conjecture, because he advocated it only on the
ground that it gave a simple explanation of natural phenomena.
Galileo in 1632 was the first to try to supply a proof of this
hypothesis.

By the beginning of the sixteenth century the printing-press
began to be active, and many of the works of the earlier
mathematicians became now for the first time accessible to all
students. This stimulated inquiry, and before the middle of
the century numerous works were issued which, though they
did not include any great discoveries, introduced a variety of
small improvements all tending to make algebra more analytical.

\nblabel{xref:214}\index{Equality, symbols for!origin of symbol}\nblabel{png234a}\nbparagraph{Record.} The sign now used to denote equality was introduced
by \emph{Robert Record}.\footnote
  {On the life and career of Robert Record, see \index{Smith, D. E.}D.~E.~Smith in \emph{The
  American Mathematical Monthly}, vol.~28, 1921, p.~296 \emph{et seq.}}
\index{Record|(}Record was born at Tenby in
Pembrokeshire about 1510, and died at London in 1558. He
entered at Oxford, and obtained a fellowship at All Souls
College in 1531; thence he migrated to Cambridge, where he
took a degree in medicine in 1545. He then returned to
Oxford and lectured there, but finally settled in London and
became physician to \index{Edward VI. of England, ref.\ to}Edward VI. and to \index{Mary of England, ref.\ to}Mary. His prosperity
must have been short-lived, for at the time of his death he was
confined in the King's Bench prison for debt.

\index{Addition, processes of!symbols for}\index{Plus!symbols for}\index{Subtraction,!symbols for}\index{Minus!z1@symbols for}\index{Negative sign}\index{Equality, symbols for}In 1540 he published an arithmetic, termed the \emph{Grounde of
Artes}, in which he employed the signs $+$ to indicate excess
and $-$ to indicate deficiency; ``$+$ whyche betokeneth too
muche, as this line $-$ plaine without a crosse line betokeneth
too little.'' \index{Equalityz@=, meanings of}In this book the equality of two ratios is indicated
by two equal and parallel lines whose opposite ends are
joined diagonally, \emph{ex.~gr.}\ by \inlinegraphics{1.75ex}{-0.25ex}{234a}. \index{Algebra.!z5@Introduction of syncopated algebra by Bhaskara!x3@Record}A few years later, in 1557, he
wrote an algebra under the title of the \emph{Whetstone of Witte}.
This is interesting as it contains the earliest introduction of the
sign = for equality, and he says he selected that particular
symbol because\footnote
  {See \emph{Whetstone of Witte}, f.~Ff, j.~v.}
than two parallel straight lines ``noe 2 thynges
can be moare equalle.'' \index{Henry C., ref.\ to}M.~Charles Henry has, however, asserted
that this sign is a recognized abbreviation for the word \emph{est}
in medieval manuscripts; and, if this be established, it would
seem to indicate a more probable origin. In this work Record
shewed how the square root of an algebraic expression could be
extracted. He also wrote an astronomy. These works give a
clear view of the knowledge of the time.\index{Record|)}

\index{Rudolff}\index{Riese}\nblabel{png235a}\nbparagraph{Rudolff. Riese.} About the same time in Germany, Rudolff
and Riese took up the subjects of algebra and arithmetic. Their
investigations form the basis of Stifel's well-known work.
\emph{Christoff Rudolff}\footnote
  {See \index{Wappler on Rudolff}E.~Wappler, \emph{Geschichte der deutschen Algebra im xv.\ Jahrhunderte},
  Zwickau, 1887.}
published his algebra in 1525; it is entitled
\emph{Die Coss}, and is founded on the writings of \index{Pacioli!ref.\ to}Pacioli, and perhaps
of Jordanus. \index{Radical, symbols for}\index{Roots, square, cube, \&c.}\index{Square root, symbols for}\index{Surds, symbols for}Rudolff introduced the sign of $\surd$ for the square
root, the symbol being a corruption of the initial letter of the
word \emph{radix}, similarly $\surd \surd \surd $ denoted the cube root, and $\surd \surd$
the fourth root. \emph{Adam Riese}\footnote
  {See two works by \index{Berlet on Riese}B.~Berlet, \emph{Ueber Adam Riese}, Annaberg, 1855; and
  \emph{Die Coss von Adam Riese}, Annaberg, 1860.}
 was born near Bamberg, Bavaria,
in 1489, of humble parentage, and after working for some years
as a miner set up a school; he died at Annaberg on March~30,
1559. He wrote a treatise on practical geometry, but his most
important book was his well-known arithmetic (which may be
described as algebraical), issued in 1536, and founded on Pacioli's
work. Riese used the symbols $+$ and $-$.

\index{Stifel|(}\index{Algebra.!z5@Introduction of syncopated algebra by Bhaskara!x4@Stifel|(}\nblabel{png235b}\nbparagraph{Stifel.\footnote
  {The authorities on Stifel are given by \index{Cantor, M., ref.\ to}Cantor chap.~lxii.}}
The methods used by Rudolff and Riese and their
results were brought into general notice through Stifel's work,
which had a wide circulation. \emph{Michael Stifel}, sometimes known
by the Latin name of \emph{Stiffelius}, was born at Esslingen in 1486,
and died at Jena on April~19, 1567. He was originally an
Augustine monk, but he accepted the doctrines of \index{Luther, ref.\ to}Luther, of
whom he was a personal friend. He tells us in his algebra that
his conversion was finally determined by noticing that the pope
\index{Leo X. of Rome, Stifel on}Leo X. was the beast mentioned in the Revelation. To shew
this, it was only necessary to add up the numbers represented by
the letters in Leo decimus (the \emph{m} had to be rejected since it
clearly stood for \emph{mysterium}) and the result amounts to exactly
ten less than 666, thus distinctly implying that it was Leo the
tenth. Luther accepted his conversion, but frankly told him he
had better clear his mind of any nonsense about the number of
the beast.

Unluckily for himself Stifel did not act on this advice.
Believing that he had discovered the true way of interpreting
the biblical prophecies, he announced that the world would come
to an end on October~3, 1533. The peasants of Holzdorf, of which
place he was pastor, aware of his scientific reputation, accepted
his assurance on this point. Some gave themselves up to
religious exercises, others wasted their goods in dissipation, but
all abandoned their work. When the day foretold had passed,
many of the peasants found themselves ruined. Furious at having
been deceived, they seized the unfortunate prophet, and he was
lucky in finding a refuge in the prison at Wittenberg, from
which he was after some time released by the personal intercession
of \index{Luther, ref.\ to}Luther.

\nblabel{png236a}Stifel wrote a small treatise on algebra, but his chief mathematical
work is his \emph{Arithmetica Integra}, published at Nuremberg
in 1544, with a preface by \index{Melanchthon, ref.\ to}Melanchthon.

\nblabel{xref:216}The first two books of the \emph{Arithmetica Integra} deal with
surds and incommensurables, and are Euclidean in form. \index{Addition, processes of!symbols for}\index{Subtraction,!symbols for}\index{Minus!z1@symbols for}\index{Negative sign}The
third book is on algebra, and is noticeable for having called
general attention to the German practice of using the signs
$+$ and $-$ to denote addition and subtraction. There are traces of
these signs being occasionally employed by Stifel as symbols of
operation and not only as abbreviations; in this use of them he
seems to have followed \index{Hoecke, G. V.}G.~V.~Hoecke. \index{Unknown quantity, word or symbol for}He not only employed the
usual abbreviations for the Italian words which represent the
unknown quantity and its powers, but in at least one case when
there were several unknown quantities he represented them
respectively by the letters $A, B, C,$ \&c.; thus re-introducing the
general algebraic notation which had fallen into disuse since the
time of \index{Jordanus!ref.\ to}Jordanus. It used to be said that Stifel was the real
inventor of \index{Logarithms}logarithms, but it is now certain that this opinion
was due to a misapprehension of a passage in which he compares
geometrical and arithmetical progressions. Stifel is said to have
indicated a formula for writing down the coefficients of the
various terms in the expansion of $(1 + x)^n$ if those in the expansion
of $(1 + x)^{n-1}$ were known.

In 1553 Stifel brought out an edition of \index{Rudolff!ref.\ to}Rudolff's \emph{Die Coss}, in
which he introduced an improvement in the algebraic notation
then current. The symbols at that time ordinarily used for the
\index{Unknown quantity, word or symbol for}unknown quantity and its powers were letters which stood for
abbreviations of the words. \index{Res used for unknown quantity}\index{Addition, processes of!symbols for}\index{Minus!z1@symbols for}\index{Negative sign}\index{Plus!symbols for}Among those frequently adopted
were $R$ or $Rj$ for \emph{radix} or \emph{res} ($x$), \emph{Z} or \emph{C} for \index{Zensus}\emph{zensus} or \index{Census}\emph{census}
($x^2$), $C$ or $K$ for \emph{cubus} ($x^3$), \&c. Thus $x^2 + 5x - 4$ would have
been written
\[
1\ Z\ \text{p.}\ 5\ R\ \text{m.}\ 4;
\]
where \emph{p} stands for plus and \emph{m} for minus. Other letters and
symbols were also employed: thus \index{Xylander!ref.\ to}Xylander (1575) would have
denoted the above expression by
\[
1Q + 5N - 4;
\]
a notation similar to this was sometimes used by \index{Vieta!ref.\ to}Vieta and even
by \index{Fermat!ref.\ to}Fermat. The advance made by Stifel was that he introduced
the symbols $1A$, $1AA$, $1AAA$, for the unknown quantity, its
square, and its cube, which shewed at a glance the relation
between them.\index{Algebra.!z5@Introduction of syncopated algebra by Bhaskara!x4@Stifel|)}\index{Stifel|)}

\index{Tartaglia|(}\nblabel{png237a}\nbparagraph{Tartaglia.} \emph{Niccolo Fontana}, generally known as \emph{Nicholas
Tartaglia}, that is, Nicholas the stammerer, was born at Brescia
in 1500, and died at Venice on December~14, 1557. After the
capture of the town by the French in 1512, most of the inhabitants
took refuge in the cathedral, and were there massacred
by the soldiers. His father, who was a postal messenger at
Brescia, was amongst the killed. The boy himself had his skull
split through in three places, while his jaws and his palate were
cut open; he was left for dead, but his mother got into the
cathedral, and finding him still alive managed to carry him off.
Deprived of all resources she recollected that dogs when wounded
always licked the injured place, and to that remedy he attributed
his ultimate recovery, but the injury to his palate produced an
impediment in his speech, from which he received his nickname.
His mother managed to get sufficient money to pay for his
attendance at school for fifteen days, and he took advantage
of it to steal a copy-book from which he subsequently taught
himself how to read and write; but so poor were they that he
tells us he could not afford to buy paper, and was obliged
to make use of the tombstones as slates on which to work his
exercises.

He commenced his public life by lecturing at Verona, but he
was appointed at some time before 1535 to a chair of mathematics
at Venice, where he was living, when he became famous
through his acceptance of a challenge from a certain \index{Fiore}\emph{Antonio del
Fiore} (or \index{Florido}\emph{Florido}). Fiore had learnt from his master, one
\index{Ferro}\emph{Scipione Ferro} (who died at Bologna in 1526), an empirical
solution of a \index{Cubic equations}cubic equation of the form $x^{3} + qx = r$. This
solution was previously unknown in Europe, and it is possible
that Ferro had found the result in an Arab work. Tartaglia, in
answer to a request from \index{Colla}Colla in 1530, stated that he could
effect the solution of a numerical equation of the form $x^3 + px^2 = r$.
Fiore, believing that Tartaglia was an impostor, challenged him
to a contest. According to this challenge each of them was to
deposit a certain stake with a notary, and whoever could solve
the most problems out of a collection of thirty propounded by
the other was to get the stakes, thirty days being allowed for
the solution of the questions proposed. \nblabel{png238a}Tartaglia was aware
that his adversary was acquainted with the solution of a cubic
equation of some particular form, and suspecting that the
questions proposed to him would all depend on the solution of
such cubic equations, set himself the problem to find a general
solution, and certainly discovered how to obtain a solution of
some if not all cubic equations. His solution is believed to have
depended on a geometrical construction,\footnote
  {See below, p.~\pageref{xref:224}.}
but led to the formula
which is often, but unjustly, described as \index{Cardan!ref.\ to}Cardan's.

When the contest took place, all the questions proposed to
Tartaglia were, as he had suspected, reducible to the solution
of a cubic equation, and he succeeded within two hours in
bringing them to particular cases of the equation $x^3 + qx = r$, of
which he knew the solution. His opponent failed to solve any
of the problems proposed to him, most of which were, as a
matter of fact, reducible to numerical equations of the form
$x^3 + px^2 = r$. Tartaglia was therefore the conqueror; he subsequently
composed some verses commemorative of his victory.

The chief works of Tartaglia are as follows: (i)~His \emph{Nova
scienza}, published in 1537: in this he investigated the fall of
bodies under gravity; and he determined the range of a \index{Projectiles}projectile,
stating that it was a maximum when the angle of
projection was $45°$, but this seems to have been a lucky guess.
(ii)~His \emph{Inventioni}, published in 1546, and containing, \emph{inter
alia}, his solution of \index{Cubic equations}cubic equations.
(iii)~His \emph{Trattato di %[**TN: 'Trattato de' in original text]
numeri e misuri}, consisting of an arithmetic, published in 1556,
and a treatise on numbers, published in 1560; in this he shewed
how the coefficients of $x$ in the expansion of $(1 + x)^n$ could be
calculated, by the use of an \index{Arithmetical!triangle}\index{Triangle, area of!arithmetical}arithmetical triangle,\footnote
  {See below, pp.~\pageref{xref:284}, \pageref{xref:285}.}
from those
in the expansion of $(1 + x)^{n-1}$ for the cases when $n$ is equal to
2, 3, 4, 5, or 6. His works were collected into a single edition
and republished at Venice in 1606.

\nblabel{png239a}The treatise on arithmetic and numbers is one of the chief
authorities for our knowledge of the early Italian \index{Algorism}algorism. It
is verbose, but gives a clear account of the arithmetical methods
then in use, and has numerous historical notes which, as far as
we can judge, are reliable, and are ultimately the authorities for
many of the statements in the \hyperref[chapter:XI]{last chapter}. It contains an
immense number of questions on every kind of problem which
would be likely to occur in mercantile arithmetic, and there
are several attempts to frame algebraical formulae suitable for
particular problems.

These problems give incidentally a good deal of information
as to the ordinary life and commercial customs of the time.
Thus we find that the interest demanded on first-class security
in Venice ranged from 5 to 12 per cent.\ a year; while the
interest on commercial transactions ranged from 20 per cent.\
a year upwards. Tartaglia illustrates the evil effects of the
law forbidding usury by the manner in which it was evaded
in farming. Farmers who were in debt were forced by their
creditors to sell all their crops immediately after the harvest;
the market being thus glutted, the price obtained was very low,
and the money-lenders purchased the corn in open market at an
extremely cheap rate. The farmers then had to borrow their
seed-corn on condition that they replaced an equal quantity, or
paid the then price of it, in the month of May, when corn was
dearest. Again, Tartaglia, who had been asked by the magistrates
at Verona to frame for them a sliding scale by which the
price of bread would be fixed by that of corn, enters into a
discussion on the principles which it was then supposed should
regulate it. In another place he gives the rules at that time
current for preparing medicines.

\index{Pacioli!ref.\ to}Pacioli had given in his arithmetic some problems of an
amusing character, and Tartaglia imitated him by inserting a
\index{Recreations, mathematical|(}\index{Puzzles|(}large collection of mathematical puzzles. He half apologizes
for introducing them by saying that it was not uncommon at
dessert to propose arithmetical questions to the company by way
of amusement, and he therefore adds some suitable problems.
He gives several questions on how to guess a number thought
of by one of the company, or the relationships caused by the
marriage of relatives, or difficulties arising from inconsistent
bequests. Other puzzles are similar to the following. ``Three
beautiful ladies have for husbands three men, who are young,
handsome, and gallant, but also jealous. The party are travelling,
and find on the bank of a river, over which they have to
pass, a small boat which can hold no more than two persons.
How can they pass, it being agreed that, in order to avoid
scandal, no woman shall be left in the society of a man unless
her husband is present?'' ``A ship, carrying as passengers
fifteen Turks and fifteen Christians, encounters a storm; and
the pilot declares that in order to save the ship and crew one-half
of the passengers must be thrown into the sea. To choose
the victims, the passengers are placed in a circle, and it is agreed
that every ninth man shall be cast overboard, reckoning from a
certain point. In what manner must they be arranged, so that
the lot may fall exclusively upon the Turks?'' ``Three men
robbed a gentleman of a vase containing 24 ounces of balsam.
Whilst running away they met in a wood with a glass-seller
of whom in a great hurry they purchased three vessels. On
reaching a place of safety they wish to divide the booty, but
they find that their vessels contain 5, 11, and 13 ounces
respectively. How can they divide the balsam into equal
portions?''

These problems---some of which are of oriental origin---form
the basis of the collections of mathematical recreations by \index{Bachet!ref.\ to}\index{Demeziriac@De Méziriac!ref.\ to}\index{Meziriac@Méziriac!ref.\ to}Bachet
de Méziriac, \index{Ozanam}Ozanam, and \index{Montucla}Montucla.\footnote
  {Solutions of these and other similar problems are given in my \emph{Mathematical
  Recreations}, chaps.~i, ii. On Bachet, see below, p.~\pageref{xref:305}. \emph{Jacques
  Ozanam}, born at Bouligneux in 1640, and died in 1717, left numerous works
  of which one, worth mentioning here, is his \emph{Récréations mathématiques et
  physiques}, two volumes, Paris, 1696. \emph{Jean Étienne Montucla}, born at
  Lyons in 1725, and died in Paris in 1799, edited and revised Ozanam's
  mathematical recreations. His history of attempts to square the circle, 1754,
  and history of mathematics to the end of the seventeenth century, in two
  volumes, 1758, are interesting and valuable works.}\index{Tartaglia|)}\index{Puzzles|)}\index{Recreations, mathematical|)}

\index{Cardan|(}\nblabel{png241a}\nbparagraph{Cardan.\footnote
  {There is an admirable account of Cardan's life in the \emph{Nouvelle biographie
  générale}, by \index{Sardou on Cardan}V.~Sardou. Cardan left an autobiography of which an analysis
  by \index{Morley on Cardan}H.~Morley was published in two volumes in London in 1854. All
  Cardan's printed works were collected by \index{Sponius on Cardan}Sponius, and published in ten
  volumes, Lyons, 1663; the works on arithmetic and geometry are contained
  in the fourth volume. It is said that there are in the Vatican several
  manuscript note-books of his which have not been yet edited.}}
The life of Tartaglia was embittered by a quarrel
with his contemporary Cardan, who published Tartaglia's solution
of a cubic equation which he had obtained under a pledge
of secrecy. \emph{Girolamo Cardan} was born at Pavia on September~24,
1501, and died at Rome on September~21, 1576. His
career is an account of the most extraordinary and inconsistent
acts. A gambler, if not a murderer, he was also an ardent
student of science, solving problems which had long baffled all
investigation; at one time of his life he was devoted to intrigues
which were a scandal even in the sixteenth century, at another
he did nothing but rave on astrology, and yet at another he
declared that philosophy was the only subject worthy of man's
attention. His was the genius that was closely allied to
madness.

He was the illegitimate son of a lawyer of Milan, and was
educated at the universities of Pavia and Padua. After taking
his degree he commenced life as a doctor, and practised his
profession at Sacco and Milan from 1524 to 1550; it was during
this period that he studied mathematics and published his chief
works. After spending a year or so in France, Scotland, and
England, he returned to Milan as professor of science, and shortly
afterwards was elected to a chair at Pavia. Here he divided his
time between debauchery, astrology, and mechanics. His two
sons were as wicked and passionate as himself: the elder was
in 1560 executed for poisoning his wife, and about the same
time Cardan in a fit of rage cut off the ears of the younger who
had committed some offence; for this scandalous outrage he
suffered no punishment, as the Pope \index{Gregory XIII. of Rome}Gregory XIII. granted him
protection. In 1562 Cardan moved to Bologna, but the scandals
connected with his name were so great that the university took
steps to prevent his lecturing, and only gave way under pressure
from Rome. In 1570 he was imprisoned for heresy on account
of his having published the horoscope of Christ, and when
released he found himself so generally detested that he determined
to resign his chair. At any rate he left Bologna in
1571, and shortly afterwards moved to Rome. Cardan was
the most distinguished astrologer of his time, and when he
settled at Rome he received a pension in order to secure his
services as astrologer to the papal court. This proved fatal to
him for, having foretold that he should die on a particular
day, he felt obliged to commit suicide in order to keep up his
reputation---so at least the story runs.

\nblabel{png242a}The chief mathematical work of Cardan is the \emph{Ars Magna}
published at Nuremberg in 1545. Cardan was much interested
in the contest between \index{Tartaglia!ref.\ to|(}Tartaglia and \index{Fiore}Fiore, and as he had
already begun writing this book he asked Tartaglia to communicate
his method of solving a cubic equation. Tartaglia
refused, whereupon \index{Algebra.!z5@Introduction of syncopated algebra by Bhaskara!x5@Cardan|(}Cardan abused him in the most violent
terms, but shortly afterwards wrote saying that a certain
Italian nobleman had heard of Tartaglia's fame and was most
anxious to meet him, and begged him to come to Milan at
once. Tartaglia came, and though he found no nobleman
awaiting him at the end of his journey, he yielded to Cardan's
importunity, and gave him the rule, Cardan on his side taking
a solemn oath that he would never reveal it. Cardan asserts
that he was given merely the result, and that he obtained
the proof himself, but this is doubtful. He seems to have
at once taught the method, and one of his pupils \index{Ferrari!ref.\ to}Ferrari
reduced the equation of the \index{Quartic equation}fourth degree to a cubic and so
solved it.

When the \emph{Ars Magna} was published in 1545 the breach of
faith was made manifest.\footnote
  {The history of the subject and of the doings of Fiore, Tartaglia, and
  Cardan are given in an Appendix to the 2nd edition of the French translation
  of \index{Ball, W. W. R., ref.\ to}my \emph{Mathematical Recreations}, Paris, 1908, vol.~ii, p.~322 \emph{et seq.}}
Tartaglia not unnaturally was very
angry, and after an acrimonious controversy he sent a challenge
to Cardan to take part in a mathematical duel. The preliminaries
were settled, and the place of meeting was to be a
certain church in Milan, but when the day arrived Cardan
failed to appear, and sent Ferrari in his stead. Both sides
claimed the victory, though I gather that Tartaglia was the
more successful; at any rate his opponents broke up the
meeting, and he deemed himself fortunate in escaping with his
life. Not only did Cardan succeed in his fraud, but modern
writers have often attributed the solution to him, so that
Tartaglia has not even that posthumous reputation which at
least is his due.\index{Tartaglia!ref.\ to|)}

The \emph{Ars Magna} is a great advance on any algebra previously
published. Hitherto algebraists had confined their
attention to those \index{Roots of equations,!imaginary}\index{Roots of equations,!negative}\index{Equations, differential!roots of imaginary}\index{Equations, differential!roots of negative}roots of equations which were positive.
Cardan discussed negative and even complex roots, and
proved that the latter would always occur in pairs, though he
declined to commit himself to any explanation as to the
meaning of these ``sophistic'' quantities which he said were
ingenious though useless. \nblabel{xref:224}Most of his analysis of \index{Cubic equations|(}cubic equations
seems to have been original; he shewed that if the three
roots were real, \index{Tartaglia!ref.\ to}Tartaglia's solution gave them in a form
which involved \index{Imaginary numbers}imaginary quantities. Except for the somewhat
similar researches of \index{Bombelli!ref.\ to}Bombelli a few years later, the theory
of imaginary quantities received little further attention from
mathematicians until \index{Bernoulli, John!ref.\ to}John Bernoulli and \index{Euler!ref.\ to}Euler took up the
matter after the lapse of nearly two centuries. \index{Gauss!ref.\ to}Gauss first put
the subject on a systematic and scientific basis, introduced the
notation of \index{Complex numbers}\index{Complex variables}complex variables, and used the symbol $i$, which had
been introduced by Euler in 1777, to denote the square root of
(-1): the modern theory is chiefly based on his researches.

\index{Equations, differential!position of roots}Cardan established the relations connecting the roots with
the coefficients of an equation. He was also aware of the
principle that underlies Descartes's ``rule of signs,'' but as he
followed the custom, then general, of writing his equations as
the equality of two expressions in each of which all the terms
were positive he was unable to express the rule concisely. He
gave a method of approximating to the root of a numerical
equation, founded on the fact that, if a function have opposite
signs when two numbers are substituted in it, the equation
obtained by equating the function to zero will have a root
between these two numbers.

Cardan's solution of a quadratic equation is geometrical
and substantially the same as that given by \index{Alkarismi!ref.\ to}Alkarismi. \nblabel{png244a}His
solution of a cubic equation is also geometrical, and may be
illustrated by the following case which he gives in chapter \textsc{xi}.
To solve the equation $x^3+6x=20$ (or any equation of the form
$x^3+qx=r$), take two cubes such that the rectangle under their
respective edges is 2 (or $\frac{1}{3}q$) and the difference of their
volumes is 20 (or $r$). Then $x$ will be equal to the difference
between the edges of the cubes. To verify this he first gives a
geometrical lemma to shew that, if from a line $AC$ a portion
$CB$ be cut off, then the cube on $AB$ will be less than the
difference between the cubes on $AC$ and $BC$ by three times
the right parallelepiped whose edges are respectively equal to
$AC$, $BC$, and $AB$---this statement is equivalent to the algebraical
identity $(a-b)^3=a^3-b^3-3ab(a-b)$---and the fact
that $x$ satisfies the equation is then obvious. To obtain the
lengths of the edges of the two cubes he has only to solve
a quadratic equation for which the geometrical solution previously
given sufficed.\index{Cubic equations|)}

Like all previous mathematicians he gives separate proofs
of his rule for the different forms of equations which can fall
under it. Thus he proves the rule independently for equations
of the form $x^3+px=q$, $x^3=px+q$, $x^3+px+q=0$, and
$x^3+q=px$. It will be noticed that with geometrical proofs
this was the natural course, but it does not appear that he was
aware that the resulting formulae were general. The equations
he considers are numerical.

Shortly after \index{Cardan!ref.\ to}Cardan came a number of mathematicians
who did good work in developing the subject, but who are
hardly of sufficient importance to require detailed mention here.
Of these the most celebrated are perhaps Ferrari and Rheticus.\index{Algebra.!z5@Introduction of syncopated algebra by Bhaskara!x5@Cardan|)}\index{Cardan|)}

\index{Ferrari|(}\nblabel{png245a}\nbparagraph{Ferrari.} \emph{Ludovico Ferraro}, usually known as \emph{Ferrari},
whose name I have already mentioned in connection with the
solution of a \index{Biquadratic equation}biquadratic equation, was born at Bologna on
Feb.~2, 1522, and died on Oct.~5, 1565. His parents were
poor and he was taken into Cardan's service as an errand boy,
but was allowed to attend his master's lectures, and subsequently
became his most celebrated pupil. He is described as
``a neat rosy little fellow, with a bland voice, a cheerful face,
and an agreeable short nose, fond of pleasure, of great natural
powers,'' but ``with the temper of a fiend.'' His manners and
numerous accomplishments procured him a place in the service
of the \index{Gonzaga, Cardinal, ref.\ to}Cardinal Ferrando Gonzago, where he managed to make
a fortune. His dissipations told on his health, and he retired
in 1565 to Bologna where he began to lecture on mathematics.
He was poisoned the same year either by his sister, who seems
to have been the only person for whom he had any affection,
or by her paramour.

Such work as Ferrari produced is incorporated in \index{Cardan!ref.\ to}Cardan's
\emph{Ars Magna} or Bombelli's \emph{Algebra}, but nothing can be definitely
assigned to him except the solution of a \index{Biquadratic equation}biquadratic
equation. \index{Colla}Colla had proposed the solution of the equation
\index{Quartic equation}$x^4+6x^2+36=60x$ as a challenge to mathematicians: this particular
equation had probably been found in some Arabic
work. Nothing is known about the history of this problem
except that Ferrari succeeded where \index{Tartaglia!ref.\ to}Tartaglia and Cardan
had failed.\index{Ferrari|)}

\index{Rheticus}\nblabel{png246a}\nbparagraph{Rheticus.} \emph{Georg Joachim Rheticus}, born at Feldkirch on
Feb.~15, 1514, and died at Kaschau on Dec.~4, 1576, was
professor at Wittenberg, and subsequently studied under
Copernicus whose works were produced under the direction of
Rheticus. Rheticus constructed various trigonometrical tables,
some of which were published by his pupil \index{Otho}Otho in 1596.
These were subsequently completed and extended by \index{Vieta!ref.\ to}Vieta
and \index{Pitiscus!ref.\ to}Pitiscus, and are the basis of those still in use. Reticus
also found the values of $\sin 2\theta$ and $\sin 3\theta$ in terms of $\sin \theta$
and $\cos \theta$, and was aware that trigonometrical ratios might be
defined by means of the ratios of the sides of a right-angled
triangle without introducing a circle.

I add here the names of some other celebrated mathematicians
of about the same time, though their works are now
of little value to any save antiquarians. \index{Maurolycus}\nbname{Franciscus
Maurolycus,} born at Messina of Greek parents in 1494, and
died in 1575, translated numerous Latin and Greek mathematical
works, and discussed the conics regarded as sections of
a cone: his works were published at Venice in 1575. \index{Borrel, J.}\nbname{Jean
Borrel,} born in 1492 and died at Grenoble in 1572, wrote an
algebra, founded on that of \index{Stifel!ref.\ to}Stifel; and a history of the
quadrature of the circle: his works were published at Lyons
in 1559. \index{Xylander}\nbname{Wilhelm Xylander,} born at Augsburg on Dec.~26,
1532, and died on Feb.~10, 1576, at Heidelberg, where since
1558 he had been professor, brought out an edition of the
works of \index{Psellus!ref.\ to}Psellus in 1556; an edition of \index{Euclid's \emph{Elements}!ref.\ to}Euclid's \emph{Elements} in
1562; an edition of the \emph{Arithmetic} of \index{Diophantus!ref.\ to}Diophantus in 1575;
and some minor works which were collected and published in
1577. \nblabel{png247a}\index{Commandino}\nbname{Frederigo Commandino,} born at Urbino in 1509,
and died there on Sept.~3, 1575, published a translation of the
works of \index{Archimedes!ref.\ to}Archimedes in 1558; selections from \index{Apollonius!ref.\ to}Apollonius and
Pappus in 1566; an edition of \index{Euclid's \emph{Elements}!ref.\ to}Euclid's \emph{Elements} in 1572; and
selections from \index{Aristarchus!ref.\ to}Aristarchus, \index{Ptolemy!ref.\ to}Ptolemy, \index{Hero of Alexandria!ref.\ to}Hero, and Pappus in
1574: all being accompanied by commentaries. \index{Peletier}\nbname{Jacques
Peletier,} born at le Mans on July~25, 1517, and died at Paris
in July 1582, wrote text-books on algebra and geometry:
most of the results of \index{Stifel!ref.\ to}Stifel and \index{Cardan!ref.\ to}Cardan are included in the
former. \index{Romanus of Louvain}\nbname{Adrian Romanus,} born at Louvain on Sept.~29,
1561, and died on May~4, 1625, professor of mathematics and
medicine at the university of Louvain, was the first to prove
the usual formula for $\sin(A+B)$. And lastly, \index{Pitiscus}\nbname{Bartholomäus
Pitiscus,} born on Aug.~24, 1561, and died at Heidelberg,
where he was professor of mathematics, on July~2, 1613,
published his \index{Trigonometry, addition formulae}\emph{Trigonometry} in 1599: \index{Expansion,!ofcosa@of $\cos (A ± B)$}%
\index{Expansion,!ofsina@of $\sin (A ± B)$}this contains the expressions
for $\sin (A ± B)$ and $\cos (A ± B)$ in terms of the trigonometrical
ratios of $A$ and $B$.

About this time also several text-books were produced
which if they did not extend the boundaries of the subject
systematized it. In particular I may mention those by Ramus
and \index{Bombelli!ref.\ to}Bombelli.

\index{Ramus|(}\nbparagraph{Ramus.\footnote
  {See the monographs by \index{Waddington on Ramus}Ch.~Waddington, Paris, 1855; and by
  \index{Desmaze on Ramus}C.~Desmaze, Paris, 1864.}}
\emph{Peter Ramus} was born at Cuth in Picardy in
1515, and was killed at Paris in the massacre of St.~Bartholomew
on Aug.~24, 1572. He was educated at the university
of Paris, and on taking his degree he astonished and charmed
the university with the brilliant declamation he delivered on
the thesis that everything \index{Aristotle!ref.\ to}Aristotle had taught was false. He
lectured---for it will be remembered that in early days there
were no professors---first at le Mans, and afterwards at Paris;
at the latter he founded the first chair of mathematics.
Besides some works on philosophy he wrote treatises on
arithmetic, algebra, geometry (founded on Euclid), astronomy
(founded on the works of \index{Copernicus!ref.\ to}Copernicus), and physics, which were
long regarded on the Continent as the standard text-books in
these subjects. They are collected in an edition of his works
published at Bâle in 1569.\index{Ramus|)}

\index{Bombelli}\index{Algebra.!z5@Introduction of syncopated algebra by Bhaskara!x6@Bombelli}\nblabel{png248a}\nbparagraph{Bombelli.} Closely following the publication of Cardan's
great work, \emph{Rafaello Bombelli} published in 1572 an algebra
which is a systematic exposition of the knowledge then current
on the subject. In the preface he traces the history of the
subject, and alludes to \index{Diophantus!ref.\ to}Diophantus who, in spite of the notice
of \index{Regiomontanus!ref.\ to}Regiomontanus, was still unknown in Europe. \index{Imaginary numbers}He discusses
radicals, real and complex. He also treats the theory of
equations, and shews that in the irreducible case of a cubic
equation the roots are all real; and he remarks that the
problem to trisect a given angle is the same as that of the
solution of a cubic equation. Finally he gave a large collection
of problems.

\index{Addition, processes of!symbols for}Bombelli's work is noticeable for his use of symbols which
indicate an approach to index notation. Following in the
steps of \index{Stifel!ref.\ to}Stifel, he introduced a \index{Unknown quantity, word or symbol for}symbol \bombelli{1} for the unknown
quantity, \bombelli{2} for its square, \bombelli{3} for its cube, and so on, and
therefore wrote $x^2+5x-4$ as
\[
1\ \bombelli{2} \text{ p.\ } 5\ \bombelli{1}\ \text{m.\ } 4
\]
\index{Stevinus!ref.\ to}\index{Algebra.!z5@Introduction of syncopated algebra by Bhaskara!x7@and Stevinus}Stevinus in 1586 employed
\circledigit{1}, \circledigit{2}, \circledigit{3}, $\ldots$ in a similar way;
and suggested, though he did not use, a corresponding notation
for \index{Exponents}\index{Indices}fractional indices. He would have written the above
expression as
\[
1\ \circledigit{2} + 5\ \circledigit{1} - 4\ \circledigit{0}.
\]
But whether the symbols were more or less convenient they
were still only abbreviations for words, and were subject to
all the rules of syntax. They merely afforded a sort of shorthand
by which the various steps and results could be expressed
concisely. The next advance was the creation of symbolic
algebra, and the chief credit of that is due to \index{Vieta!ref.\ to}Vieta.

\section*{The development of symbolic algebra.\nblabel{png249a}}

We have now reached a point beyond which any considerable
development of algebra, so long as it was strictly
syncopated, could hardly proceed. It is evident that Stifel
and Bombelli and other writers of the sixteenth century had
introduced or were on the point of introducing some of the
ideas of symbolic algebra. But so far as the credit of inventing
symbolic algebra can be put down to any one man
we may perhaps assign it to \index{Vieta!ref.\ to}Vieta, while we may say that
\index{Harriot!ref.\ to}Harriot and \index{Descartes!ref.\ to}Descartes did more than any other writers to
bring it into general use. It must be remembered, however,
that it took time before all these innovations became generally
known, and they were not familiar to mathematicians until the
lapse of some years after they had been published.

\index{Vieta|(}\nblabel{png249b}\nbparagraph{Vieta.\footnote
  {The best account of Vieta's life and works is that by A.~De~Morgan in
  the \emph{English Cyclopaedia}, London, vol.~vi, 1858.}}
\emph{Franciscus Vieta (François Viète)} was born in
1540 at Fontenay near la Rochelle, and died in Paris in 1603.
He was brought up as a lawyer and practised for some time
at the Parisian bar; he then became a member of the provincial
parliament in Brittany; and finally in 1580, through
the influence of the \index{Rohan, ref.\ to}\index{Derohan@De Rohan, ref.\ to}Duke de~Rohan, he was made master of
requests, an office attached to the parliament at Paris; the
rest of his life was spent in the public service. He was a
firm believer in the right divine of kings, and probably a
zealous catholic. After 1580 he gave up most of his leisure
to mathematics, though his great work, \emph{In Artem Analyticam
Isagoge}, in which he explained how algebra could be applied
to the solution of geometrical problems, was not published till
1591.

His mathematical reputation was already considerable,
when one day the ambassador from the Low Countries remarked
to \index{Henry A@Henry IV. of France, ref.\ to}Henry IV. that France did not possess any
geometricians capable of solving a problem which had been
propounded in 1593 by his countryman \index{Romanus of Louvain!ref.\ to|(}Adrian Romanus to
all the mathematicians of the world, and which required the
solution of an equation of the 45th degree. The king thereupon
summoned Vieta, and informed him of the challenge.
Vieta saw that the equation was satisfied by the chord of a
circle (of unit radius) which subtends an angle $2\pi/45$ at the
centre, and in a few minutes he gave back to the king two
solutions of the problem written in pencil. In explanation of
this feat I should add that Vieta had previously discovered
how to form the equation connecting $\sin n\theta$ with $\sin \theta$ and
$\cos \theta$. Vieta in his turn asked Romanus to give a geometrical
construction to describe a circle which should touch three
given circles. This was the problem which \index{Apollonius!ref.\ to}Apollonius had
treated in his \emph{De Tactionibus,} a lost book which Vieta at
a later time conjecturally restored. Romanus solved the
problem by the use of conic sections, but failed to do it by
Euclidean geometry. Vieta gave a Euclidean solution which
so impressed Romanus that he travelled to Fontenay, where
the French court was then settled, to make Vieta's acquaintance---an
acquaintanceship which rapidly ripened into warm
friendship.\index{Romanus of Louvain!ref.\ to|)}

Henry was much struck with the ability shown by Vieta in
this matter. The Spaniards had at that time a \index{Ciphers, discoveries of}cipher containing
nearly 600 characters, which was periodically changed, and
which they believed it was impossible to decipher. A despatch
having been intercepted, the king gave it to Vieta, and asked
him to try to read it and find the key to the system. Vieta
succeeded, and for two years the French used it, greatly to
their profit, in the war which was then raging. So convinced
was \index{Philip II. of Spain, ref.\ to}Philip II. that the cipher could not be discovered, that when
he found his plans known he complained to the Pope that the
French were using sorcery against him, ``contrary to the practice
of the Christian faith.''

{\nbstretchyspace \index{Algebra.!z6@Introduction of symbolic algebra by Vieta|(}Vieta wrote numerous works on algebra and geometry.
The most important are the \emph{In Artem Analyticam Isagoge},
Tours, 1591; the \emph{Supplementum Geometriae}, and a collection
of geometrical problems, Tours, 1593; and the \emph{De Numerosa}
\emph{Potestatum Resolutione}, Paris, 1600.} All of these were printed
for private circulation only, but they were collected by \index{Schooten, van!ref.\ to}\index{Vans@Van Schooten!ref.\ to}F.~van
Schooten and published in one volume at Leyden in 1646.
Vieta also wrote the \emph{De Aequationum Recognitione et Emendatione},
which was published after his death in 1615 by \index{Anderson on Vieta}Alexander
Anderson.

\nblabel{png251a}The \emph{In Artem} is the earliest work on symbolic algebra. It
also introduced the \index{Unknown quantity, word or symbol for}use of \index{Letters in diagrams!to indicate magnitudes}letters for both known and unknown
(positive) quantities, a notation for the powers of quantities,
and enforced the advantage of working with \index{Homogeneity, Vieta on}homogeneous
equations. To this an appendix called \emph{Logistice Speciosa} was
added on addition and multiplication of algebraical quantities,
and on the powers of a binomial up to the sixth. Vieta
implies that he knew how to form the coefficients of these six
expansions by means of the \index{Arithmetical!triangle}\index{Triangle, area of!arithmetical}arithmetical triangle as \index{Tartaglia!ref.\ to}Tartaglia
had previously done, but \index{Pascal!ref.\ to}Pascal gave the general rule for
forming it for any order, and \index{Stifel!ref.\ to}Stifel had already indicated the
method in the expansion of $(1+x)^n$ if those in the expansion
of $(1+x)^{n-1}$ were known; \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton was the first to give the
general expression for the coefficient of $x^p$ in the expansion of
$(1+x)^n$. Another appendix known as \emph{Zetetica} on the solution
of equations was subsequently added to the \emph{In Artem}.

The \emph{In Artem} is memorable for two improvements in
algebraic notation which were introduced here, though it is
probable that Vieta took the idea of both from other authors.

One of these improvements was that he denoted the known
quantities by the consonants $B, C, D$, \&c., and the unknown
quantities by the vowels $A, E, I$, \&c. Thus in any problem
he was able to use a number of unknown quantities. In this
particular point he seems to have been forestalled by \index{Jordanus!ref.\ to}Jordanus
and by Stifel. The present custom of using the letters at the
beginning of the alphabet $a, b, c$, \&c., to represent known
quantities and those towards the end, $x, y, z$, \&c., to represent
the unknown quantities was introduced by \index{Descartes!ref.\ to}Descartes in 1637.

The other improvement was this. Till this time it had been
generally the custom to introduce new symbols to represent the
square, cube, \&c., of quantities which had already occurred in
the equations; thus, if $R$ or $N$ stood for $x$, $Z$ or $C$ or $Q$ stood
for $x^2$, and $C$ or $K$ for $x^3$, \&c. So long as this was the case the
chief advantage of algebra was that it afforded a concise statement of results every statement of which was reasoned out.
\index{Unknown quantity, word or symbol for}But when Vieta used $A$ to denote the unknown quantity $x$, he
sometimes employed $A$ \emph{quadratus}, $A$ \emph{cubus}, \dots to represent $x^2,
x^3$, \dots, which at once showed the connection between the
different powers; and later the successive powers of \emph{A} were
commonly denoted by the abbreviations $Aq, Ac, Aqq$, \&c. Thus
Vieta would have written the equation
\[
3BA^2-DA+A^3=Z,
\]
as \emph{B 3 in A quad.\ --- D plano in A + A cubo aequatur Z solido}.
It will be observed that the dimensions of the constants ($B, D, $
and $Z$) are chosen so as to make the equation \index{Homogeneity, Vieta on}homogeneous:
this is characteristic of all his work. It will be also noticed
that he does not use a \index{Equality, symbols for}\index{Equalityz@=, meanings of}sign for equality; and in fact the particular
sign = which we use to denote equality was employed by
him to represent \index{Difference between, sign for}``the difference between.'' Vieta's notation is
not so convenient as that previously used by \index{Stifel!ref.\ to}Stifel, \index{Bombelli!ref.\ to}Bombelli,
and \index{Stevinus!ref.\ to}Stevinus, but it was more generally adopted.

These two steps were almost essential to any further progress
in algebra. In both of them Vieta had been forestalled, but it
was his good luck in emphasising their importance to be the
means of making them generally known at a time when opinion
was ripe for such an advance.

\nblabel{png252a}The \emph{De Aequationum Recognitione et Emendatione} is mostly
on the theory of equations. It was not published till twelve
years after Vieta's death, and it is possible that the editor made
additions to it. \index{Equations, differential!theory of}Vieta here indicated how from a given equation
another could be obtained whose roots were equal to those of
the original increased %[**TN: 'increase' in original text]
by a given quantity, or multiplied by a
given quantity; he used this method to get rid of the coefficient
of $x$ in a quadratic equation and of the coefficient of
$x^2$ in a \index{Cubic equations|(}cubic equation, and was thus enabled to give the general
algebraic solution of both. It would seem that he knew that
the first member of an algebraical equation $\phi(x)=0$ could be
resolved into linear factors, and that the coefficients of $x$ could
be expressed as functions of the roots; perhaps the discovery
of both these theorems should be attributed to him.

His solution of a cubic equation is as follows. First reduce
the equation to the form $x^3+3a^2x=2b^3$. Next let $x=a^2/y-y$,
and we get $y^6 + 2b^3y^3=a^6$, which is a quadratic in $y^3$. Hence $y$
can be found, and therefore $x$ can be determined.\index{Cubic equations|)}

His solution of a \index{Biquadratic equation}biquadratic is similar to that known as
\index{Ferrari!ref.\ to}Ferrari's, and essentially as follows. He first got rid of the
term involving $x^3$, thus reducing the equation to the form
\index{Quartic equation}$x^4 + a^2 x^2 + b^3x= c^4$. He then took the forms involving $x^2$ and $x$
to the right-hand side of the equation and added $x^2y^2+\frac{1}{4}y^4$ to
each side, so that the equation became
\[
(x^2+\frac{1}{2}y^2)^2=x^2(y^2-a^2)-b^3x+\frac{1}{4}y^4+c^4.
\]
He then chose $y$ so that the right-hand side of this equality is
a perfect square. Substituting this value of $y$, he was able to
take the square root of both sides, and thus obtain two quadratic
equations for $x$, each of which can be solved.

The \emph{De Numerosa Potestatum Resolutione} deals with numerical
equations. In this a method for approximating to the
values of positive roots is given, but it is prolix and of little
use, though the principle (which is similar to that of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's
rule) is correct. Negative roots are uniformly rejected. This
work is hardly worthy of Vieta's reputation.

Vieta's trigonometrical researches are included in various
tracts which are collected in \index{Schooten, van!ref.\ to}\index{Vans@Van Schooten!ref.\ to}Van Schooten's edition. Besides
some trigonometrical tables he gave the general expression for
the sine (or chord) of an angle in terms of the sine and cosine
of its submultiples. \index{Delambre}Delambre considers this as the completion
of the Arab system of trigonometry. We may take it then
that from this time the results of elementary trigonometry were
familiar to mathematicians. Vieta also elaborated the theory
of right-angled spherical triangles.

Among Vieta's miscellaneous tracts will be found a proof
that each of the famous geometrical problems of the \index{Angle,!trisection of}\index{Trisection of angle}trisection
of an angle and the \index{Cube, duplication of}duplication of the cube depends on the
solution of a cubic equation. There are also some papers
connected with an angry controversy with \index{Clavius}Clavius, in 1594,
on the subject of the reformed calendar, in which Vieta was
not well advised.\index{Algebra.!z6@Introduction of symbolic algebra by Vieta|)}

Vieta's works on geometry are good, but they contain
nothing which requires mention here. He applied algebra
and trigonometry to help him in investigating the properties
of figures. He also, as I have already said, laid great stress
on the desirability of always working with homogeneous
equations, so that if a square or a cube were given it should
be denoted by expressions like $a^{2}$ or $b^{3}$, and not by terms like
$m$ or $n$ which do not indicate the dimensions of the quantities
they represent. He had a lively dispute with \index{Scaliger}Scaliger on the
latter publishing a solution of the quadrature of the circle,
and Vieta succeeded in showing the mistake into which his
rival had fallen. He gave a solution of his own which as far
as it goes is correct, and stated that the area of a square is to
that of the circumscribing circle as

\[
\surd\tfrac{1}{2} ×
\surd(\tfrac{1}{2} + \surd\tfrac{1}{2}) ×
\surd\{\tfrac{1}{2} + \surd(\tfrac{1}{2} + \surd\tfrac{1}{2})\}
\ldots \textit{ad inf.} : 1.
\]

This is one of the earliest attempts to find the \index{P1@$\pi$,!z1@value of}value of $\pi$ by
means of an infinite series. He was well acquainted with the
extant writings of the Greek geometricians, and introduced the
curious custom, which during the seventeenth and eighteenth
centuries became fashionable, of restoring lost classical works.
He himself produced a conjectural restoration of the \emph{De Tactionibus} of \index{Apollonius!ref.\ to}Apollonius.\index{Vieta|)}

\index{Girard|(}\nblabel{png254a}\nbparagraph{Girard.} Vieta's results in trigonometry and the theory of
equations were extended by \emph{Albert Girard}, a Dutch mathematician,
who was born in Lorraine in 1595, and died on
December~9, 1632.

\index{Trigonometrical functions|(}In 1626 Girard published at the Hague a short treatise on
trigonometry, to which were appended tables of the values of
the trigonometrical functions. This work contains the earliest
use of the abbreviations \index{Sine}\index{Secant}\index{Tangent (trigonometrical)}\emph{sin}, \emph{tan}, \emph{sec} for sine, tangent, and
secant. The \index{Supplemental triangle}supplemental \index{Polar triangle}triangles in spherical trigonometry
are also discussed; their properties seem to have been discovered
by Girard and Snell at about the same time. Girard also gave
the expression for the area of a spherical triangle in terms of
the \index{Spherical excess}spherical excess---this was discovered independently by
\index{Cavalieri!ref.\ to}Cavalieri. In 1627 Girard brought out an edition of \index{Marolois}Marolois's
Geometry with considerable additions.\index{Trigonometrical functions|)}

\index{Algebra.!z6@Introduction of symbolic algebra by Vieta!q1@Girard}Girard's algebraical investigations are contained in his \emph{Invention
nouvelle en l'algèbre}, published at Amsterdam in 1629.\footnote
  {It was reissued by B.~de~Haan at Leyden in 1884.} %[**TN: 're-issued' changed to 'reissued' for consistency]
This
contains the earliest use of \index{Brackets, introduction of}brackets; a \index{Negative sign!geometrical interpretation}geometrical interpretation of the negative sign; the statement that the number of
roots of an algebraical question is equal to its degree; the
distinct recognition of imaginary roots; the theorem, known as
\index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's rule, for finding the sum of like powers of the roots
of an equation; and (in the opinion of some writers) implies
also a knowledge that the first member of an algebraical equation
$\phi(x) = 0$ could be resolved into linear factors. Girard's
investigations were unknown to most of his contemporaries,
and exercised no appreciable influence on the development of
mathematics.

The invention of \index{Logarithms|(}logarithms by Napier of Merchiston in
1614, and their introduction into England by Briggs and others,
have been already mentioned in chapter~\hyperref[chapter:XI]{\textsc{xi}}. A few words on
these mathematicians may be here added.\index{Girard|)}

\index{Napier of Merchiston|(}\nblabel{png255a}\nbparagraph{Napier.\footnote
  {See the \emph{Napier Tercentenary Memorial Volume}, Edinburgh, 1915. An
  edition of all his works was issued at Edinburgh in 1839: a bibliography
  of his writings is appended to a translation of the \emph{Constructio} by \index{Macdonald on Napier}W.~R.~Macdonald,
  Edinburgh, 1889.}}
\emph{John Napier} was born at Merchiston in 1550,
and died on April~4, 1617. He spent most of his time on the
family estate near Edinburgh, and took an active part in the
political and religious controversies of the day; the business of
his life was to show that the Pope was Antichrist, but his
favourite amusement was the study of mathematics and science.

As soon as the use of \index{Exponents}exponents became common in algebra
the introduction of logarithms would naturally follow, but
Napier reasoned out the result without the use of any symbolic
notation to assist him, and the invention of logarithms was the
result of the efforts of many years with a view to abbreviate
the processes of multiplication and division. It is likely that
Napier's attention may have been partly directed to the
desirability of facilitating computations by the stupendous
arithmetical efforts of some of his contemporaries, who seem
to have taken a keen pleasure in surpassing one another in
the extent to which they carried multiplications and divisions.
The trigonometrical tables by \index{Rheticus!ref.\ to}Rheticus, which were published
in 1596 and 1613, were calculated in a most laborious way:
\index{Vieta!ref.\ to}Vieta himself delighted in arithmetical calculations which must
have taken days of hard work, and of which the results often
served no useful purpose: \index{Van Ceulen}\index{Ceulen, van}L.~van Ceulen (1539--1610) practically
devoted his life to finding a numerical approximation to the
value of \index{P1@$\pi$,!z1@value of}$\pi$, finally in 1610 obtaining it correct to 35 places of
decimals: while, to cite one more instance, \index{Cataldi}P.~A.~Cataldi (1548--1626),
who is chiefly known for his invention in 1613 of the
form of \index{Continued fractions}\index{Fractions, continued}continued fractions, must have spent years in numerical
calculations.

In regard to Napier's other work I may again mention that
in his \index{Rabdologia@\emph{Rabdologia}, the}\emph{Rabdologia}, published in 1617, he introduced an \index{Rods, Napier's}improved
form of rod by the use of which the product of two
numbers can be found in a mechanical way, or the quotient of
one number by another. He also invented two other rods
called ``virgulae,'' by which square and cube roots can be
extracted. I should add that in spherical trigonometry he
discovered certain formulae known as Napier's analogies, and
enunciated the ``rule of circular parts'' for the solution of
right-angled spherical triangles.\index{Napier of Merchiston|)}

\index{Briggs|(}\nblabel{png256a}\nbparagraph{Briggs.} The name of Briggs is inseparably associated with
the history of logarithms. \emph{Henry Briggs}\footnote
  {See pp.~27--30 of \index{Ball, W. W. R., ref.\ to}my \emph{History of the Study of Mathematics at Cambridge},
  Cambridge, 1889.}
was born near
Halifax in 1561: he was educated at St.~John's College,
Cambridge, took his degree in 1581, and obtained a fellowship
in 1588: he was elected to the \index{Gresham, Sir Thos., ref.\ to}Gresham professorship of
geometry in 1596, and in 1619 or 1620 became Savilian
professor at Oxford, a chair which he held until his death on
January~26, 1631. It may be interesting to add that the
chair of geometry founded by Sir Thomas Gresham was the
earliest professorship of mathematics established in Great
Britain. Some twenty years earlier \index{Savile, Sir Hen.}Sir Henry Savile had
given at Oxford open lectures on Greek geometry and geometricians,
and in 1619 he endowed the chairs of geometry
and astronomy in that university which are still associated
with his name. Both in London and at Oxford Briggs was
the first occupant of the chair of geometry. He began his
lectures at Oxford with the ninth proposition of the first book
of Euclid---that being the furthest point to which Savile had
been able to carry his audiences. At Cambridge the Lucasian
chair was established in 1663, the earliest occupants being
\index{Barrow!ref.\ to}Barrow and \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton.

The almost immediate adoption throughout Europe of
logarithms for astronomical and other calculations was mainly
the work of Briggs, who undertook the tedious work of calculating
and preparing tables of logarithms. Amongst others he
convinced \index{Kepler!ref.\ to}Kepler of the advantages of Napier's discovery, and
the spread of the use of logarithms was rendered more rapid by
the zeal and reputation of Kepler, who by his tables of 1625
and 1629 brought them into vogue in Germany, while \index{Cavalieri!ref.\ to}Cavalieri
in 1624 and \index{Wingate, E.}Edmund Wingate in 1626 did a similar service for
Italian and French mathematicians respectively. Briggs also
was instrumental in bringing into common use the \index{Division,!processes of}method of
long division now generally employed.\index{Logarithms|)}\index{Briggs|)}

\index{Harriot|(}\nblabel{png257a}\nbparagraph{Harriot.} \emph{Thomas Harriot}, who was born at Oxford in
1560, and died in London on July~2, 1621, did a great deal to
extend and codify the theory of equations. The early part of
his life was spent in America with \index{Raleigh, Sir Walter, ref.\ to}Sir Walter Raleigh; while
there he made the first survey of Virginia and North Carolina,
the \index{Maps}maps of these being subsequently presented to \index{Elizabeth of England, ref.\ to}Queen
Elizabeth. On his return to England he settled in London,
and gave up most of his time to mathematical studies.

The majority of the propositions I have assigned to \index{Vieta!ref.\ to}Vieta
are to be found in \index{Algebra.!z6@Introduction of symbolic algebra by Vieta!q2@and Harriot}Harriot's writings, but it is uncertain
whether they were discovered by him independently of Vieta
or not. In any case it is probable that Vieta had not fully
realised all that was contained in the propositions he had
enunciated. Some of the consequences of these, with extensions
and a systematic exposition of the theory of equations,
were given by Harriot in his \emph{Artis Analyticae Praxis}, which
was first printed in 1631. The \emph{Praxis} is more analytical than
any algebra that preceded it, and marks an advance both in
symbolism and notation, though negative and imaginary roots
are rejected. It was widely read, and proved one of the most
powerful instruments in bringing analytical methods into general
use. Harriot was the first to use the signs $>$ and $<$ to represent
\index{Greater than, symbol for}greater than and \index{Less than, symbol for}less than. When he denoted the unknown
quantity by $a$ he represented $a^{2}$ by $aa$, $a^{3}$ by $aaa$, and so on.
This is a distinct improvement on Vieta's notation. The same
symbolism was used by \index{Wallis!ref.\ to}Wallis as late as 1685, but concurrently
with the modern \index{Indices}\index{Exponents}index notation which was introduced by
\index{Descartes!ref.\ to}Descartes. I need not allude to the other investigations of
Harriot, as they are comparatively of small importance;
extracts from some of them were published by \index{Rigaud, ref.\ to}S.~P.~Rigaud
in 1833.\index{Harriot|)}

\index{Oughtred|(}\nblabel{png258a}\nbparagraph{Oughtred.} Among those who contributed to the general
adoption in England of these various improvements and
additions to algorism and algebra was \emph{William Oughtred},\footnote
  {See \emph{William Oughtred}, by \index{Cajori, F.}F.~Cajori, Chicago, 1916. A complete
  edition of Oughtred's works was published at Oxford in 1677.}
who
was born at Eton on March~5, 1574, and died at his vicarage %[**TN: "1575" in original text]
of Albury in Surrey on June~30, 1660: it is sometimes said
that the cause of his death was the excitement and delight
which he experienced ``at hearing the House of Commons [or
Convention] had voted the King's return''; a recent critic adds
that it should be remembered ``by way of excuse that he
[Oughtred] was then eighty-six years old,'' but perhaps the
story is sufficiently discredited by the date of his death.
Oughtred was educated at Eton and King's College, Cambridge,
of the latter of which colleges he was a fellow and for some time
mathematical lecturer.

His \emph{Clavis Mathematicae} published in 1631 is a good systematic
text-book on arithmetic, and it contains practically all that
was then known on the subject. In this work he introduced the
symbol $×$ for multiplication. He also introduced the \index{Proportion, symbols for}symbol
$\nbproportion$ in proportion: previously to his time a proportion such as
$a : b = c : d$ was usually written as $a-b-c-d$; he denoted it
by $a \ctrdot b\ \nbproportion\ c \ctrdot d$. \index{Wallis!ref.\ to}Wallis says that some found fault with the
book on account of the style, but that they only displayed their
own incompetence, for Oughtred's ``words be always full but
not redundant.'' Pell makes a somewhat similar remark.

\index{Trigonometrical functions}Oughtred also wrote a treatise on trigonometry published in
1657, in which abbreviations for \index{Sine}\emph{sine}, \index{Cosine}\emph{cosine}, \&c., were employed.
This was really an important advance, but the works of \index{Girard!ref.\ to}Girard
and Oughtred, in which they were used, were neglected and soon
forgotten, and it was not until \index{Euler!ref.\ to}Euler reintroduced contractions
for the trigonometrical functions that they were generally adopted.
\index{Ratio, symbols for}In this work the colon (\emph{i.e.}\ the symbol $:$) was used to denote a ratio.

We may say roughly that henceforth elementary arithmetic,
algebra, and trigonometry were treated in a manner which is
not substantially different from that now in use; and that the
subsequent improvements introduced were additions to the
subjects as then known, and not a rearrangement of them
on new foundations.\index{Oughtred|)}

\section*{The origin of the more common symbols in algebra.\nblabel{png259a}}

\index{Symbols, algebraical|(}\index{Algebraa@Algebra,!symbols in|(}It may be convenient if I collect here in one place the
scattered remarks I have made on the introduction of the
various symbols for the more common operations in algebra.\footnote
  {See also two articles by \index{Henry C., ref.\ to}C.~Henry in the June and July numbers of the
  \emph{Revue Archéologique}, 1879, vol.~xxxvii, pp.~324--333, vol.~xxxviii, pp.~1--10.}

The later Greeks, the Hindoos, and \index{Jordanus!ref.\ to}Jordanus indicated
\emph{addition} by mere juxtaposition. It will be observed that this
is still the custom in arithmetic, where, for instance, $2\frac{1}{2}$ stands
for $2 + \frac{1}{2}$. The Italian algebraists, when they gave up expressing
every operation in words at full length and introduced syncopated
algebra, usually denoted \index{Addition, processes of!symbols for}\index{Plus!symbols for}\emph{plus} by its initial letter $P$ or $p$,
a line being sometimes drawn through the letter to show that it
was a contraction, or a symbol of operation, and not a quantity.
The practice, however, was not uniform; \index{Pacioli!ref.\ to}Pacioli, for example,
sometimes denoted plus by $\bar{p}$, and sometimes by $e$, and \index{Tartaglia!ref.\ to}Tartaglia
commonly denoted it by $\phi$. The German and English algebraists,
on the other hand, introduced the sign $+$ almost as soon as they
used algorism, but they spoke of it as \emph{signum additorum} and
employed it only to indicate excess; they also used it with a
special meaning in solutions by the method of false assumption.
\index{Widman!ref.\ to}Widman used it as an abbreviation for excess in 1489: by 1630
it was part of the recognised notation of algebra, and was
used as a symbol of operation.

\index{Minus!z1@symbols for}\index{Negative sign}\index{Subtraction,!symbols for}\emph{Subtraction} was indicated by Diophantus by an inverted and
truncated $\psi$. The Hindoos denoted it by a dot. The Italian
algebraists when they introduced syncopated algebra generally
denoted \emph{minus} by $M$ or $m$, a line being sometimes drawn through
the letter; but the practice was not uniform---Pacioli, for example,
denoting it sometimes by $\bar{m}$, and sometimes by \emph{de} for
\emph{demptus}. The German and English algebraists introduced the
present symbol which they described as \emph{signum subtractorum}.
It is most likely that the vertical bar in the symbol for plus
was superimposed on the symbol for minus to distinguish the
two. It may be noticed that Pacioli and Tartaglia found the
sign $-$ already used to denote a division, a ratio, or a proportion
indifferently. The present sign for minus was in general use by
about the year 1630, and was then employed as a symbol of
operation.

\index{Vieta!ref.\ to}Vieta, Schooten, and others among their contemporaries
employed the sign $=$ written between two quantities to denote
the difference between them; thus $a = b$ means with them what
we denote by $a \nbdiffc b$.
On the other hand, \index{Barrow!ref.\ to}Barrow wrote $\nbdiffm$ for
the same purpose. I am not aware when or by whom the current
symbol $\nbdiffc$ was first used with this signification.

\index{Oughtred!ref.\ to}Oughtred in 1631 used the \index{Multiplication, processes of}\index{Product, symbols for}sign $×$ to indicate \emph{multiplication};
\index{Harriot!ref.\ to}Harriot in 1631 denoted the operation by a dot; \index{Descartes!ref.\ to}Descartes in
1637 indicated it by juxtaposition. I am not aware of any
symbols for it which were in previous use. \index{Leibnitz!ref.\ to}Leibnitz in 1686
employed the sign $\frown$ to denote multiplication.

\nblabel{xref:241}\index{Division,!symbols for}\emph{Division} was ordinarily denoted by the Arab way of
writing the quantities in the form of a \index{Fractions, continued!symbols for}fraction by means of
a line drawn between them in any of the \index{Quotient!symbols for}forms $a - b$, $a/b$, or
$\dfrac{a}{b}$. \index{Ratio, symbols for}Oughtred in 1631 employed a dot to denote either division
or a ratio. Leibnitz in 1686 employed the sign $\smile$ to denote
division. The colon (or symbol $:$), used to denote a \emph{ratio},
occurs on the last two pages of Oughtred's \emph{Canones Sinuum},
published in 1657. I believe that the current symbol for
division $÷$ is only a combination of the $-$ and the symbol $:$ for
a ratio; it was used by \index{Rahn}Johann Heinrich Rahn at Zürich in %[**TN: Zurich in original]
1659, and by \index{Pell!ref.\ to}John Pell in London in 1668. \index{Proportion, symbols for}The symbol $\nbbarrowdiv$
was used by Barrow and other writers of his time to indicate
continued proportion.

\index{Equality, symbols for}\index{Equalityz@=, meanings of}The current symbol for \emph{equality} was introduced by \index{Record!ref.\ to}Record
in 1557; \index{Xylander!ref.\ to}Xylander in 1575 denoted it by two parallel vertical
lines; but in general till the year 1600 the word was written at
length; and from then until the time of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton, say about
1680, it was more frequently represented by $\nbinftyleft$ or by $\nbinftyright$ than
by any other symbol. Either of these latter signs was used as
a contraction for the first two letters of the word \emph{aequalis}.

The symbol $\nbproportion$ to denote \emph{proportion}, or the equality of two
ratios, was introduced by Oughtred in 1631, and was brought
into common use by \index{Wallis!ref.\ to}Wallis in 1686. There is no object in
having a symbol to indicate the equality of two ratios which is
different from that used to indicate the equality of other things,
and it is better to replace it by the sign $=$.

\index{Greater than, symbol for}\index{Less than, symbol for}The sign $>$ for \emph{is greater than} and the sign $<$ for \emph{is less than}
were introduced by \index{Harriot!ref.\ to}Harriot in 1631, but \index{Oughtred!ref.\ to}Oughtred simultaneously
invented the symbols $\nboughtredgreater$ and $\nboughtredless$ for the same purpose; and
these latter were frequently used till the beginning of the
eighteenth century, \emph{ex.~gr.}\ by \index{Barrow!ref.\ to}Barrow.

The symbols $\nbne$ for \emph{is not equal to},
$\nbngt$ for \emph{is not greater than}, and %[**TN: Added word 'for']
$\nbnlt$ for \emph{is not less than}, are, I believe, now rarely used outside
Great Britain; they were employed, if not invented, by \index{Euler!ref.\ to}Euler.
The symbols $\geqq$ and $\leqq$ were introduced by \index{Bouguer, P.}P.~Bouguer in 1734.

\index{Vinculum, introduction of}The vinculum was introduced by \index{Vieta!ref.\ to}Vieta in 1591; and \index{Brackets, introduction of}brackets
were first used by \index{Girard!ref.\ to}Girard in 1629.

\index{Roots, square, cube, \&c.}\index{Square root, symbols for}The symbol \index{Surds, symbols for}$\surd$ to denote the square root was introduced by
Rudolff in 1526; a similar notation had been used by Bhaskara
and by \index{Chuquet!ref.\ to}Chuquet.

The different methods of representing the power to which
a magnitude was raised have been already briefly alluded to.
The earliest known attempt to frame a symbolic notation was
made by \index{Bombelli!ref.\ to}Bombelli in 1572, when he represented the unknown
quantity by \bombelli{1}, its square by \bombelli{2}, its cube by \bombelli{3}, \&c. In
1586 \index{Stevinus!ref.\ to}Stevinus used \circledigit{1}, \circledigit{2}, \circledigit{3}, \&c., in a similar way; and
suggested, though he did not use, a corresponding notation
for fractional \index{Exponents}\index{Indices}indices. In 1591 Vieta improved on this by
denoting the different powers of \emph{A} by \emph{A, A quad., A cub.,} \&c.,
so that he could indicate the powers of different magnitudes;
Harriot in 1631 further improved on Vieta's notation by
writing $aa$ for $a^{2}$, $aaa$ for $a^{3}$, \&c., and this remained in use for
fifty years concurrently with the index notation. In 1634
\index{Herigonus}P.~Herigonus, in his \emph{Cursus mathematicus}, published in five
volumes at Paris in 1634--1637, wrote $a, a2, a3, \dots$ for $a, a^{2},
a^{3} \dots$.

The idea of using exponents to mark the power to which a
quantity was raised was due to \index{Descartes!ref.\ to}Descartes, and was introduced
by him in 1637; but he used only positive integral indices
$a^{1}, a^{2}, a^{3},$ \dots. \index{Wallis!ref.\ to}Wallis in 1659 explained the meaning of negative
and fractional indices in expressions such as $a^{-1}$, $ax^{1/2}$, \&c.; the
latter conception having been foreshadowed by \index{Oresmus!ref.\ to}Oresmus and
perhaps by Stevinus. Finally the idea of an index unrestricted
in magnitude, such as the $n$ in the expression $a^{n}$, is, I believe,
due to \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton, and was introduced by him in connection with
the binomial theorem in the letters for \index{Leibnitz!ref.\ to}Leibnitz written in
1676.

\index{Infinity, symbol for}The symbol $\infty$ for infinity was first employed by \index{Wallis!ref.\ to}Wallis in
1655 in his \emph{Arithmetica Infinitorum}; but does not occur again
until 1713, when it is used in \index{Bernoulli, James!ref.\ to}James Bernoulli's \emph{Ars Conjectandi}.
This sign was sometimes employed by the Romans
to denote the number 1000, and it has been conjectured that
this led to its being applied to represent any very large
number.\index{Algebraa@Algebra,!symbols in|)}\index{Symbols, algebraical|)}

There are but few special \index{Symbols, algebraical!trigonometrical}symbols in trigonometry; I may,
however, add here the following note which contains all that I
have been able to learn on the subject. \index{Angle,!sexagesimal division}The current \index{Sexagesimal angles}sexagesimal
division of angles is derived from the Babylonians through the
Greeks. The Babylonian unit angle was the angle of an
equilateral triangle; following their usual practice this was
divided into sixty equal parts or degrees, a degree was subdivided
into sixty equal parts or minutes, and so on; it is said
that 60 was assumed as the base of the system in order that the
number of degrees corresponding to the circumference of a circle
should be the same as the number of days in a year which it is
alleged was taken (at any rate in practice) to be 360.

The word \emph{sine} was used by \index{Regiomontanus!ref.\ to}Regiomontanus and was derived
from the Arabs; the terms \index{Secant}\emph{secant} and \index{Tangent (trigonometrical)}\emph{tangent} were introduced
by \index{Finck}Thomas Finck (born in Denmark in 1561 and died in 1646)
in his \emph{Geometriae Rotundi}, Bâle, 1583; the word \index{Cosecant}\emph{cosecant}
was (I believe) first used by \index{Rheticus!ref.\ to}Rheticus in his \emph{Opus Palatinum},
1596; the terms \emph{cosine} and \index{Cotangent}\emph{cotangent} were first employed by
\index{Gunter, E.}E.~Gunter in his \emph{Canon Triangulorum}, London, 1620. The
abbreviations \emph{sin}, \emph{tan}, \emph{sec} were used in 1626 by \index{Girard!ref.\ to}Girard, and
those of \emph{cos} and \emph{cot} by \index{Oughtred!ref.\ to}Oughtred in 1657; but these contractions
did not come into general use till \index{Euler!ref.\ to}Euler reintroduced them in
1748. The idea of trigonometrical \emph{functions} originated with
\index{Bernoulli, John!ref.\ to}John Bernoulli, and this view of the subject was elaborated in
1748 by Euler in his \emph{Introductio in Analysin Infinitorum}.

\nbChapter{XIII}{The Close of the Renaissance}{}{Circ.\ 1586--1637.}
  {See footnote to chapter~\hyperref[chapter:XII]{xii}.}
\index{Trigonometry.!z7@Treated by most of the mathematicians of the renaissance, chapters \textsc{xii} and \textsc{xiii}}\index{Renaissance, the mathematics of, chapters \textsc{xii, xiii}}\index{Geometry.!z5@Geometry of the renaissance; characterized by a free use of algebra and trigonometry, chapters \textsc{xii, xiii}}\nblabel{png264a}

\nblabel{png264b}\First{The} closing years of the renaissance were marked by a revival
of interest in nearly all branches of mathematics and science.
As far as pure mathematics is concerned we have already seen
that during the last half of the sixteenth century there had been
a great advance in algebra, theory of equations, and trigonometry;
and we shall shortly see (in the second section of this
chapter) that in the early part of the seventeenth century
some new processes in geometry were invented. If, however,
we turn to applied mathematics it is impossible not to be
struck by the fact that even as late as the middle or end of the
sixteenth century no marked progress in the theory had been
made from the time of \index{Archimedes!ref.\ to}Archimedes. Statics (of solids) and
hydrostatics remained in much the state in which he had left
them, while dynamics as a science did not exist. It was
\index{Stevinus!ref.\ to}Stevinus who gave the first impulse to the renewed study of
statics, and \index{Galileo!ref.\ to}Galileo who laid the foundation of dynamics; and
to their works the first section of this chapter is devoted.

\section*{The development of mechanics and experimental methods.\nblabel{png264c}}

\index{Stevinus|(}\nblabel{png264d}\nbparagraph{Stevinus.\footnote
  {An analysis of his works is given in the \emph{Histoire des sciences mathématiques
  et physiques chez les Belges}, by L.~A.~J.~Quetelet, Brussels, 1866,
  pp.~144--168; see also \emph{Notice historique sur la vie et les ouvrages de Stevinus},
  by \index{Gothals@Göthals on Stevinus}J.~V.~Göthals, Brussels, 1841; and \emph{Les travaux de Stevinus}, by
  \index{Steichen on Stevinus}M.~Steichen, Brussels, 1846. The works of Stevinus were collected by
  Snell, translated into Latin, and published at Leyden in 1608 under the title
  \emph{Hypomnemata Mathematica}.}}
\emph{Simon Stevinus} was born at Bruges in 1548,
and died at the Hague in 1620. We know very little of his life
save that he was originally a merchant's clerk at Antwerp, and at
a later period of his life was the friend of \index{Maurice of Orange, ref.\ to}Prince Maurice of
Orange, by whom he was made quartermaster-general of the
Dutch army.

\index{Mechanics.!z2@Development of,!x1@by Stevinus and Galileo|(}To his contemporaries he was best known for his works on
fortifications and military engineering, and the principles he
laid down are said to be in accordance with those which are now
usually accepted. To the general populace he was also well
known on account of his invention of a carriage which was
propelled by sails; this ran on the sea-shore, carried twenty-eight
people, and easily outstripped horses galloping by the
side; his model of it was destroyed in 1802 by the French
when they invaded Holland. It was chiefly owing to the
influence of \index{Stevinus!ref.\ to}Stevinus that the Dutch and French began a
proper system of \index{Book-keeping}\index{Double entry, book-keeping by}book-keeping in the national accounts.

I have already alluded to the introduction in his \emph{Arithmetic},
published in 1585, of \index{Exponents}\index{Indices}exponents to mark the power to which
quantities were raised; for instance, he wrote $3x^2 - 5x + 1$
as $3\;\circledigit{2} -5\;\circledigit{1} + 1\;\circledigit{0}$. His notation for \index{Decimal fractions}decimal fractions was
of a similar character. He further suggested the use of
fractional (but not negative) exponents. In the same book he
likewise suggested a \index{Decimal measures}decimal system of weights and measures.

He also published a geometry which is ingenious though it
does not contain many results which were not previously
known; in it some theorems on \index{Perspective}perspective are enunciated.

\nblabel{png265a}It is, however, on his \index{Hydrostatics. Developed by!z2@Stevinus|(}\emph{Statics and Hydrostatics}, published (in
Flemish) at Leyden in 1586, that his fame rests. In this
work he enunciates the \index{Forces!triangle of}\index{Triangle of forces}triangle of forces---a theorem which
some think was first propounded by \index{Vinci, Leonardo da!ref.\ to}\index{Leonardo da Vinci!ref.\ to}Leonardo da Vinci.
\index{Stevinus!ref.\ to}Stevinus regards this as the fundamental proposition of the
subject. Previous to the publication of his work the science of
statics had rested on the theory of the lever, but subsequently it
became usual to commence by proving the possibility of representing
forces by straight lines, and thus reducing many
theorems to geometrical propositions, and in particular to
obtaining in this way a proof of the \index{Forces!parallelogram of}\index{Parallelogram of forces}parallelogram (which is
equivalent to the \index{Triangle of forces}triangle) of forces. \index{Stevinus!ref.\ to}Stevinus is not clear in
his arrangement of the various propositions or in their logical
sequence, and the new treatment of the subject was not definitely
established before the appearance in 1687 of \index{Varignon!ref.\ to}Varignon's work
on mechanics. Stevinus also found the force which must be
exerted along the line of greatest slope to support a given
weight on an inclined plane---a problem the solution of which
had been long in dispute. He further distinguishes between
stable and unstable equilibrium. In hydrostatics he discusses
the question of the pressure which a fluid can exercise, and
explains the so-called hydrostatic paradox.\index{Hydrostatics. Developed by!z2@Stevinus|)}

His method\footnote
  {\emph{Hypomnemata Mathematica}, vol.~iv, \emph{de Statica}, prop.~19.}
of finding the \index{Force, component of, in a given direction}resolved part of a force in a
given direction, as illustrated by the case of a weight resting on
an inclined plane, is a good specimen of his work and is worth
quoting.

He takes a wedge $ABC$ whose base $AC$ %[**TN: 'AE' in original text]
is horizontal [and
whose sides $BA$, $BC$ are in the ratio of 2 to 1]. A thread
connecting a number of small equal equidistant weights is placed
over the wedge as indicated in the \figureref{267a}{figure} below (which %[**TN: Replaced 'on the next page' with 'below']
I reproduce from his demonstration) so that the number of these
weights on $BA$ is to the number on $BC$ in the same proportion
as $BA$ is to $BC$. This is always possible if the dimensions of the
wedge be properly chosen, and he places four weights resting on
$BA$ and two on $BC$; but we may replace these weights by a
heavy uniform chain $TSLVT$ without altering his argument.
He says in effect, that experience shews that such a chain will
remain at rest; if not, we could obtain perpetual motion. Thus
the effect in the direction $BA$ of the weight of the part $TS$ of
the chain must balance the effect in the direction $BC$ of the
weight of the part $TV$ of the chain. Of course $BC$ may be
vertical, and if so the above statement is equivalent to saying
that the effect in the direction $BA$ of the weight of the chain on
it is diminished in the proportion of $BC$ to $BA$; in other words,
if a weight $W$ rests on an inclined plane of inclination $\alpha$ the
component of $W$ down the line of greatest slope is $W \sin \alpha$.

\begin{center}
\labelincludegraphics[width=3in]{267a}
\end{center}

\index{Stevinus!ref.\ to}Stevinus was somewhat dogmatic in his statements, and
allowed no one to differ from his conclusions, ``and those,'' says
he, in one place, ``who cannot see this, may the Author of nature
have pity upon their unfortunate eyes, for the fault is not
in the thing, but in the sight which we are unable to give them.''\index{Stevinus|)}

\index{Galileo|(}\nblabel{png267a}\nbparagraph{Galileo.\footnote
  {See the biography of Galileo, by \index{Fahie, J. J.}J.~J.~Fahie, London, 1903. An
  edition of Galileo's works was issued in 16 volumes by \index{Albe@Albèri on Galileo}E.~Albèri, Florence,
  1842--1856. A good many of his letters on various mathematical subjects
  have been since discovered, and a new and complete edition is in process of
  issue by the Italian Government, Florence; vols.~i to xix and a bibliography,
  1890--1907.}}
Just as the modern treatment of statics originates
with Stevinus, so the foundation of the science of dynamics is
due to Galileo. \emph{Galileo Galilei} was born at Pisa on February~18,
1564, and died near Florence on January~8, 1642. His
father, a poor descendant of an old and noble Florentine house,
was himself a fair mathematician and a good musician. Galileo
was educated at the monastery of Vallombrosa, where his
literary ability and mechanical ingenuity attracted considerable
attention. He was persuaded to become a novitiate of the order
in 1579, but his father, who had other views, at once removed
him, and sent him in 1581 to the university of Pisa to study
medicine. \index{Pendulum, motion of}It was there that he noticed that the great bronze
lamp, hanging from the roof of the cathedral, performed its
oscillations in equal times, and independently of whether the
oscillations were large or small---a fact which he verified by
counting his pulse. He had been hitherto kept in ignorance of
mathematics, but one day, by chance hearing a lecture on
geometry (by \index{Ricci}Ricci), he was so fascinated by the science that
thenceforward he devoted his leisure to its study, and finally
got leave to discontinue his medical studies. He left the
university in 1585, and almost immediately commenced his
original researches.

\index{Hydrostatics. Developed by!z3@Galileo}He published in 1586 an account of the hydrostatic balance,
and in 1588 an essay on the centre of gravity in solids; these
were not printed till later. The fame of these works secured for
him in 1589 the appointment to the mathematical chair at Pisa---the
stipend, as was then the case with most professorships,
being very small. \nblabel{png268a}During the next three years he carried on,
from the leaning tower, that series of experiments on falling
bodies which established the first principles of dynamics.
Unfortunately, the manner in which he promulgated his discoveries,
and the ridicule he threw on those who opposed him,
gave not unnatural offence, and in 1591 he was obliged to
resign his position.

At this time he seems to have been much hampered by want
of money. Influence was, however, exerted on his behalf with
the Venetian senate, and he was appointed professor at Padua, a
chair which he held for eighteen years, 1592--1610. His
lectures there seem to have been chiefly on mechanics and
hydrostatics, and the substance of them is contained in his
treatise on mechanics, which was published in 1612. In these
lectures he repeated his Pisan experiments, and demonstrated
that falling bodies did not (as was then commonly believed)
descend with velocities proportional, amongst other things, to
their weights. He further shewed that, if it were assumed that
they descended with a uniformly accelerated motion, it was
possible to deduce the relations connecting velocity, space, and
time which did actually exist. At a later date, by observing the
times of descent of bodies sliding down inclined planes, he
shewed that this hypothesis was true. He also proved that the
path of a \index{Projectiles}projectile is a parabola, and in doing so implicitly
used the principles laid down in the first two \index{Motion, laws of}laws of motion as
enunciated by \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton. He gave an accurate definition of
momentum which some writers have thought may be taken to
imply a recognition of the truth of the third law of motion.
The laws of motion are, however, nowhere enunciated in a precise
and definite form, and \index{Galileo!ref.\ to}Galileo must be regarded rather as
preparing the way for \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton than as being himself the creator
of the science of dynamics.

In statics he laid down the principle that in machines what
was gained in power was lost in speed, and in the same ratio.
In the statics of solids he found the force which can support
a given weight on an inclined plane; in \index{Hydrostatics. Developed by!z3@Galileo}hydrostatics he propounded
the more elementary theorems on pressure and on
floating bodies; while among hydrostatical instruments he used,
and perhaps invented, the \index{Thermometer, invention of}thermometer, though in a somewhat
imperfect form.\index{Mechanics.!z2@Development of,!x1@by Stevinus and Galileo|)}

\nblabel{png269a}It is, however, as an \index{Astronomy.!z16@Modern theory of, created by Copernicus!q1@Galileo}astronomer that most people regard
\index{Galileo!ref.\ to}Galileo, and though, strictly speaking, his astronomical researches
lie outside the subject-matter of this book, it may be interesting
to give the leading facts. It was in the spring of 1609
that Galileo heard that a tube containing \index{Lenses, construction of}lenses had been made
by an optician, \index{Lippershey}Hans Lippershey, of Middleburg, which served
to magnify objects seen through it. This gave him the clue,
and he constructed a \index{Telescopes}telescope of that kind which still bears his
name, and of which an ordinary opera-glass is an example.
Within a few months he had produced instruments which were
capable of magnifying thirty-two diameters, and within a year
he had made and published observations on the solar spots, the
lunar mountains, Jupiter's satellites, the phases of Venus, and
Saturn's ring. The discovery of the \index{Microscope}microscope followed naturally
from that of the \index{Telescopes}telescope. Honours and emoluments were
showered on him, and he was enabled in 1610 to give up his
professorship and retire to Florence. In 1611 he paid a temporary
visit to Rome, and exhibited in the gardens of the
Quirinal the new worlds revealed by the telescope.

\index{Astronomy.!z16@Modern theory of, created by Copernicus!q1@Galileo}It would seem that \index{Galileo!ref.\ to}Galileo had always believed in the
\index{Copernicus!ref.\ to}Copernican system, but was afraid of promulgating it on
account of the ridicule it excited. The existence of Jupiter's
satellites seemed, however, to make its truth almost certain, and
he now boldly preached it. The orthodox party resented his
action, and in February 1616 the Inquisition declared that to
suppose the sun the centre of the solar system was false, and
opposed to Holy Scripture. The edict of March~5, 1616, which
carried this into effect, has never been repealed, though it has
been long tacitly ignored. It is well known that towards the
middle of the seventeenth century the Jesuits evaded it by
treating the theory as an hypothesis from which, though false,
certain results would follow.

\index{Planetary motions}In January 1632 \index{Galileo!ref.\ to}Galileo published his dialogues on the
system of the world, in which in clear and forcible language he
expounded the Copernican theory. In these, apparently through
jealousy of \index{Kepler!ref.\ to}Kepler's fame, he does not so much as mention
\index{Kepler's laws}Kepler's laws (the first two of which had been published in
1609, and the third in 1619); he rejects Kepler's hypothesis
that the \index{Tides, theory of}tides are caused by the attraction of the moon, and
tries to explain their existence (which he alleges is a confirmation
of the Copernican hypothesis) by the statement that
different parts of the earth rotate with different velocities. He
was more successful in showing that mechanical principles
would account for the fact that a stone thrown straight up
falls again to the place from which it was thrown---a fact
which previously had been one of the chief difficulties in the
way of any theory which supposed the earth to be in
motion.

The publication of this book was approved by the papal
censor, but substantially was contrary to the edict of 1616.
\index{Galileo!ref.\ to}Galileo was summoned to Rome, forced to recant, do penance,
and was released only on promise of obedience. The documents
recently printed show that he was threatened with the torture,
but probably there was no intention of carrying the threat into
effect.

When released he again took up his work on mechanics, and
by 1636 had finished a book which was published under the
title \emph{Discorsi intorno a due nuove scienze} at Leyden in 1638.
In 1637 he lost his sight, but with the aid of pupils he continued
his experiments on mechanics and hydrostatics, and in
particular on the possibility of using a \index{Pendulum, motion of}pendulum to regulate a
\index{Clocks}clock, and on the theory of impact.

An anecdote of this time has been preserved which, though
probably not authentic, is sufficiently interesting to bear repetition.
According to one version of the story, Galileo was
interviewed by some members of a Florentine guild who wanted
their pumps altered so as to raise water to a height which was
greater than thirty feet; and thereupon he remarked that it
might be desirable to first find out why the water rose at all.
A bystander intervened and said there was no difficulty about
that, because nature abhorred a vacuum. Yes, said Galileo, but
apparently it is only a vacuum which is less than thirty feet.
His favourite pupil \index{Torricelli!ref.\ to}Torricelli was present, and thus had his
attention directed to the question, which he subsequently
elucidated.

\index{Experiments, necessity of}Galileo's work may, I think, be fairly summed up by saying
that his researches on mechanics are deserving of high praise,
and that they are memorable for clearly enunciating the fact
that science must be founded on laws obtained by experiment;
his astronomical observations and his deductions therefrom were
also excellent, and were expounded with a literary skill which
leaves nothing to be desired; but though he produced some of
the evidence which placed the Copernican theory on a satisfactory
basis, he did not himself make any special advance in
the theory of astronomy.\index{Galileo|)}

\index{Bacon, Francis}\nblabel{png272a}\nbparagraph{Francis Bacon.\footnote
  {See his life by J.~Spedding, London, 1872--74. The best edition of his
  works is that by \index{Ellis, R. L., on Fr. Bacon}Ellis, \index{Spedding on Francis Bacon}Spedding,
  and \index{Heath, D. D., on Bacon}Heath, in 7 volumes, London, second
  edition, 1870.}}
\index{Experiments, necessity of}The necessity of an experimental foundation
for science was also advocated with considerable effect by
\index{Galileo!ref.\ to}Galileo's contemporary \index{Verulam, Lord}\index{Bacon, Francis!ref.\ to}\emph{Francis Bacon} (Lord Verulam), who was
born at London on Jan.~22, 1561, and died on April~9, 1626.
He was educated at Trinity College, Cambridge. His career in
politics and at the bar culminated in his becoming Lord Chancellor,
with the title of Lord Verulam. The story of his subsequent
degradation for accepting bribes is well known.

His chief work is the \emph{Novum Organum}, published in 1620,
in which he lays down the principles which should guide those
who are making experiments on which they propose to found
a theory of any branch of physics or applied mathematics. He
gave rules by which the results of induction could be tested,
hasty generalisations avoided, and experiments used to check
one another. The influence of this treatise in the eighteenth
century was great, but it is probable that during the preceding
century it was little read, and the remark repeated by several
French writers that Bacon and \index{Descartes!ref.\ to}Descartes are the creators of
modern philosophy rests on a misapprehension of Bacon's
influence on his contemporaries; any detailed account of this
book belongs, however, to the history of scientific ideas rather
than to that of mathematics.

Before leaving the subject of applied mathematics I may
add a few words on the writings of \index{Guldinus|(}Guldinus, Wright, and
\index{Snell!ref.\ to}Snell.

\nbparagraph{Guldinus.} \emph{Habakkuk Guldinus}, born at St.~Gall on June~12,
1577, and died at Grätz on Nov.~3, 1643, was of Jewish
descent, but was brought up as a Protestant; he was converted
to Roman Catholicism, and became a Jesuit, when he took the
Christian name of Paul, and it was to him that the Jesuit
colleges at Rome and Grätz owed their mathematical reputation.
\nblabel{xref:252}The two theorems known by the name of \index{Pappus!ref.\ to|(}Pappus (to which I
have alluded above) were published by Guldinus in the fourth
book of his \emph{De Centro Gravitatis}, Vienna, 1635--1642. Not
only were the rules in question taken without acknowledgment
from Pappus, but (according to \index{Montucla!ref.\ to}Montucla) the proof of them
given by Guldinus was faulty, though he was successful in
applying them to the determination of the volumes and surfaces
of certain solids. The theorems were, however, previously
unknown, and their enunciation excited considerable interest.\index{Pappus!ref.\ to|)}\index{Guldinus|)}

\index{Wright|(}\nblabel{png273a}\nbparagraph{Wright.\footnote
  {See pp.~25--27 of \index{Ball, W. W. R., ref.\ to}my \emph{History of the Study of Mathematics at Cambridge},
  Cambridge, 1889.}}
\index{Navigation, science of}I may here also refer to \emph{Edward Wright}, who is
worthy of mention for having put the art of navigation on a
scientific basis. Wright was born in Norfolk about 1560, and
died in 1615. He was educated at Caius College, Cambridge,
of which society he was subsequently a fellow. He seems to
have been a good sailor, and he had a special talent for the construction
of instruments. About 1600 he was elected lecturer
on mathematics by the East India Company; he then settled in
London, and shortly afterwards was appointed mathematical
tutor to \index{Henry b@Henry of Wales, ref.\ to}Henry, Prince of Wales, the son of \index{James I. of England, ref.\ to}James I\@. His
mechanical ability may be illustrated by an \index{Orrery}orrery of his
construction by which it was possible to predict eclipses; it was
shewn in the Tower as a curiosity as late as 1675.

\index{Maps|(}In the maps in use before the time of \index{Mercator, G.}\index{Mercators@Mercator's projection}Gerard Mercator a
degree, whether of latitude or longitude, had been represented
in all cases by the same length, and the course to be pursued
by a vessel was marked on the map by a straight line joining
the ports of arrival and departure. Mercator had seen that
this led to considerable errors, and had realised that to make
this method of tracing the course of a ship at all accurate the
space assigned on the map to a degree of latitude ought gradually
to increase as the latitude increased. Using this principle,
he had empirically constructed some charts, which were published
about 1560 or 1570. Wright set himself the problem to determine
the theory on which such maps should be drawn, and
succeeded in discovering the law of the scale of the maps,
though his rule is strictly correct for small arcs only. The
result was published in the second edition of Blundeville's
\emph{Exercises}.\index{Maps|)}

\index{Navigation, science of}In 1599 Wright published his \emph{Certain Errors in Navigation
Detected and Corrected}, in which he explained the theory and
inserted a table of meridional parts. The reasoning shews considerable
geometrical power. \index{Stars, lists of}In the course of the work he gives
the declinations of thirty-two stars, explains the phenomena of
the dip, parallax, and \index{Refraction}refraction, and adds a table of magnetic
declinations; he assumes the earth to be stationary. In the
following year he published some maps constructed on his
principle. In these the northernmost point of \index{Australia, map of}Australia is
shewn; the latitude of London is taken to be $51° 32'$.\index{Wright|)}

\index{Snell}\nblabel{png274a}\nbparagraph{Snell.} A contemporary of Guldinus and Wright was
\emph{Willebrod Snell}, whose name is still well known through his
discovery in 1619 of the law of refraction in \index{Optics (geometrical).!Discussed by (among others) Euclid!z4@Snell}optics. Snell was
born at Leyden in 1581, occupied a chair of mathematics at the
university there, and died there on Oct.~30, 1626. He was one
of those infant prodigies who occasionally appear, and at the
age of twelve he is said to have been acquainted with the
standard mathematical works. I will here only add that in
\index{Geodesy}geodesy he laid down the principles for determining the length
of the arc of a meridian from the measurement of any base line,
and in spherical trigonometry he discovered the properties of the
\index{Polar triangle}\index{Supplemental triangle}polar or supplemental triangle.

\section*{Revival of interest in pure geometry.\nblabel{png274b}}

The close of the sixteenth century was marked not only by
the attempt to found a theory of dynamics based on laws derived
from experiment, but also by a revived interest in geometry.
This was largely due to the influence of \index{Kepler!ref.\ to}Kepler.

\index{Kepler|(}\nblabel{png274c}\nbparagraph{Kepler.\footnote
  {See \emph{Johann Kepplers Leben und Wirken}, by \index{Von Breitschwert on Kepler}\index{Breitschwert on Kepler}J.~L.~E.~von Breitschwert,
  Stuttgart, 1831; and \index{Wolf}R.~Wolf's \emph{Geschichte der Astronomie}, Munich, 1877.
  A complete edition of \index{Kepler!ref.\ to}Kepler's works was published by \index{Frisch on Kepler}C.~Frisch at Frankfort,
  in 8 volumes, 1858--71; and an analysis of the mathematical part of his chief
  work, the \emph{Harmonice Mundi}, is given by \index{Chasles, M., ref.\ to}Chasles in his \emph{Aperçu historique}.
  See also \index{Cantor, M., ref.\ to}Cantor, vol.~ii, part~xv.}}
\index{Kepler!ref.\ to}\emph{Johann Kepler}, one of the founders of modern
astronomy, was born of humble parents near Stuttgart on
Dec.~27, 1571, and died at Ratisbon on Nov.~15, 1630. He
was educated under \index{Mastlin@Mästlin}Mästlin at Tübingen. In 1593 he was
appointed professor at Grätz, where he made the acquaintance
of a wealthy widow, whom he married, but found too late that
he had purchased his freedom from pecuniary troubles at the
expense of domestic happiness. In 1599 he accepted an appointment
as assistant to \index{Tycho Brahe}Tycho Brahe, and in 1601 succeeded
his master as astronomer to the emperor \index{Rudolph II. of Germany, ref.\ to}Rudolph II\@. But his
career was dogged by bad luck: first his stipend was not paid;
next his wife went mad and then died, and a second marriage in
1611 did not prove fortunate; while, to complete his discomfort,
he was expelled from his chair, and narrowly escaped condemnation
for heterodoxy. During this time he depended for his
income on telling fortunes and casting horoscopes, for, as he
says, ``nature which has conferred upon every animal the means
of existence has designed \index{Astrology}astrology as an adjunct and ally to
astronomy.'' He seems, however, to have had no scruple in
charging heavily for his services, and to the surprise of his contemporaries
was found at his death to possess a considerable
hoard of money. He died while on a journey to try and
recover for the benefit of his children some of the arrears of his
stipend.

In describing \index{Galileo!ref.\ to}Galileo's work I alluded briefly to the three
laws in astronomy that \index{Kepler!ref.\ to}Kepler had discovered, and in connection
with which his name will be always associated. I may further
add that he suggested that the planets might be retained in
their orbits by magnetic vortices, but this was little more than
a crude conjecture. I have also already mentioned the prominent
part he took in bringing logarithms into general use on the continent.
These are familiar facts; but it is not known so generally
that Kepler was also a geometrician and algebraist of considerable
power, and that he, \index{Desargues!ref.\ to}Desargues, and perhaps Galileo, may
be considered as forming a connecting link between the mathematicians
of the renaissance and those of modern times.

Kepler's work in geometry consists rather in certain general
principles enunciated, and illustrated by a few cases, than in any
systematic exposition of the subject. \index{Conic Sections (Geometrical).!Discussed by most of the Greek geometricians after Menaechmus!x3@interest in, revived by writings of Kepler}\nblabel{png276a}In a short chapter on
conics inserted in his \emph{Paralipomena}, published in 1604, he lays
down what has been called the \index{Continuity, principle of}principle of continuity, and
gives as an example the statement that a parabola is at once the
limiting case of an ellipse and of a hyperbola; he illustrates the
same doctrine by reference to the foci of conics (the word \index{Focus of a conic}\emph{focus}
was introduced by him); and he also explains that \index{Parallel lines}parallel lines
should be regarded as meeting at infinity. He introduced the
use of the \index{Eccentric angle}eccentric angle in discussing properties of the ellipse.

\index{Quadrature of curves}\nblabel{png276b}In his \emph{Stereometria}, which was published in 1615, he determines
the volumes of certain vessels and the areas of certain
surfaces, by means of \index{Infinitesimals, use of}infinitesimals instead of by the long and
tedious method of exhaustions. These investigations as well
as those of 1604 arose from a dispute with a wine merchant as
to the proper way of gauging the contents of a cask. This
use of infinitesimals was objected to by \index{Guldinus!ref.\ to}Guldinus and other
writers as inaccurate, but though the methods of \index{Kepler!ref.\ to}Kepler are
not altogether free from objection he was substantially correct,
and by applying the law of continuity to infinitesimals he prepared
the way for \index{Cavalieri!ref.\ to}Cavalieri's \index{Indivisibles, method of}method of indivisibles, and the
\index{Calculus, infinitesimal}infinitesimal calculus of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton and \index{Leibnitz!ref.\ to}Leibnitz.

\index{Astronomy.!z16@Modern theory of, created by Copernicus!q2@and Kepler|(}Kepler's work on astronomy lies outside the scope of this
book. I will mention only that it was founded on the observations
of \index{Tycho Brahe}Tycho Brahe,\footnote
  {For an account of Tycho Brahe, born at Knudstrup in 1546 and died at
  Prague in 1601, see his life by \index{Dreyer on Tycho Brahe}J.~L.~E.~Dreyer, Edinburgh, 1890.}
whose assistant he was. \nblabel{png276c}His three laws
of \index{Planetary motions|(}planetary motion were the result of many and laborious
efforts to reduce the phenomena of the solar system to certain
simple rules. The first two were published in 1609, and stated
that the \index{Elliptic orbits of planets}planets describe ellipses round the sun, the sun being
in a focus; and that the line joining the sun to any planet
\index{Areas, conservation of}sweeps over equal areas in equal times. The third was published
in 1619, and stated that the squares of the periodic times
of the planets are proportional to the cubes of the major axes of
their orbits. The laws were deduced from observations on the
motions of Mars and the earth, and were extended by analogy
to the other planets. I ought to add that he attempted to
explain why these motions took place by a hypothesis which is
not very different from \index{Descartes!ref.\ to}Descartes's theory of vortices. He suggested
that the tides were caused by the attraction of the moon.
\index{Kepler!ref.\ to}Kepler also devoted considerable time to the elucidation of the
theories of vision and refraction in optics.\index{Planetary motions|)}\index{Astronomy.!z16@Modern theory of, created by Copernicus!q2@and Kepler|)}

While the conceptions of the geometry of the Greeks were
being extended by \index{Kepler!ref.\ to}Kepler, a Frenchman, whose works until
recently were almost unknown, was inventing a new method of
investigating the subject---a method which is now known as
projective geometry. This was the discovery of \index{Desargues!ref.\ to}Desargues,
whom I put (with some hesitation) at the close of this period,
and not among the mathematicians of modern times.\index{Kepler|)}

\index{Geometry.!z8@Modern synthetic geometry, originated with Desargues|(}\index{Desargues|(}\nblabel{png277a}\nbparagraph{Desargues.\footnote
  {See \emph{Oeuvres de Desargues}, by \index{Poudra on Desargues}M.~Poudra, 2 vols., Paris, 1864; and a note
  in the \emph{Bibliotheca Mathematica}, 1885, p.~90.}}
\emph{Gérard Desargues}, born at Lyons in 1593, and
died in 1662, was by profession an engineer and architect, but
he gave some courses of gratuitous lectures in Paris from 1626
to about 1630 which made a great impression upon his contemporaries.
Both Descartes and \index{Pascal!ref.\ to}Pascal had a high opinion of his
work and abilities, and both made considerable use of the
theorems he had enunciated.

\index{Conic Sections (Geometrical).!Discussed by most of the Greek geometricians after Menaechmus!x4@and Desargues}\nblabel{png277b}In 1636 Desargues issued a work on \index{Perspective}perspective; but most
of his researches were embodied in his \emph{Brouillon proiect} on
conics, published in 1639, a copy of which was discovered
by \index{Chasles, M., ref.\ to}Chasles in 1845. I take the following summary of it from
\index{Taylor, C., on conics}C.~Taylor's work on conics. Desargues commences with a
statement of the doctrine of continuity as laid down by
Kepler: thus the points at the opposite ends of a straight
line are regarded as coincident, parallel lines are treated as
meeting at a point at infinity, and parallel planes on a line at
infinity, while a straight line may be considered as a circle whose
centre is at infinity. The theory of involution of six points,
with its special cases, is laid down, and the projective property
of pencils in involution is established. The theory of polar lines
is expounded, and its analogue in space suggested. A tangent
is defined as the limiting case of a secant, and an asymptote as
a tangent at infinity. Desargues shows that the lines which join
four points in a plane determine three pairs of lines in involution
on any transversal, and from any conic through the four
points another pair of lines can be obtained which are in
involution with any two of the former. He proves that the
points of intersection of the diagonals and the two pairs of
opposite sides of any quadrilateral inscribed in a conic are a
conjugate triad with respect to the conic, and when one of the
three points is at infinity its polar is a diameter; but he fails to
explain the case in which the quadrilateral is a parallelogram,
although he had formed the conception of a straight line which
was wholly at infinity. The book, therefore, may be fairly said
to contain the fundamental theorems on involution, \index{Homology}homology,
poles and polars, and \index{Perspective}perspective.

The influence exerted by the lectures of Desargues on
\index{Descartes!ref.\ to}Descartes, \index{Pascal!ref.\ to}Pascal, and the French geometricians of the
seventeenth century was considerable; but the subject of
projective geometry soon fell into oblivion, chiefly because the
analytical geometry of \index{Descartes!ref.\ to}Descartes was so much more powerful as
a method of proof or discovery.

The researches of \index{Kepler!ref.\ to}Kepler and Desargues will serve to remind
us that as the geometry of the Greeks was not capable of
much further extension, mathematicians were now beginning
to seek for new methods of investigation, and were extending
the conceptions of geometry. The invention of analytical
geometry and of the infinitesimal calculus temporarily diverted
attention from pure geometry, but at the beginning of the
last century there was a revival of interest in it, and since
then it has been a favourite subject of study with many
mathematicians.\index{Desargues|)}\index{Geometry.!z8@Modern synthetic geometry, originated with Desargues|)}


\section*{Mathematical knowledge at the close of the renaissance.\nblabel{png278a}}

Thus by the beginning of the seventeenth century we may
say that the fundamental principles of arithmetic, algebra,
theory of equations, and trigonometry had been laid down, and
the outlines of the subjects as we know them had been traced.
It must be, however, remembered that there were no good
elementary text-books on these subjects; and a knowledge of
them was therefore confined to those who could extract it from
the ponderous treatises in which it lay buried. Though much of
the modern algebraical and trigonometrical notation had been
introduced, it was not familiar to mathematicians, nor was it
even universally accepted; and it was not until the end of the
seventeenth century that the language of these subjects was
definitely fixed. Considering the absence of good text-books,
I am inclined rather to admire the rapidity with which it came
into universal use, than to cavil at the hesitation to trust to it
alone which many writers showed.

If we turn to applied mathematics, we find, on the other
hand, that the science of statics had made but little advance
in the eighteen centuries that had elapsed since the time of
\index{Archimedes!ref.\ to}Archimedes, while the foundations of dynamics were laid by
\index{Galileo!ref.\ to}Galileo only at the close of the sixteenth century. In fact, as
we shall see later, it was not until the time of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton that the
science of mechanics was placed on a satisfactory basis. The
fundamental conceptions of mechanics are difficult, but the
ignorance of the principles of the subject shown by the
mathematicians of this time is greater than would have been
anticipated from their knowledge of pure mathematics.

With this exception, we may say that the principles of
analytical geometry and of the infinitesimal calculus were needed
before there was likely to be much further progress. The
former was employed by \index{Descartes!ref.\ to}Descartes in 1637, the latter was
invented by \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton some thirty or forty years later, and
their introduction may be taken as marking the commencement
of the period of modern mathematics.

\nbpart{THIRD PERIOD}{Modern Mathematics}{modern}
\nblabel{png280a}

\emph{The history of modern mathematics begins with the invention
of analytical geometry and the infinitesimal calculus. The
mathematics is far more complex than that produced in either of
the preceding periods; but, during the seventeenth and eighteenth
centuries, it may be generally described as characterized by the
development of analysis, and its application to the phenomena
of nature.}
\medskip

I continue the chronological arrangement of the subject.
Chapter~\hyperref[chapter:XV]{\textsc{xv}} contains the history of the forty years from 1635
to 1675, and an account of the mathematical discoveries of
\index{Descartes!ref.\ to}Descartes, \index{Cavalieri!ref.\ to}Cavalieri, \index{Pascal!ref.\ to}Pascal, \index{Wallis!ref.\ to}Wallis, Fermat, and Huygens.
Chapter~\hyperref[chapter:XVI]{\textsc{xvi}} is given up to a discussion of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's researches.
Chapter~\hyperref[chapter:XVII]{\textsc{xvii}} contains an account of the works of \index{Leibnitz!ref.\ to}Leibnitz and
his followers during the first half of the eighteenth century
(including D'Alembert), and of the contemporary English school
to the death of Maclaurin. The works of \index{Euler!ref.\ to}Euler, Lagrange,
Laplace, and their contemporaries form the subject-matter of
chapter~\hyperref[chapter:XVIII]{\textsc{xviii}}.

Lastly, in chapter~\hyperref[chapter:XIX]{\textsc{xix}} I have added some notes on a few of
the mathematicians of recent times; but I exclude all detailed
reference to living writers, and partly because of this, partly
for other reasons there given, the account of contemporary
mathematics does not profess to cover the subject.

\nbChapter{XIV}{The History of Modern Mathematics}{History of Modern Mathematics}{}{}
\index{Geometry.!z6@Analytical geometry!x3@discussed or used by nearly all the mathematicians considered in the third period, chapters \textsc{xiv-xix}|(}\nblabel{png281a}

\nblabel{png281b}\First{The} division between this period and that treated in the
last six chapters is by no means so well defined as that which
separates the history of Greek mathematics from the mathematics
of the middle ages. The methods of analysis used in
the seventeenth century and the kind of problems attacked
changed but gradually; and the mathematicians at the beginning
of this period were in immediate relations with those at
the end of that last considered. For this reason some writers
have divided the history of mathematics into two parts only,
treating the schoolmen as the lineal successors of the Greek
mathematicians, and dating the creation of modern mathematics
from the introduction of the Arab text-books into
Europe. The division I have given is, I think, more convenient,
for the introduction of analytical geometry and of the
infinitesimal calculus revolutionized the development of the subject,
and therefore it seems preferable to take their invention as
marking the commencement of modern mathematics.

The time that has elapsed since these methods were invented
has been a period of incessant intellectual activity in
all departments of knowledge, and the progress made in mathematics
has been immense. The greatly extended range of
knowledge, the mass of materials to be mastered, the absence
of perspective, and even the echoes of old controversies, combine
to increase the difficulties of an author. As, however, the
leading facts are generally known, and the works published
during this time are accessible to any student, I may deal more
concisely with the lives and writings of modern mathematicians
than with those of their predecessors, and confine myself more
strictly than before to those who have materially affected the
progress of the subject.

To give a sense of unity to a history of mathematics it is
necessary to treat it chronologically, but it is possible to do
this in two ways. We may discuss separately the development
of different branches of mathematics during a certain period
(not too long), and deal with the works of each mathematician
under such heads as they may fall. Or we may describe in
succession the lives and writings of the mathematicians of a
certain period, and deal with the development of different subjects
under the heads of those who studied them. Personally,
I prefer the latter course; and not the least advantage of this,
from my point of view, is that it adds a human interest to the
narrative. No doubt as the subject becomes more complex
this course becomes more difficult, and it may be that when the
history of mathematics in the nineteenth century is written it
will be necessary to deal separately with the separate branches
of the subject, but, as far as I can, I continue to present the
history biographically.

Roughly speaking, we may say that five distinct stages in
the history of modern mathematics can be discerned.

\nblabel{png282a}\emph{First} of all, there is the invention of analytical geometry by
\index{Descartes!ref.\ to}Descartes in 1637; and almost at the same time the introduction
of the method of indivisibles, by the use of which
areas, volumes, and the positions of centres of mass can be
determined by summation in a manner analogous to that effected
nowadays by the aid of the integral calculus. The method of
indivisibles was soon superseded by the integral calculus. Analytical
geometry, however, maintains its position as part of the
necessary training of every mathematician, and for all purposes
of research is incomparably more potent than the geometry of
the ancients. The latter is still, no doubt, an admirable intellectual
training, and it frequently affords an elegant demonstration
of some proposition the truth of which is already known,
but it requires a special procedure for every particular problem
attacked. The former, on the other hand, lays down a few simple
rules by which any property can be at once proved or disproved.

\nblabel{png283a}In the \emph{second} place, we have the invention, some thirty
years later, of the \index{Fluxional calculus}fluxional or differential calculus. Wherever
a quantity changes according to some continuous law---and most
things in nature do so change---the differential calculus enables
us to measure its rate of increase or decrease; and, from its rate
of increase or decrease, the integral calculus enables us to find
the original quantity. Formerly every separate function of $x$
such as $(1+x)^{n}$, $\log(1+x)$, $\sin x$, $\tan^{-1} x$, \&c., could be expanded
in ascending powers of $x$ only by means of such special
procedure as was suitable for that particular problem; but, by
the aid of the calculus, the expansion of any function of $x$ in
ascending powers of $x$ is in general reducible to one rule which
covers all cases alike. So, again, the theory of maxima and
minima, the determination of the lengths of curves and the
areas enclosed by them, the determination of surfaces, of volumes,
and of centres of mass, and many other problems, are each reducible
to a single rule. The theories of differential equations,
of the calculus of variations, of finite differences, \&c., are the
developments of the ideas of the calculus.

These two subjects---analytical geometry and the calculus---became
the chief instruments of further progress in mathematics.
In both of them a sort of machine was constructed: to solve a
problem, it was only necessary to put in the particular function
dealt with, or the equation of the particular curve or surface
considered, and on performing certain simple operations the
result came out. The validity of the process was proved once
for all, and it was no longer requisite to invent some special
method for every separate function, curve, or surface.

\nblabel{png283b}In the \emph{third} place, \index{Huygens!ref.\ to}Huygens, following \index{Galileo!ref.\ to}Galileo, laid the
foundation of a satisfactory treatment of dynamics, and \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton
reduced it to an exact science. \index{Historical methods}The latter mathematician proceeded
to apply the new \index{Analytical geometry,!x1@origin of}\index{Geometry.!z6@Analytical geometry}analytical methods not only to numerous
problems in the mechanics of solids and fluids on the earth,
but to the solar system; the whole of mechanics terrestrial and
celestial was thus brought within the domain of mathematics.
There is no doubt that \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton used the calculus to obtain many
of his results, but he seems to have thought that, if his demonstrations
were established by the aid of a new science which was
at that time generally unknown, his critics (who would not
understand the fluxional calculus) would fail to realise the truth
and importance of his discoveries. He therefore determined to
give geometrical proofs of all his results. He accordingly cast
the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia} into a geometrical form, and thus presented it to
the world in a language which all men could then understand.
The theory of mechanics was extended, systematized, and put
in its modern form by \index{Lagrange!ref.\ to}Lagrange and \index{Laplace!ref.\ to}Laplace towards the end
of the eighteenth century.

\nblabel{png284a}In the \emph{fourth} place, we may say that during this period
the chief branches of \index{Physics, mathematical|(}physics have been brought within the
scope of mathematics. This extension of the domain of mathematics
was commenced by \index{Huygens!ref.\ to}Huygens and \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton when they
propounded their theories of light; but it was not until the
beginning of the last century that sufficiently accurate observations
were made in most physical subjects to enable mathematical
reasoning to be applied to them.

Numerous and far-reaching conclusions have been obtained
in physics by the application of mathematics to the results of
observations and experiments, but we now want some more
simple hypotheses from which we can deduce those laws which
at present form our starting-point. If, to take one example,
we could say in what electricity consisted, we might get some
simple laws or hypotheses from which by the aid of mathematics
all the observed phenomena could be deduced, in the
same way as \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton deduced all the results of physical astronomy
from the law of gravitation. All lines of research seem,
moreover, to indicate that there is an intimate connection between
the different branches of physics, \emph{e.g.} between light, heat,
elasticity, electricity, and magnetism. The ultimate explanation
of this and of the leading facts in physics seems to demand a
study of molecular physics; a knowledge of molecular physics
in its turn seems to require some theory as to the \index{Matter, constitution of}constitution
of matter; it would further appear that the key to the constitution
of matter is to be found in electricity or chemical physics.
So the matter stands at present; the connection between the
different branches of physics, and the fundamental laws of those
branches (if there be any simple ones), are riddles which are yet
unsolved. This history does not pretend to treat of problems
which are now the subject of investigation; the fact also that
mathematical physics is mainly the creation of the nineteenth
century would exclude all detailed discussion of the subject.\index{Physics, mathematical|)}

\nblabel{png285a}\emph{Fifthly}, this period has seen an immense extension of pure
mathematics. Much of this is the creation of comparatively
recent times, and I regard the details of it as outside the limits
of this book, though in chapter~\hyperref[chapter:XIX]{\textsc{xix}} I have allowed myself to
mention some of the subjects discussed. The most striking
features of this extension are the critical discussion of
fundamental principles, the developments of higher geometry,
of higher arithmetic or the theory of numbers, of higher
algebra (including the theory of forms), and of the theory
of equations, also the discussion of functions of double and
multiple periodicity, and the creation of a theory of functions.

This hasty summary will indicate the subjects treated and
the limitations I have imposed on myself. The history of the
origin and growth of analysis and its application to the
material universe comes within my purview. The extensions
in the latter half of the nineteenth century of pure mathematics
and of the application of mathematics to physical
problems open a new period which lies beyond the limits of
this book; and I allude to these subjects only so far as they
may indicate the directions in which the future history of
mathematics appears to be developing.

\nbChapter{XV}{History of Mathematics from Descartes to Huygens}{History of Mathematics}{Circ.\ 1635--1675.}
  {See Cantor, part~xv, vol.~ii, pp.~599--844; other authorities for the
  mathematicians of this period are mentioned in the footnotes.}
\index{Analytical geometry,!x2@on development of, see chapters \textsc{xv-xix}|(}\index{Cantor, M., ref.\ to}\nblabel{png286a}

\nblabel{png286b}\First{I propose} in this chapter to consider the history of mathematics
during the forty years in the middle of the seventeenth century.
I regard \index{Descartes!ref.\ to}Descartes, \index{Cavalieri!ref.\ to}Cavalieri, \index{Pascal!ref.\ to}Pascal, \index{Wallis!ref.\ to}Wallis, \index{Fermat!ref.\ to}Fermat, and
Huygens as the leading mathematicians of this time. I shall
treat them in that order, and I shall conclude with a brief list of
the more eminent remaining mathematicians of the same date.

I have already stated that the mathematicians of this period---and
the remark applies more particularly to Descartes, Pascal,
and Fermat---were largely influenced by the teaching of \index{Kepler!ref.\ to}Kepler
and \index{Desargues!ref.\ to}Desargues, and I would repeat again that I regard these
latter and \index{Galileo!ref.\ to}Galileo as forming a connecting link between the
writers of the renaissance and those of modern times. I should
also add that the mathematicians considered in this chapter were
contemporaries, and, although I have tried to place them roughly
in such an order that their chief works shall come in a chronological
arrangement, it is essential to remember that they were
in relation one with the other, and in general were acquainted
with one another's researches as soon as these were published.

\index{Descartes|(}\nblabel{png286c}\nbparagraph{Descartes.\footnote
  {See \emph{Descartes}, by \index{Haldane, E. S., on Descartes}E.~S.~Haldane, London, 1905; and \emph{Descartes Savant},
  by G.~Milhaud, Paris, 1921. A complete edition of his works, edited by \index{Adam, C.}C.~Adam
  and \index{Tanner, P.}P.~Tanner, is in process of issue by the French Government; vols.~i--ix,
  1897--1904. A tolerably complete account of Descartes's mathematical
  and physical investigations is given in \index{Ersch and Gruber on Descartes}Ersch and Gruber's \emph{Encyclopädie}.
  The most complete edition of his works is that by \index{Cousin on Descartes}Victor Cousin in 11 vols.,
  Paris, 1824--26. Some minor papers subsequently discovered were printed by
  \index{De Careil on Descartes}\index{Careil on Descartes}F.~de~Careil, Paris, 1859.}}
 Subject to the above remarks, we may consider
\index{Descartes!ref.\ to}Descartes as the first of the modern school of mathematics.
\emph{René Descartes} was born near Tours on March~31, 1596, and
died at Stockholm on February~11, 1650; thus he was a contemporary
of \index{Galileo!ref.\ to}Galileo and \index{Desargues!ref.\ to}Desargues. His father, who, as the
name implies, was of a good family, was accustomed to spend
half the year at Rennes when the local parliament, in which he
held a commission as councillor, was in session, and the rest of
the time on his family estate of \emph{Les Cartes} at La Haye. René,
the second of a family of two sons and one daughter, was sent
at the age of eight years to the Jesuit School at La Flêche, and
of the admirable discipline and education there given he speaks
most highly. On account of his delicate health he was permitted
to lie in bed till late in the mornings; this was a custom
which he always followed, and when he visited \index{Pascal!ref.\ to}Pascal in 1647
he told him that the only way to do good work in mathematics
and to preserve his health was never to allow any one to make
him get up in the morning before he felt inclined to do so; an
opinion which I chronicle for the benefit of any schoolboy into
whose hands this work may fall.

On leaving school in 1612 Descartes went to Paris to be
introduced to the world of fashion. Here, through the medium
of the Jesuits, he made the acquaintance of \index{Mydorge, ref.\ to}Mydorge, and
renewed his schoolboy friendship with \index{Mersenne!ref.\ to}Mersenne, and together
with them he devoted the two years of 1615 and 1616 to the
study of mathematics. At that time a man of position usually
entered either the army or the church; Descartes chose the
former profession, and in 1617 joined the army of Prince
\index{Maurice of Orange, ref.\ to}Maurice of Orange, then at Breda. Walking through the streets
there he saw a placard in Dutch which excited his curiosity,
and stopping the first passer, asked him to translate it into
either French or Latin. The stranger, who happened to be
\index{Beeckman, I., ref.\ to}Isaac Beeckman, the head of the Dutch College at Dort, offered
to do so if \index{Descartes!ref.\ to}Descartes would answer it; the placard being, in fact,
a challenge to all the world to solve a certain geometrical
problem.\footnote
  {Some doubt has been recently expressed as to whether the story is
  well founded: see \emph{L'Intermédiaire des Mathématiciens}, Paris, 1909, vol.~xvi,
  pp.~12--13.}
Descartes worked it out within a few hours, and a
warm friendship between him and Beeckman was the result. This
unexpected test of his mathematical attainments made the uncongenial
life of the army distasteful to him, and though, under
family influence and tradition, he remained a soldier, he continued
to occupy his leisure with mathematical studies. He was
accustomed to date the first ideas of his new philosophy and of
his analytical geometry from three dreams which he experienced
on the night of November~10, 1619, at Neuberg, when campaigning
on the Danube, and he regarded this as the critical day of
his life, and one which determined his whole future.

He resigned his commission in the spring of 1621, and
spent the next five years in travel, during most of which time
he continued to study pure mathematics. In 1626 we find
him settled at Paris, ``a little well-built figure, modestly clad in
green taffety, and only wearing sword and feather in token of
his quality as a gentleman.'' During the first two years there
he interested himself in general society, and spent his leisure in
the construction of optical instruments; but these pursuits were
merely the relaxations of one who failed to find in philosophy
that theory of the universe which he was convinced finally
awaited him.

In 1628 \index{Berulle, Cardinal, ref.\ to}\index{De Berulle, Cardinal, ref.\ to}Cardinal de~Berulle, the founder of the Oratorians,
met Descartes, and was so much impressed by his conversation
that he urged on him the duty of devoting his life to the
examination of truth. Descartes agreed, and the better to
secure himself from interruption moved to Holland, then at the
height of its power. There for twenty years he lived, giving up
all his time to philosophy and mathematics. Science, he says,
may be compared to a tree; metaphysics is the root, physics is
the trunk, and the three chief branches are mechanics, medicine,
and morals, these forming the three applications of our knowledge,
namely, to the external world, to the human body, and
to the conduct of life.

He spent the first four years, 1629 to 1633, of his stay in
Holland in writing \emph{Le Monde}, which embodies an attempt to
give a physical theory of the universe; but finding that its
publication was likely to bring on him the hostility of the
church, and having no desire to pose as a martyr, he abandoned
it: the incomplete manuscript was published in 1664. He
then devoted himself to composing a treatise on universal
science; this was published at Leyden in 1637 under the title
\emph{Discours de la méthode pour bien conduire sa raison et chercher
la vérité dans les sciences}, and was accompanied with three
appendices (which possibly were not issued till 1638) entitled
\emph{La Dioptrique}, \emph{Les Météores}, and \emph{La Géométrie}; it is from the
last of these that the invention of analytical geometry dates.
In 1641 he published a work called \emph{Meditationes}, in which he
explained at some length his views of philosophy as sketched
out in the \emph{Discours}. In 1644 he issued the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia
Philosophiae}, the greater part of which was devoted to physical
science, especially the laws of motion and the theory of vortices.
In 1647 he received a pension from the French court in honour
of his discoveries. He went to Sweden on the invitation of the
Queen in 1649, and died a few months later of inflammation of
the lungs.

In appearance, \index{Descartes!ref.\ to}Descartes was a small man with large head,
projecting brow, prominent nose, and black hair coming down
to his eyebrows. His voice was feeble. In disposition he was
cold and selfish. Considering the range of his studies he was
by no means widely read, and he despised both learning and
art unless something tangible could be extracted therefrom.
He never married, and left no descendants, though he had one
illegitimate daughter, who died young.

\index{Philosophy, treatment of|(}\nblabel{png289a}As to his philosophical theories, it will be sufficient to say
that he discussed the same problems which have been debated
for the last two thousand years, and probably will be debated
with equal zeal two thousand years hence. It is hardly necessary to say that the problems themselves are of importance
and interest, but from the nature of the case no solution ever
offered is capable either of rigid proof or of disproof; all
that can be effected is to make one explanation more probable
than another, and whenever a philosopher like \index{Descartes!ref.\ to}Descartes
believes that he has at last finally settled a question it has
been possible for his successors to point out the fallacy in
his assumptions. I have read somewhere that philosophy has
always been chiefly engaged with the inter-relations of God,
Nature, and Man. The earliest philosophers were Greeks
who occupied themselves mainly with the relations between
God and Nature, and dealt with Man separately. The
Christian Church was so absorbed in the relation of God to
Man as entirely to neglect Nature. Finally, modern philosophers concern themselves chiefly with the relations between
Man and Nature. Whether this is a correct historical
generalization of the views which have been successively
prevalent I do not care to discuss here, but the statement as
to the scope of modern philosophy marks the limitations of
Descartes's writings.\index{Philosophy, treatment of|)}

\index{Geometry.!z6@Analytical geometry|(}\index{Analytical geometry,!x1@origin of|(}\nblabel{png290a}Descartes's chief contributions to mathematics were his
analytical geometry and his theory of vortices, and it is on his
researches in connection with the former of these subjects that
his mathematical reputation rests.

Analytical geometry does not consist merely (as is sometimes
loosely said) in the application of algebra to geometry; that had
been done by \index{Archimedes!ref.\ to}Archimedes and many others, and had become the
usual method of procedure in the works of the mathematicians
of the sixteenth century. The great advance made by Descartes
was that he saw that a point in a plane could be completely
determined if its distances, say $x$ and $y$, from two fixed lines
drawn at right angles in the plane were given, with the convention
familiar to us as to the interpretation of positive and negative
values; and that though an equation $f(x, y)=0$ was indeterminate and could be satisfied by an infinite number of values of
$x$ and $y$, yet these values of $x$ and $y$ determined the \index{Coord@Co-ordinates}co-ordinates
of a number of points which form a curve, of which the equation
$f(x,y)=0$ expresses some geometrical property, that is, a
property true of the curve at every point on it. \index{Descartes!ref.\ to}Descartes
asserted that a point in space could be similarly determined by
three co-ordinates, but he confined his attention to plane
curves.

It was at once seen that in order to investigate the properties
of a curve it was sufficient to select, as a definition, any
characteristic geometrical property, and to express it by means
of an equation between the (current) co-ordinates of any point
on the curve, that is, to translate the definition into the
language of analytical geometry. The equation so obtained
contains implicitly every property of the curve, and any
particular property can be deduced from it by ordinary algebra
without troubling about the geometry of the figure. This
may have been dimly recognized or foreshadowed by earlier
writers, but \index{Descartes!ref.\ to}Descartes went further and pointed out the very
important facts that two or more curves can be referred to one
and the same system of co-ordinates, and that the points in
which two curves intersect can be determined by finding the
roots common to their two equations. I need not go further
into details, for nearly everyone to whom the above is intelligible
will have read analytical geometry, and is able to appreciate the
value of its invention.

Descartes's \emph{Géométrie} is divided into three books: the first
two of these treat of analytical geometry, and the third includes
an analysis of the algebra then current. It is somewhat difficult
to follow the reasoning, but the obscurity was intentional.
``Je n'ai rien omis,'' says he, ``qu'à dessein \ldots j'avois prévu
que certaines gens qui se vantent de sçavoir tout n'auroient
pas manqué de dire que je n'avois rien écrit qu'ils n'eussent
sçu auparavant, si je me fusse rendu assez intelligible pour
eux.''

The first book commences with an explanation of the
principles of analytical geometry, and contains a discussion
of a certain problem which had been propounded by \index{Pappus!ref.\ to}Pappus in
the seventh book of his \mytextgreek{Sunagwg'h} and of which some particular
cases had been considered by \index{Euclid!ref.\ to}Euclid and \index{Apollonius!ref.\ to}Apollonius. The
general theorem had baffled previous geometricians, and it
was in the attempt to solve it that \index{Descartes!ref.\ to}Descartes was led to the
invention of analytical geometry. \index{Conics (Analytical).!Invention of, by Descartes|(}The full enunciation of the
problem is rather involved, but the most important case is to
find the locus of a point such that the product of the
perpendiculars on $m$ given straight lines shall be in a constant
ratio to the product of the perpendiculars on $n$ other given
straight lines. The ancients had solved this geometrically
for the case $m=1$, $n = 1$, and the case $m=1$, $n=2$. Pappus
had further stated that, if $m=n=2$, the locus is a conic,
but he gave no proof; Descartes also failed to prove this by
pure geometry, but he shewed that the curve is represented
by an equation of the second degree, that is, is a conic;
subsequently \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton gave an elegant solution of the problem
by pure geometry.

\index{Curves, classification of}In the second book Descartes divides curves into two
classes, namely, geometrical and mechanical curves. He
defines geometrical curves as those which can be generated
by the intersection of two lines each moving parallel to one
co-ordinate axis with ``commensurable'' velocities; by which
terms he means that $dy/dx$ is an algebraical function, as, for
example, is the case in the ellipse and the cissoid. He calls a
curve mechanical when the ratio of the velocities of these lines
is ``incommensurable''; by which term he means that $dy/dx$ is
a transcendental function, as, for example, is the case in the
cycloid and the quadratrix. Descartes confined his discussion
to geometrical curves, and did not treat of the theory of
mechanical curves. The classification into algebraical and transcendental
curves now usual is due to \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton.\footnote
  {See below, page~\pageref{xref:340}.}

Descartes also paid particular attention to the theory of the
\index{Tangent (geometrical)}tangents to curves---as perhaps might be inferred from his
system of classification just alluded to. The then current
definition of a tangent at a point was a straight line through
the point such that between it and the curve no other straight
line could be drawn, that is, the straight line of closest contact.
\index{Descartes!ref.\ to}Descartes proposed to substitute for this a statement equivalent
to the assertion that the tangent is the limiting position of the
secant; \index{Fermat!ref.\ to}Fermat, and at a later date \index{Maclaurin!ref.\ to}Maclaurin and \index{Lagrange!ref.\ to}Lagrange,
adopted this definition. \index{Barrow!ref.\ to}Barrow, followed by \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton and
\index{Leibnitz!ref.\ to}Leibnitz, considered a curve as the limit of an inscribed
polygon when the sides become indefinitely small, and stated
that a side of the polygon when produced became in the limit a
tangent to the curve. \index{Roberval!ref.\ to}Roberval, on the other hand, defined a
tangent at a point as the direction of motion at that instant of a
point which was describing the curve. The results are the same
whichever definition is selected, but the controversy as to which
definition was the correct one was none the less lively. In his
letters Descartes illustrated his theory by giving the general
rule for drawing tangents and normals to a roulette.

The method used by Descartes to find the tangent or normal
at any point of a given curve was substantially as follows. He
determined the centre and radius of a circle which should cut
the curve in two consecutive points there. The tangent to the
circle at that point will be the required tangent to the curve.
In modern text-books it is usual to express the condition that
two of the points in which a straight line (such as $y = mx + c$) cuts
the curve shall coincide with the given point: this enables us to
determine $m$ and $c$, and thus the equation of the tangent there
is determined. Descartes, however, did not venture to do this,
but selecting a circle as the simplest curve and one to which he
knew how to draw a tangent, he so fixed his circle as to make it
touch the given curve at the point in question, and thus reduced
the problem to drawing a tangent to a circle. I should note
in passing that he only applied this method to curves which are
symmetrical about an axis, and he took the centre of the circle
on the axis.\index{Analytical geometry,!x1@origin of|)}\index{Geometry.!z6@Analytical geometry|)}

The obscure style deliberately adopted by Descartes
diminished the circulation and immediate appreciation of these
books; but a Latin translation of them, with explanatory
notes, was prepared by \index{De Beaune, ref.\ to}\index{Beaune, De, ref.\ to}F.~de Beaune, and an edition of this,
with a commentary by \index{Schooten, van!ref.\ to}\index{Vans@Van Schooten!ref.\ to}F.~van Schooten, issued in 1659, was
widely read.

\index{Algebra.!z8@Developed by (amongst others) Descartes}\nblabel{png294a}The third book of the \emph{Géométrie} contains an analysis of the
algebra then current, and it has affected the language of the
subject by fixing the custom of employing the letters at the
beginning of the alphabet to denote known quantities, and those
at the end of the alphabet to denote \index{Unknown quantity, word or symbol for}unknown quantities.\footnote
  {On the origin of the custom of using $x$ to represent an unknown
  example, see a note by \index{Eneström, ref.\ to}G.~Eneström in the \emph{Bibliotheca Mathematica}, 1885,
  p.~43.}
Descartes further introduced the system of \index{Exponents}\index{Indices}indices now in use;
very likely it was original on his part, but I would here remind
the reader that the suggestion had been made by previous
writers, though it had not been generally adopted. It is
doubtful whether or not Descartes recognised that his letters
might represent any quantities, positive or negative, and that it
was sufficient to prove a proposition for one general case. He
was the earliest writer to realize the advantage to be obtained
by taking all the terms of an equation to one side of it, though
\index{Stifel!ref.\ to}Stifel and \index{Harriot!ref.\ to}Harriot had sometimes employed that form by choice.
He realised the meaning of negative quantities and used them
freely. \index{Descartes!rule of signs of}\index{Roots of equations,!position of}In this book he made use of the rule for finding a limit
to the number of positive and of negative roots of an algebraical
equation, which is still known by his name; and introduced the
method of indeterminate coefficients for the solution of equations.
He believed that he had given a method by which algebraical
equations of any order could be solved, but in this he was
mistaken. It may be also mentioned that he enunciated the
theorem, commonly attributed to \index{Euler!ref.\ to}Euler, on the relation between
the numbers of faces, edges, and angles of a polyhedron: this is
in one of the papers published by \index{Careil on Descartes!ref.\ to}Careil.

Of the two other appendices to the \emph{Discours} one was devoted
to \emph{optics}. The chief interest of this consists in the statement
given of the law of \index{Refraction}refraction. This appears to have been taken
from \index{Snell!ref.\ to}Snell's work, though, unfortunately, it is enunciated in a
way which might lead a reader to suppose that it is due to the
researches of \index{Descartes!ref.\ to}Descartes. Descartes would seem to have repeated
Snell's experiments when in Paris in 1626 or 1627, and it is
possible that he subsequently forgot how much he owed to the
earlier investigations of Snell. \index{Optics (geometrical).!Discussed by (among others) Euclid!z5@Descartes}A large part of the \index{Optics (geometrical).!z11@(physical)}optics is
devoted to determining the \index{Lenses, construction of}best shape for the lenses of a
\index{Telescopes}telescope, but the mechanical difficulties in grinding a surface of
glass to a required form are so great as to render these investigations
of little practical use. \index{Light,!physical theories of}Descartes seems to have been
doubtful whether to regard the rays of light as proceeding from
the eye and so to speak touching the object, as the Greeks
had done, or as proceeding from the object, and so affecting the
eye; but, since he considered the \index{Light,!velocity of}velocity of light to be infinite,
he did not deem the point particularly important.

\index{Rainbow, explanation of}The other appendix, on \emph{meteors}, contains an explanation of
numerous atmospheric phenomena, including the rainbow; the
explanation of the latter is necessarily incomplete, since
Descartes was unacquainted with the fact that the refractive
index of a substance is different for lights of different colours.

Descartes's physical theory of the universe, embodying most
of the results contained in his earlier and unpublished \emph{Le Monde},
is given in his \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia}, 1644, and rests on a metaphysical
basis. \index{Motion, laws of}He commences with a discussion on motion; and then
lays down ten laws of nature, of which the first two are almost
identical with the first two laws of motion as given by \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton;
the remaining eight laws are inaccurate. He next proceeds to
discuss the nature of matter which he regards as uniform in
kind though there are three forms of it. He assumes that the
matter of the universe must be in motion, and that the motion
must result in a number of \index{Cartesian vortices}\index{Vortices, Cartesian}vortices. \index{Planetary motions}He states that the sun is
the centre of an immense whirlpool of this matter, in which the
planets float and are swept round like straws in a whirlpool of
water. Each planet is supposed to be the centre of a secondary
whirlpool by which its satellites are carried: these secondary
whirlpools are supposed to produce variations of density in the
surrounding medium which constitute the primary whirlpool,
and so cause the planets to move in ellipses and not in circles.
All these assumptions are arbitrary and unsupported by any
investigation. It is not difficult to prove that on his hypothesis
the sun would be in the centre of these ellipses, and not at a
focus (as \index{Kepler!ref.\ to}Kepler had shewn was the case), and that the weight
of a body at every place on the surface of the earth except the
equator would act in a direction which was not vertical; but it
will be sufficient here to say that \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton in the second book of
his \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia}, 1687, considered the theory in detail, and shewed
that its consequences are not only inconsistent with each of
\index{Kepler's laws}Kepler's laws and with the fundamental laws of mechanics, but
are also at variance with the laws of nature assumed by \index{Descartes!ref.\ to}Descartes.
Still, in spite of its crudeness and its inherent defects, the
theory of \index{Cartesian vortices}vortices marks a fresh era in astronomy, for it was
an attempt to explain the phenomena of the whole universe by
the same mechanical laws which experiment shews to be true
on the earth.\index{Descartes|)}\index{Conics (Analytical).!Invention of, by Descartes|)}

\index{Cavalieri|(}\nblabel{png296a}\nbparagraph{Cavalieri.\footnote
  {Cavalieri's life has been written by \index{Frisi on Cavalieri}P.~Frisi, Milan, 1778; by \index{Predari on Cavalieri}F.~Predari,
  Milan, 1843; by \index{Piola on Cavalieri}Gabrio Piola, Milan, 1844; and by \index{Favaro, A., ref.\ to}A.~Favaro,
  Bologna, 1888. An analysis of his works is given in \index{Marie, ref.\ to}Marie's \emph{Histoire des
  Sciences}, Paris, 1885--8, vol.~iv, pp.~69--90.}}
Almost contemporaneously with the publication
in 1637 of Descartes's geometry, the principles of the integral
calculus, so far as they are concerned with summation, were
being worked out in Italy. This was effected by what was
called the \index{Indivisibles, method of}principle of indivisibles, and was the invention of
Cavalieri. It was applied by him and his contemporaries to
numerous problems connected with the quadrature of curves and
surfaces, the determination of volumes, and the positions of
\index{Centres of mass}\index{Gravity,!centres of}\index{Mass, centres of}centres of mass. It served the same purpose as the tedious
\index{Exhaustions, method of}method of exhaustions used by the Greeks; in principle the
methods are the same, but the notation of indivisibles is more
concise and convenient. It was, in its turn, superseded at the
beginning of the eighteenth century by the integral calculus.

\emph{Bonaventura Cavalieri} was born at Milan in 1598, and died
at Bologna on November~27, 1647. He became a Jesuit at an
early age; on the recommendation of the Order he was in 1629
made professor of mathematics at Bologna; and he continued
to occupy the chair there until his death. I have already
mentioned Cavalieri's name in connection with the introduction of the use of \index{Logarithms}logarithms into Italy, and have alluded to
his discovery of the expression for the \index{Spherical trigonometry}area of a spherical
triangle in terms of the spherical excess. He was one of the
most influential mathematicians of his time, but his subsequent
reputation rests mainly on his invention of the principle of
indivisibles.

\nblabel{png297a}The principle of indivisibles had been used by \index{Kepler!ref.\ to}Kepler in
1604 and 1615 in a somewhat crude form. It was first stated
by \index{Cavalieri!ref.\ to}Cavalieri in 1629, but he did not publish his results till
1635. In his early enunciation of the principle in 1635
Cavalieri asserted that a line was made up of an infinite
number of points (each without magnitude), a surface of an
infinite number of lines (each without breadth), and a volume
of an infinite number of surfaces (each without thickness). To
meet the objections of \index{Guldinus!ref.\ to}Guldinus and others, the statement
was recast, and in its final form as used by the mathematicians
of the seventeenth century it was published in Cavalieri's
\emph{Exercitationes Geometricae} in 1647; the third exercise is
devoted to a defence of the theory. This book contains the
earliest demonstration of the properties of \index{Pappus!ref.\ to}Pappus.\footnote
  {See above, pp.~\pageref{xref:101}, \pageref{xref:252}.}
Cavalieri's
works on indivisibles were reissued with his later corrections in
1653.

The method of indivisibles rests, in effect, on the assumption
that any magnitude may be divided into an infinite number of
small quantities which can be made to bear any required ratios
(\emph{ex.~gr.}\ equality) one to the other. The analysis given by
Cavalieri is hardly worth quoting except as being one of the
first steps taken towards the formation of an infinitesimal
calculus. One example will suffice. Suppose it be required to
find the area of a right-angled triangle. Let the base be made
up of, or contain $n$ points (or indivisibles), and similarly let the
other side contain $na$ points, then the ordinates at the successive
points of the base will contain $a, 2a \ldots, na$ points. Therefore
the number of points in the area is $a+2a+\ldots+na$; the sum
of which is $\frac{1}{2}n^{2}a+\frac{1}{2}na$. Since $n$ is very large, we may neglect
$\frac{1}{2}na$, for it is inconsiderable compared with $\frac{1}{2}n^{2}a$. Hence the
area is equal to $\frac{1}{2}(na)n$, that is, $\frac{1}{2} × \text{ altitude } × \text{ base}$. There is
no difficulty in criticizing such a proof, but, although the form
in which it is presented is indefensible, the substance of it is
correct.

It would be misleading to give the above as the only
specimen of the method of indivisibles, and I therefore quote
another example, taken from a later writer, which will fairly
illustrate the use of the method when modified and corrected by
the \index{Limits, method of}method of limits. \index{Quadrature of curves!parabolas|(}\index{Parabola,!quadrature of|(}Let it be required to find the area
outside a parabola $APC$ and bounded by the curve, the tangent
at $A$, and a line $DC$ parallel to $AB$ the diameter at $A$. Complete
the parallelogram $ABCD$. Divide $AD$ into $n$ equal parts,
let $AM$ contain $r$ of them, and let $MN$ be the $(r+1)$th part.
Draw $MP$ and $NQ$ parallel to $AB$, and draw $PR$
 parallel to $AD$.
Then when $n$ becomes indefinitely large, the curvilinear area
$APCD$ will be the limit of the sum of all parallelograms like $PN$.
Now
\[
\text{area } PN : \text{ area } BD = MP \ctrdot MN : DC \ctrdot AD.
\]
\begin{center}
\labelincludegraphics{298a}
\end{center}
But by the properties of the parabola
\begin{DPgather*}
                        MP : DC = AM^{2} : AD^{2} = r^{2} :n^{2},\\
\lintertext{and} 			  MN : AD = 1 : n.\\
\lintertext{Hence}			MP \ctrdot MN : DC \ctrdot AD = r^{2} : n^{3}.\\
\lintertext{Therefore}  \text{area } PN : \text{area } BD = r^{2} : n^{3}.
\end{DPgather*}
Therefore, ultimately,
\begin{DPalign*}
\text{area } APCD : \text{area } BD &= 1^{2} + 2^{2} + \ldots + (n - 1)^{2} : n^{3}\\
                                    &= \tfrac{1}{6}n (n-1) (2n-1) : n^{3}\\
\lintertext{\rlap{which, in the limit,}}   &= 1 : 3.
\end{DPalign*}

\index{Parabola,!quadrature of|)}\index{Quadrature of curves!parabolas|)}
It is perhaps worth noticing that \index{Cavalieri!ref.\ to}Cavalieri and his successors
always used the method to find the ratio of two areas, volumes,
or magnitudes of the same kind and dimensions, that is, they
never thought of an area as containing so many units of area.
The idea of comparing a magnitude with a unit of the same
kind seems to have been due to \index{Wallis!ref.\ to}Wallis.

It is evident that in its direct form the method is applicable
to only a few curves. Cavalieri proved that, if $m$ be
a positive integer, then the limit, when $n$ is infinite, of
$(1^{m} + 2^{m}+ \dots+n^{m})/n^{m+1}$ is $1/(m + 1)$, which is equivalent to
saying that he found the integral to $x$ of $x^{m}$ from $x = 0$ to
$x = 1$; he also discussed the quadrature of the hyperbola.\index{Cavalieri|)}

\index{Pascal|(}\nblabel{png299a}\nbparagraph{Pascal.\footnote
  {See \emph{Pascal} by \index{Bertrand}J.~Bertrand, Paris, 1891; and \emph{Pascal, sein Leben und
  seine Kämpfe}, by \index{Dreydorff on Pascal}J.~G.~Dreydorff, Leipzig, 1870. Pascal's life, written by
  his sister \index{Perier@Périer on Pascal}Mme.\ Périer, was edited by \index{Faugere on Pascal}A.~P.~Faugère, Paris, 1845, and has
  formed the basis for several works. An edition of his writings was published
  in five volumes at the Hague in 1779, second edition, Paris, 1819; some
  additional pamphlets and letters were published in three volumes at Paris
  in 1858.}}
Among the contemporaries of \index{Descartes!ref.\ to}Descartes none
displayed greater natural genius than Pascal, but his mathematical
reputation rests more on what he might have done
than on what he actually effected, as during a considerable part
of his life he deemed it his duty to devote his whole time
to religious exercises.

\emph{Blaise Pascal} was born at Clermont on June~19, 1623, and
died at Paris on Aug.~19, 1662. His father, a local judge at
Clermont, and himself of some scientific reputation, moved to
Paris in 1631, partly to prosecute his own scientific studies,
partly to carry on the education of his only son, who had
already displayed exceptional ability. Pascal was kept at home
in order to ensure his not being overworked, and with the same
object it was directed that his education should be at first
confined to the study of languages, and should not include any
mathematics. This naturally excited the boy's curiosity, and
one day, being then twelve years old, he asked in what geometry
consisted. His tutor replied that it was the science of constructing
exact figures and of determining the proportions
between their different parts. Pascal, stimulated no doubt by
the injunction against reading it, gave up his play-time to this
new study, and in a few weeks had discovered for himself many
properties of figures, and in particular the proposition that the
sum of the angles of a triangle is equal to two right angles. I
have read somewhere, but I cannot lay my hand on the authority,
that his proof merely consisted in turning the angular points of
a triangular piece of paper over so as to meet in the centre of
the inscribed circle: a similar demonstration can be got by
turning the angular points over so as to meet at the foot of the
perpendicular drawn from the biggest angle to the opposite side.
His father, struck by this display of ability, gave him a copy of
\index{Euclid's \emph{Elements}!ref.\ to}Euclid's \emph{Elements}, a book which Pascal read with avidity and
soon mastered.

At the age of fourteen he was admitted to the weekly
meetings of \index{Roberval!ref.\ to}Roberval, \index{Mersenne!ref.\ to}Mersenne, \index{Mydorge, ref.\ to}Mydorge, and other French
geometricians; from which, ultimately, the \index{French Academy}\index{Academy,!the French}French Academy
sprung. At sixteen Pascal wrote an essay on conic sections;
and in 1641, at the age of eighteen, he constructed the first
\index{Calculating machine}\index{Arithmetical!machine}arithmetical machine, an instrument which, eight years later, he
further improved. His correspondence with \index{Fermat!ref.\ to}Fermat about this
time shews that he was then turning his attention to analytical
geometry and physics. He repeated \index{Torricelli!ref.\ to}Torricelli's experiments, by
which the pressure of the atmosphere could be estimated as a
weight, and he confirmed his theory of the cause of barometrical
variations by obtaining at the same instant readings at different
altitudes on the hill of Puy-de-Dôme.

In 1650, when in the midst of these researches, Pascal
suddenly abandoned his favourite pursuits to study religion, or,
as he says in his \emph{Pensées}, ``to contemplate the greatness and the
misery of man''; and about the same time he persuaded the
younger of his two sisters to enter the \index{Port-Royal, society of|(}Port Royal society.

\index{Triangle, area of!arithmetical|(}In 1653 he had to administer his father's estate. He now
took up his old life again, and made several experiments on the
pressure exerted by gases and liquids; it was also about this
period that he invented the \index{Arithmetical!triangle|(}arithmetical triangle, and together
with \index{Fermat!ref.\ to}Fermat created the calculus of probabilities. He was
meditating marriage when an accident again turned the current
of his thoughts to a religious life. He was driving a four-in-hand
on November~23, 1654, when the horses ran away; the
two leaders dashed over the parapet of the bridge at Neuilly,
and Pascal was saved only by the traces breaking. Always
somewhat of a mystic, he considered this a special summons to
abandon the world. He wrote an account of the accident on
a small piece of parchment, which for the rest of his life he
wore next to his heart, to perpetually remind him of his
covenant; and shortly moved to Port Royal, where he continued
to live until his death in 1662. Constitutionally delicate, he
had injured his health by his incessant study; from the age of
seventeen or eighteen he suffered from insomnia and acute
dyspepsia, and at the time of his death was physically worn
out.

{\nbstretchyspace His famous \emph{Provincial Letters} directed against the Jesuits,
and his \emph{Pensées,} were written towards the close of his life, and
are the first example of that finished form which is characteristic
of the best French literature.} The only mathematical work
that he produced after retiring to Port Royal was the essay on
the \index{Cycloid}cycloid in 1658. He was suffering from sleeplessness and
toothache when the idea occurred to him, and to his surprise his
teeth immediately ceased to ache. Regarding this as a divine
intimation to proceed with the problem, he worked incessantly
for eight days at it, and completed a tolerably full account of
the geometry of the cycloid.\index{Port-Royal, society of|)}

I now proceed to consider his mathematical works in rather
greater detail.

\index{Geometry.!z8@Modern synthetic geometry, originated with Desargues!x3@continued by Pascal}\index{Conic Sections (Geometrical).!Discussed by most of the Greek geometricians after Menaechmus!x5@and subsequently by Pascal}\nblabel{png302a}His early essay on the \emph{geometry of conics}, written in 1639,
but not published till 1779, seems to have been founded on the
teaching of \index{Desargues!ref.\ to}Desargues. Two of the results are important as
well as interesting. The first of these is the theorem known
now as ``Pascal's theorem,'' namely, that if a hexagon be
inscribed in a conic, the points of intersection of the opposite
sides will lie in a straight line. The second, which is really due
to Desargues, is that if a quadrilateral be inscribed in a conic,
and a straight line be drawn cutting the sides taken in order in
the points $A, B, C,$ and $D$, and the conic in $P$ and $Q$, then
\[
PA \ctrdot PC:PB \ctrdot PD=QA \ctrdot QC:QB \ctrdot QD.
\]

\nblabel{xref:284}\nblabel{png302b}Pascal employed his \emph{arithmetical triangle} in 1653, but no
account of his method was printed till 1665. The triangle is
constructed as in the \figureref{pascal}{figure} below, each horizontal line
being formed from the one above it by making every number
in it equal to the sum of those above and to the left of it in the
row immediately above it; \emph{ex.~gr.}\ the fourth number in the
fourth line,
namely, 20, is equal to $1+3+6+10$. The numbers
in each line are what are now called \index{Numbers,!figurate}\index{Figurate numbers}\emph{figurate} numbers. Those
in the first line are called numbers of the first order; those in
the second line, natural numbers or numbers of the second
order; those in the third line, numbers of the third order, and
so on. It is easily shewn that the $m$th number in the $n$th row
is $(m+n-2)!/(m-1)!(n-1)!$

\begin{table}[ht]
\nblabel{fig:pascal}
\makeatletter
\def\@hline{\advance\@linelen\@halfwidth
  \ifnum \@xarg <\z@ \hskip -\@linelen \hskip\@halfwidth
    \else\hskip-\@halfwidth\fi
  \vrule \@height \@halfwidth \@depth \@halfwidth \@width \@linelen
  \ifnum \@xarg <\z@ \hskip -\@linelen \hskip\@halfwidth
    \else\hskip-\@halfwidth\fi}
\def\@upline{%
  \hb@xt@\z@{\advance\@linelen\@halfwidth\hskip -\@halfwidth
   \vrule \@width \@wholewidth \@height \@linelen \@depth\z@\hss}}
\makeatother
  \centering
  \setlength{\unitlength}{1cm}
\thicklines
  \begin{picture}(0,0)(0,2.65)
    {%
      \put(0, 5){\line( 1, 0){5}}
      \put(5, 5){\line(-1,-1){5}}
      \put(0, 0){\line( 0, 1){5}}
    }%
  \end{picture}%
  \begin{tabular}{cccccc}%
    \hspace{0.5 cm} & \hspace{0.5 cm} & \hspace{0.5 cm} & \hspace{0.5 cm} & \hspace{0.5 cm} & \\
    1  &  1  &  1  &  1  &  1  &  \ldots\rule{0pt}{0.75cm} \\
    1  &  2  &  3  &  4  &  5  &  \ldots\rule{0pt}{0.75cm} \\
    1  &  3  &  6  &  10 &  15 &  \ldots\rule{0pt}{0.75cm} \\
    1  &  4  &  10 &  20 &  35 &  \ldots\rule{0pt}{0.75cm} \\
    1  &  5  &  15 &  35 &  70 &  \ldots\rule{0pt}{0.75cm} \\
    \vdots & \vdots & \vdots & \vdots & \vdots & \rule{0pt}{0.75cm}
  \end{tabular}
\end{table}

Pascal's\nblabel{xref:285} arithmetical triangle, to any required order, is got
by drawing a diagonal downwards from right to left as in the
\figureref{pascal}{figure}. The numbers in any diagonal give the coefficients of the
expansion of a binomial; for example, the figures in the fifth
diagonal, namely, 1, 4, 6, 4, 1, are the coefficients in the
expansion $(a+b)^{4}$. Pascal used the triangle partly for this
purpose, and partly to find the numbers of combinations of $m$
 things taken $n$ at a time, which he stated, correctly, to be
$(n+1)(n+2)(n+3)\ldots m/(m-n)!$\index{Arithmetical!triangle|)}\index{Triangle, area of!arithmetical|)}

\nblabel{png303a}Perhaps as a mathematician Pascal is best known in connection
with his correspondence with \index{Fermat!ref.\ to}Fermat in 1654, in which he
laid down the principles of the \index{Probabilities, theory of|(}\emph{theory of probabilities}. This
correspondence arose from a problem proposed by a gamester,
the \index{Demere@De Méré, ref.\ to}\index{Mere@Méré, de, ref.\ to}Chevalier de~Méré, to Pascal, who communicated it to
Fermat. The problem was this. Two players of equal skill
want to leave the table before finishing their game. Their
scores and the number of points which constitute the game
being given, it is desired to find in what proportion they should
divide the stakes. Pascal and Fermat agreed on the answer,
but gave different proofs. The following is a translation of
Pascal's solution. That of Fermat is given later.

\begin{nbQuote}
The following is my method for determining the share of each player
when, for example, two players play a game of three points and each
player has staked 32 pistoles.

Suppose that the first player has gained two points, and the second
player one point; they have now to play for a point on this condition,
that, if the first player gain, he takes all the money which is at stake,
namely, 64 pistoles; while, if the second player gain, each player has two
points, so that they are on terms of equality, and, if they leave off playing,
each ought to take 32 pistoles. Thus, if the first player gain, then
64 pistoles belong to him, and, if he lose, then 32 pistoles belong to him.
If therefore the players do not wish to play this game, but to separate
without playing it, the first player would say to the second, ``I am
certain of 32 pistoles even if I lose this game, and as for the other 32
pistoles perhaps I shall have them and perhaps you will have them;
the chances are equal. Let us then divide these 32 pistoles equally,
and give me also the 32 pistoles of which I am certain.'' Thus the first
player will have 48 pistoles and the second 16 pistoles.

Next, suppose that the first player has gained two points and the
second player none, and that they are about to play for a point; the
condition then is that, if the first player gain this point, he secures the
game and takes the 64 pistoles, and, if the second player gain this point,
then the players will be in the situation already examined, in which the
first player is entitled to 48 pistoles and the second to 16 pistoles.
Thus, if they do not wish to play, the first player would say to the second,
``If I gain the point I gain 64 pistoles; if I lose it, I am entitled to
48 pistoles. Give me then the 48 pistoles of which I am certain, and
divide the other 16 equally, since our chances of gaining the point are
equal.'' Thus the first player will have 56 pistoles and the second player
8 pistoles.

Finally, suppose that the first player has gained one point and the
second player none. If they proceed to play for a point, the condition is
that, if the first player gain it, the players will be in the situation first
examined, in which the first player is entitled to 56 pistoles; if the first
player lose the point, each player has then a point, and each is entitled
to 32 pistoles. Thus, if they do not wish to play, the first player would
say to the second, ``Give me the 32 pistoles of which I am certain, and
divide the remainder of the 56 pistoles equally, that is, divide 24 pistoles
equally.'' Thus the first player will have the sum of 32 and 12
pistoles, that is, 44 pistoles, and consequently the second will have 20
pistoles.
\end{nbQuote}

Pascal proceeds next to consider the similar problems when
the game is won by whoever first obtains $m + n$ points, and
one player has $m$ while the other has $n$ points. The answer
is obtained by using the arithmetical triangle. The general
solution (in which the skill of the players is unequal) is given
in many modern text-books on algebra, and agrees with Pascal's
result, though of course the notation of the latter is different
and less convenient.

Pascal made an illegitimate use of the new theory in
the seventh chapter of his \emph{Pensées}. In effect, he puts his
argument that, as the value of eternal happiness must be
infinite, then, even if the probability of a religious life ensuring
eternal happiness be very small, still the expectation (which is
measured by product of the two) must be of sufficient magnitude
to make it worth while to be religious. The argument,
if worth anything, would apply equally to any religion which
promised eternal happiness to those who accepted its doctrines.
If any conclusion may be drawn from the statement, it is the
undesirability of applying mathematics to questions of morality
of which some of the data are necessarily outside the range
of an exact science. It is only fair to add that no one
had more contempt than Pascal for those who changed
their opinions according to the prospect of material benefit,
and this isolated passage is at variance with the spirit of his
writings.\index{Probabilities, theory of|)}

\nblabel{png305a}The last mathematical work of Pascal was that on the \index{Cycloid}\emph{cycloid}
in 1658. The cycloid is the curve traced out by a point on the
circumference of a circular hoop which rolls along a straight
line. \index{Galileo!ref.\ to}Galileo, in 1630, had called attention to this curve, the
shape of which is particularly graceful, and had suggested that
the arches of bridges should be built in this form.\footnote
  {The bridge, by \index{Essex, ref.\ to}Essex, across the Cam in the grounds of Trinity College,
  Cambridge, has cycloidal arches. On the history of the cycloid before Galileo,
  see \index{Gunther@Günther, S.}S.~Günther, \emph{Bibliotheca Mathematica}, 1887, vol.~i, pp.~7--14.}
Four years
later, in 1634, \index{Roberval!ref.\ to}Roberval found the area of the cycloid; \index{Descartes!ref.\ to}Descartes
thought little of this solution and defied him to find its tangents,
the same challenge being also sent to \index{Fermat!ref.\ to}Fermat who at once
solved the problem. Several questions connected with the
curve, and with the surface and volume generated by its
revolution about its axis, base, or the tangent at its vertex,
were then proposed by various mathematicians. These and
some analogous questions, as well as the positions of the centres
of the mass of the solids formed, were solved by Pascal in 1658,
and the results were issued as a challenge to the world. \index{Wallis!ref.\ to}Wallis
succeeded in solving all the questions except those connected
with the centre of mass. Pascal's own solutions were effected
by the method of indivisibles, and are similar to those which
a modern mathematician would give by the aid of the integral
calculus. He obtained by summation what are equivalent to
the integrals of $\sin\phi$, $\sin^2\phi$, and $\phi\sin\phi$, one limit being either
0 or $\frac{1}{2}\pi$. He also investigated the geometry of the Archimedean
spiral. These researches, according to \index{D'Alembert!ref.\ to}D'Alembert,
form a connecting link between the geometry of \index{Archimedes!ref.\ to}Archimedes and
the infinitesimal calculus of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton.\index{Pascal|)}

\index{Wallis|(}\nblabel{png306a}\nbparagraph{Wallis.\footnote
  {See \index{Ball, W. W. R., ref.\ to}my \emph{History of the Study of Mathematics at Cambridge}, pp.~41--46.
  An edition of Wallis's mathematical works was published in three volumes at
  Oxford, 1693--98.}}
\emph{John Wallis} was born at Ashford on November~22,
1616, and died at Oxford on October~28, 1703. He was educated
at Felstead school, and one day in his holidays, when fifteen
years old, he happened to see a book of arithmetic in the
hands of his brother; struck with curiosity at the odd signs
and symbols in it he borrowed the book, and in a fortnight,
with his brother's help, had mastered the subject. As it was
intended that he should be a doctor, he was sent to Emmanuel
College, Cambridge, while there he kept an ``act'' on the
doctrine of the circulation of the blood; that is said to have
been the first occasion in Europe on which this theory was
publicly maintained in a disputation. His interests, however,
centred on mathematics.

\index{Ciphers, discoveries of}He was elected to a fellowship at Queens' College, Cambridge,
and subsequently took orders, but on the whole adhered to the
Puritan party, to whom he rendered great assistance in deciphering
the royalist despatches. He, however, joined the moderate
Presbyterians in signing the remonstrance against the execution
of \index{Charles I. of England, ref.\ to}Charles I., by which he incurred the lasting hostility of the
Independents. In spite of their opposition he was appointed in
1649 to the Savilian chair of geometry at Oxford, where he
lived until his death on October~28, 1703. Besides his mathematical
works he wrote on theology, logic, and philosophy, and
was the first to devise a system for teaching deaf-mutes. I
confine myself to a few notes on his more important mathematical
writings. They are notable partly for the introduction of the
use of infinite series as an ordinary part of analysis, and partly
for the fact that they revealed and explained to all students the
principles of the new methods of analysis introduced by his
contemporaries and immediate predecessors.

\index{Conics (Analytical).!q2@treated by Wallis}In 1655 \index{Wallis!ref.\ to}Wallis published a treatise on \emph{conic sections} in which
they were defined analytically. I have already mentioned that
the \emph{Géométrie} of \index{Descartes!ref.\ to}Descartes is both difficult and obscure, and to
many of his contemporaries, to whom the method was new, it
must have been incomprehensible. This work did something to
make the method intelligible to all mathematicians: it is the
earliest book in which these curves are considered and defined
as curves of the second degree.

\nblabel{png307a}The most important of Wallis's works was his \emph{Arithmetica
Infinitorum}, which was published in 1656. In this treatise
the methods of analysis of Descartes and \index{Cavalieri!ref.\ to}Cavalieri were
systematised and greatly extended, but their logical exposition
is open to criticism. It at once became the standard book
on the subject, and is constantly referred to by subsequent
writers. It is prefaced by a short tract on conic sections.
\nblabel{png307b}He commences by proving the \index{Exponents}\index{Indices}law of indices; shews that
$x^0, x^{-1}, x^{-2} \ldots$ represents $1, 1/x, 1/x^2 \ldots$; that $x^{1/2}$ represents the
\index{Roots, square, cube, \&c.}\index{Square root, symbols for}\index{Surds, symbols for}square root of $x$, that $x^{2/3}$ represents the cube root of $x^{2}$, and
generally that $x^{-n}$ represents the reciprocal of $x^{n}$, and that
$x^{p/q}$ represents the $q$th root of $x^p$.

Leaving the numerous algebraical applications of this discovery
he next proceeds to find, by the method of indivisibles,
the area enclosed between the curve $y=x^{m}$, the axis of $x$, and
any ordinate $x=h$; and he proves that the ratio of this area
to that of the parallelogram on the same base and of the
same altitude is equal to the ratio $1:m+1$. He apparently
assumed that the same result would be true also for the
curve $y=ax^{m}$, where $a$ is any constant, and $m$ any number
positive or negative; but he only discusses the case of the
\index{Parabola,!quadrature of}\index{Quadrature of curves!parabolas}parabola in which $m=2$, and that of the hyperbola in which
$m=-1$: in the latter case his interpretation of the result
is incorrect. He then shews that similar results might be
written down for any curve of the form $y = \sum ax^m$; and hence
that, if the ordinate $y$ of a curve can be expanded in powers
\index{Infinite series, quadrature of curves in}of the abscissa $x$, its \index{Quadrature of curves}quadrature can be determined: thus he
says that if the equation of a curve were $y = x^0 + x^1 + x^2 + \ldots$,
its area would be $x + \frac{1}{2}x^2 + \frac{1}{3}x^3 + \ldots$. He then applies this
to the quadrature of the curves $y = (x-x^2)^0$, $y = (x-x^2)^1$,
$y = (x-x^2)^2$, $y = (x-x^2)^3$, etc.\ taken between the limits $x = 0$ and
$x = 1$; and shews that the areas are respectively $1$, $\frac{1}{6}$, $\frac{1}{30}$, $\frac{1}{140}$,
etc. He next considers curves of the form $y = x^{-m}$ and establishes
the theorem that the area bounded by the curve, the
axis of $x$, and the ordinate $x = 1$, is to the area of the rectangle
on the same base and of the same altitude as $m : m+1$. This
is equivalent to finding the value of $\int_0^1 x^{1/m}\,dx$. He illustrates
this by the parabola in which $m = 2$. He states, but does
not prove, the corresponding result for a curve of the form
$y = x^{p/q}$.

\nblabel{png308a}\index{Wallis!ref.\ to}Wallis shewed considerable ingenuity in reducing the equations
of curves to the forms given above, but, as he was unacquainted
with the binomial theorem, he could not effect the quadrature of
the circle, whose equation is $y = (x-x^2)^{1/2}$, since he was unable to
expand this in powers of $x$. He laid down, however, the principle
of \index{Interpolation, method of}interpolation. Thus, as the ordinate of the circle $y = (x-x^2)^{1/2}$
is the geometrical mean between the ordinates of the curves
$y = (x-x^2)^0$ and $y = (x-x^2)^1$, it might be supposed that, as an
approximation, the area of the semicircle $\int_0^1 (x-x^2)^{1/2}\,dx$, which
is \index{P1@$\pi$,!z1@value of}$\frac{1}{8}\pi$, might be taken as the geometrical mean between the
values of
\[
  \int_0^1 (x-x^2)^0\,dx \quad \text{and}\quad  \int_0^1 (x-x^2)^1\,dx,
\]
that is, 1 and $\frac{1}{6}$; this is equivalent to taking $4\sqrt{\frac{2}{3}}$ or $3 \cdot 26\ldots$
as the value of $\pi$. But, Wallis argued, we have in fact a
series 1, $\frac{1}{6}$, $\frac{1}{30}$, $\frac{1}{140}$, $\ldots$, and therefore the term interpolated
between 1 and $\frac{1}{6}$ ought to be so chosen as to obey the law
of this series. This, by an elaborate method, which I need not
describe in detail, leads to a value for the interpolated term
which is equivalent to taking
\[
\pi=2\frac{2 \ctrdot 2 \ctrdot 4 \ctrdot 4 \ctrdot 6 \ctrdot 6 \ctrdot 8 \ctrdot 8 \ldots}
          {1 \ctrdot 3 \ctrdot 3 \ctrdot 5 \ctrdot 5 \ctrdot 7 \ctrdot 7 \ctrdot 9 \ldots}
\]
The mathematicians of the seventeenth century constantly used
interpolation to obtain results which we should attempt to obtain
by direct analysis.

In this work also the formation and properties of continued
fractions are discussed, the subject having been brought into
prominence by Brouncker's use of these fractions.

A few years later, in 1659, \index{Wallis!ref.\ to}Wallis published a tract containing
the solution of the problems on the \index{Cycloid}cycloid which had
been proposed by \index{Pascal!ref.\ to}Pascal. \nblabel{png309a}In this he incidentally explained
how the principles laid down in his \emph{Arithmetica Infinitorum}
 could be used for the \index{Rectification of curves}\index{Curves, rectification of}rectification of algebraic curves; and
gave a solution of the problem to rectify the semi-cubical
\index{Parabola,!rectification of}parabola $x^{3}=ay^{2}$, which had been discovered in 1657 by his
pupil \index{Neil}William Neil. Since all attempts to rectify the ellipse
and hyperbola had been (necessarily) ineffectual, it had been
supposed that no curves could be rectified, as indeed \index{Descartes!ref.\ to}Descartes
had definitely asserted to be the case. The logarithmic spiral
had been rectified by \index{Torricelli!ref.\ to}Torricelli, and was the first curved line
(other than the circle) whose length was determined by mathematics,
but the extension by Neil and Wallace to an algebraical
curve was novel. The cycloid was the next curve rectified; this
was done by \index{Wren!ref.\ to}Wren in 1658.

Early in 1658 a similar discovery, independent of that of
Neil, was made by \index{Heuraet, van}\index{Vanh@Van Heuraët}van Heuraët,\footnote
  {On van~Heuraët, see the \emph{Bibliotheca Mathematica}, 1887, vol.~i,
  pp.~76--80.}
and this was published by
\index{Schooten, van!ref.\ to}\index{Vans@Van Schooten!ref.\ to}van~Schooten in his edition of Descartes's \emph{Geometria} in 1659.
Van~Heuraët's method is as follows. He supposes the curve
to be referred to rectangular axes; if this be so, and if $(x,y)$
be the co-ordinates of any point on it, and $n$ the length of the
normal, and if another point whose co-ordinates are $(x,\eta)$ be
taken such that $\eta:h=n:y$ where $h$ is a constant; then, if $ds$
be the element of the length of the required curve, we have by
similar triangles $ds:dx=n:y$. Therefore $hds=\eta dx$. Hence,
if the area of the locus of the point $(x,\eta)$ can be found, the
first curve can be rectified. In this way \index{Heuraet, van}\index{Vanh@Van Heuraët}van~Heuraët effected
the rectification of the curve $y^{3}=ax^{2}$; but added that the
rectification of the parabola $y^{2}=ax$ is impossible since it
requires the quadrature of the hyperbola. The solutions given
by Neil and \index{Wallis!ref.\ to}Wallis are somewhat similar to that given by van~Heuraët,
though no general rule is enunciated, and the analysis
is clumsy. A third method was suggested by \index{Fermat!ref.\ to}Fermat in 1660,
but it is inelegant and laborious.

The theory of the \index{Collision of bodies}collision of bodies was propounded by
the Royal Society in 1668 for the consideration of mathematicians.
Wallis, \index{Wren!ref.\ to}Wren, and Huygens sent correct and
similar solutions, all depending on what is now called the
conservation of momentum; but, while Wren and \index{Huygens!ref.\ to}Huygens
confined their theory to perfectly elastic bodies, Wallis considered
also imperfectly elastic bodies. This was followed in
1669 by a work on statics \index{Centres of mass}\index{Gravity,!centres of}\index{Mass, centres of}(centres of gravity), and in 1670 by
one on dynamics: these provide a convenient synopsis of what
was then known on the subject.

\nblabel{png310a}In 1685 Wallis published an \index{Algebra.!z8@Developed by (amongst others) Descartes!x1@Wallis}\emph{Algebra}, preceded by a
\index{Algebraa@Algebra,!histories of}historical account of the development of the subject, which
contains a great deal of valuable information. The second
edition, issued in 1693 and forming the second volume of his
\emph{Opera}, was considerably enlarged. This algebra is noteworthy
as containing the first systematic use of formulae. A given
magnitude is here represented by the numerical ratio which
it bears to the unit of the same kind of magnitude: thus,
when Wallis wants to compare two lengths he regards each as
containing so many units of length. This perhaps will be
made clearer if I say that the relation between the space
described in any time by a particle moving with a uniform
velocity would be denoted by Wallis by the formula $s=vt$,
where $s$ is the number representing the ratio of the space
described to the unit of length; while previous writers would
have denoted the same relation by stating what is equivalent\index{Newtons@Newton's \emph{Principia}!ref.\ to}\footnote
  {See \emph{ex.~gr.}\ Newton's \emph{Principia}, bk.~i, sect.~i, lemma~10 or 11.}
to the proposition $s_1:s_2=v_1t_1:v_2t_2$. It is curious to note that
\index{Wallis!ref.\ to}Wallis rejected as absurd the now usual idea of a negative
number as being less than nothing, but accepted the view that
it is something greater than infinity. The latter opinion may
be tenable and not inconsistent with the former, but it is hardly
a more simple one.\index{Wallis|)}

\index{Fermat|(}\nblabel{png311a}\nbparagraph{Fermat.\footnote
  {The best edition of Fermat's works is that in three volumes, edited by
  \index{Tannery, S. P., ref.\ to}S.~P.~Tannery and \index{Henry C., ref.\ to}C.~Henry, and published by the French government,
  1891--6. Of earlier editions, I may mention one of his papers and correspondence,
  printed at Toulouse in two volumes, 1670 and 1679: of which a
  summary, with notes, was published by E.~Brassinne at Toulouse in 1853,
  and a reprint was issued at Berlin in 1861.}}
While \index{Descartes!ref.\ to}Descartes was laying the foundations of
analytical geometry, the same subject was occupying the attention
of another and not less distinguished Frenchman. This
was Fermat. \emph{Pierre de Fermat}, who was born near Montauban
in 1601, and died at Castres on January~12, 1665, was the son
of a leather-merchant; he was educated at home; in 1631 he
obtained the post of councillor for the local parliament at
Toulouse, and he discharged the duties of the office with scrupulous
accuracy and fidelity. There, devoting most of his leisure
to mathematics, he spent the remainder of his life---a life which,
but for a somewhat acrimonious dispute with Descartes on the
validity of certain analysis used by the latter, was unruffled by
any event which calls for special notice. The dispute was chiefly
due to the obscurity of Descartes, but the tact and courtesy of
Fermat brought it to a friendly conclusion. Fermat was a good
scholar, and amused himself by conjecturally restoring the work
of \index{Apollonius!ref.\ to}Apollonius on plane loci.

Except a few isolated papers, Fermat published nothing in
his lifetime, and gave no systematic exposition of his methods.
Some of the most striking of his results were found after his
death on loose sheets of paper or written in the margins of
works which he had read and annotated, and are unaccompanied
by any proof. It is thus somewhat difficult to estimate the
character of his investigations. He was constitutionally modest
and retiring, and does not seem to have intended his papers
to be published. It is probable that he revised his notes as
occasion required, and that his published works represent the
final form of his researches, and therefore cannot be dated much
earlier than 1660. I shall consider separately (i)~his investigations
in the \index{Numbers, theory of.!Treatment of, by Pythagoras!q3@by Fermat|(}theory of numbers; (ii)~his use in geometry of
analysis and of infinitesimals; and (iii)~his method of treating
questions of probability.

\nblabel{png312a}(i)~\emph{The theory of numbers} appears to have been the favourite
study of Fermat. He prepared an edition of \index{Diophantus!ref.\ to}Diophantus, and
the notes and comments thereon contain numerous theorems of
considerable elegance. Most of the proofs of Fermat are lost,
and it is possible that some of them were not rigorous---an
induction by analogy and the intuition of genius sufficing to
lead him to correct results. The following examples will illustrate these investigations.

(a) If $p$ be a prime and $a$ be prime to $p$, then $a^{p-1}-1$ is
divisible by $p$, that is, $a^{p-1}-1\equiv 0 \pmod{p}$. A proof of this,
first given by \index{Euler!ref.\ to}Euler, is well known. A more general theorem
is that $a^{\phi(n)}-1\equiv 0 \pmod{n}$ %[**TN: 'a^{\phi(n)-1}-1\equiv 0 \pmod{n}' in original text]
where $a$ is prime to $n$, and $\phi(n)$ is
the number of integers less than $n$ and prime to it.

(b) An odd prime can be expressed as the difference of two
square integers in one and only one way. Fermat's proof is as
follows. Let $n$ be the prime, and suppose it equal to $x^{2}-y^{2}$,
that is, to $(x+y)(x-y)$. Now, by hypothesis, the only integral
factors of $n$ are $n$ and unity, hence $x+y=n$ and $x-y=1$.
Solving these equations we get $x=\frac{1}{2}(n+1)$ and $y=\frac{1}{2}(n-1)$.

(c) He gave a proof of the statement made by Diophantus
that the sum of the squares of two integers cannot be of the
form $4n-1$; and he added a corollary which I take to mean
that it is impossible that the product of a square and a prime
of the form $4n-1$ [even if multiplied by a number prime to the
latter], can be either a square or the sum of two squares.
For example, $44$ is a multiple of $11$ (which is of the form
$4 × 3 - 1$) by $4$, hence it cannot be expressed as the sum of two
squares. He also stated that a number of the form $a^{2}+b^{2}$,
where $a$ is prime to $b$, cannot be divided by a prime of the
form $4n-1$.

(d) Every prime of the form $4n+1$ is expressible, and that
in one way only, as the sum of two squares. This problem was
first solved by \index{Euler!ref.\ to}Euler, who shewed that a number of the form
$2^{m}(4n+1)$ can be always expressed as the sum of two squares.

(e) If $a, b, c,$ be integers, such that $a^{2}+b^{2}= c^{2}$, then $ab$
cannot be a square. \index{Lagrange!ref.\ to}Lagrange gave a solution of this.

(f) The determination of a number $x$ such that $x^{2}n+1$ may
be a square, where $n$ is a given integer which is not a square.
Lagrange gave a solution of this.

(g) There is only one integral solution of the equation
$x^{2}+2= y^{3}$; and there are only two integral solutions of the
equation $x^{2}+4=y^{3}$. The required solutions are evidently for
the first equation $x=5$, and for the second equation $x=2$ and
$x=11$. This question was issued as a challenge to the English
mathematicians \index{Wallis!ref.\ to}Wallis and \index{Digby}Digby.

(h) No integral values of $x, y, z$ can be found to satisfy the
equation $x^{n}+y^{n}=z^{n},$ if $n$ be an integer greater than 2. This
proposition\footnote
  {On this curious proposition, see \index{Mordell, L. J.}L.~J.~Mordell, \emph{Fermat's Last Theorem},
  Cambridge, 1921; \index{Dickson, L. E.}L.~E.~Dickson, \emph{History of the Theory of Numbers}, vol.~ii
  chap.~26, Washington, 1920.}
has acquired extraordinary celebrity from the fact
that no general demonstration of it has been given, but there
is no reason to doubt that it is true.

Probably Fermat discovered its truth first for the case $n=3$,
and then for the case $n=4$. His proof for the former of these
cases is lost, but that for the latter is extant, and a similar
proof for the case of $n=3$ can be given. These proofs depend
upon shewing that, if three integral values of $x, y, z$ can be
found which satisfy the equation, then it will be possible to find
three other and smaller integers which also satisfy it: in this
way, finally, we shew that the equation must be satisfied by three
values which obviously do not satisfy it. Thus no integral solution
is possible. This method is inapplicable to the general case.

Fermat's discovery of the general theorem was made later. A
proof can be given on the assumption that a number can be
resolved into the product of powers of primes in one and only
one way. It is possible that Fermat's argument rested on some
such supposition, but this is an unsupported conjecture. The
assumption is true of real integers, but is not necessarily true
for algebraic integers---an algebraic integer being defined as a
root of an algebraic equation $x^{n}+a_{1}x^{n-1} + \ldots + a_{n} = 0$, whose
coefficients, $a$, are arithmetical integers; for instance, $a + b \sqrt{ -m},
$ where $a, b,$ and $m$ are arithmetical integers, is an algebraic
integer. Thus, admitting the use of these generalised integers,
$21$ can be expressed in three ways as the product of primes,
namely, of $3$ and $7$, or of $4+\sqrt{-5}$ and $4 - \sqrt{-5}$, or of
$1 + 2\sqrt{-5} $ and $1-2\sqrt{-5}$; and similarly there are values of $n$
for which Fermat's equation leads to expressions which can be
factorised in more than one way.

In 1823 \index{Legendre!ref.\ to}Legendre obtained a proof for the case of $n = 5$; in
1832 \index{Dirichlet, Lejeune!ref.\ to}Dirichlet gave one for $n=14$, and in 1840 \index{Lame@Lamé}Lamé and
\index{Lebesgue}Lebesgue gave proofs for $n = 7$. In 1849 \index{Kummer!ref.\ to}Kummer proved
the truth of the theorem for all numbers which satisfy certain
Bernoullian conditions. The only numbers less than $100$ which
do not do so are $37, 59, 67,$ and the theorem can, by other
arguments, be proved for these three cases. It has now been extended
far beyond 100, and no exception to it has yet been found.
The following extracts, from a letter now in the university
library at Leyden, will give an idea of Fermat's methods; the
letter is undated, but it would appear that, at the time Fermat
wrote it, he had proved the proposition $(h)$ above only for the
case when $n= 3$.

\begin{nbQuote}
Je ne m'en servis au commencement que pour demontrer les propositions
negatives, comme par exemple, qu'il n'y a auc\=u nombre moindre
de l'unité qu'un multiple de $3$ qui soit composé d'un quarré et du triple
d'un autre quarré. Qu'il n'y a aucun triangle rectangle de nombres dont
l'aire soit un nombre quarré. La preuve se fait par
\mytextgreek{>apagwg`hn t`hn e>is >ad'unaton}
en cette manière. S'il y auoit aucun triangle rectangle en
nombres entiers, qui eust son aire esgale à un quarré, il y auroit un
autre triangle moindre que celuy la qui auroit la mesme proprieté. S'il
y en auoit un second moindre que le premier qui eust la mesme proprieté
il y en auroit par un pareil raisonnement un troisieme moindre
que ce second qui auroit la mesme proprieté et enfin un quatrieme, un
cinquieme etc.\ a l'infini en descendant. Or est il qu'estant donné un
nombre il n'y en a point infinis en descendant moindres que celuy la,
j'entens parler tousjours des nombres entiers. D'ou on conclud qu'il est
donc impossible qu'il y ait aucun triangle rectangle dont l'aire soit
quarré. Vide foli\={u} post sequens\ldots.

Je fus longtemps sans pouuoir appliquer ma methode aux questions
affirmatiues, parce que le tour et le biais pour y venir est beaucoup plus
malaisé que celuy dont je me sers aux negatives. De sorte que lors qu'il
me falut demonstrer que tout nombre premier qui surpasse de l'unité un
multiple de 4, est composé de deux quarrez je me treuuay en belle peine.
Mais enfin une meditation diverses fois reiterée me donna les lumieres qui
me manquoient. Et les questions affirmatiues passerent par ma methode
a l'ayde de quelques nouueaux principes qu'il y fallust joindre par
necessité. Ce progres de mon raisonnement en ces questions affirmatives
estoit tel. Si un nombre premier pris a discretion qui surpasse de l'unité
un multiple de 4 n'est point composé de deux quarrez il y aura un nombre
premier de mesme nature moindre que le donné; et ensuite un troisieme
encore moindre, etc.\ en descendant a l'infini jusques a ce que uous arriviez
au nombre 5, qui est le moindre de tous ceux de cette nature, lequel il s'en
suivroit n'estre pas composé de deux quarrez, ce qu'il est pourtant d'ou on
doit inferer par la deduction a l'impossible que tous ceux de cette nature
sont par consequent composez de 2 quarrez.

Il y a infinies questions de cette espece. Mais il y en a quelques
autres qui demandent de nouveaux principes pour y appliquer la descente,
et la recherche en est quelques fois si mal aisée, qu'on n'y peut venir
qu'auec une peine extreme. Telle est la question suiuante que \index{Bachet!ref.\ to}Bachet sur
\index{Diophantus!ref.\ to}Diophante avoüe n'avoir jamais peu demonstrer, sur le suject de laquelle
M$^{\text{r}}$. \index{Descartes!ref.\ to}Descartes fait dans une de ses lettres la mesme declaration, jusques la
qu'il confesse qu'il la juge si difficile, qu'il ne voit point de voye pour la
resoudre. Tout nombre est quarré, ou composé de deux, de trois, ou de
quatre quarréz. Je l'ay enfin rangée sous ma methode et je demonstre
que si un nombre donné n'estoit point de cette nature il y en auroit un
moindre qui ne le seroit pas non plus, puis un troisieme moindre que le
second \&{}c.\ a l'infini, d'ou l'on infere que tous les nombres sont de cette
nature\ldots.

{\nbstretchyspace J'ay ensuite consideré certaines questions qui bien que negatives ne
restent pas de receuoir tres-grande difficulté, la methode pour y pratiquer
la descente estant tout a fait diuerse des precedentes comme il sera aisé
d'esprouuer.} Telles sont les suiuantes. Il n'y a aucun cube diuisible en
deux cubes. Il n'y a qu'un seul quarré en entiers qui augmenté du binaire
fasse un cube, ledit quarré est 25. II n'y a que deux quarrez en entiers
lesquels augmentés de 4 fassent cube, lesdits quarrez sont 4 et 121\ldots.

Apres auoir couru toutes ces questions la plupart de diuerses (\emph{sic}) nature
et de differente façon de demonstrer, j'ay passé a l'inuention des regles
generales pour resoudre les equations simples et doubles de Diophante.
On propose par exemple $2 \text{ quarr.\,} + 7957$ esgaux a un quarré (hoc est
$2xx + 7967\propto\text{quadr.}$) J'ay une regle generale pour resoudre cette equation
si elle est possible, ou decouvrir son impossibilité. Et ainsi en tons les cas et
en tous nombres tant des quarrez que des unitez. On propose cette
equation double $2x + 3$ et $3x + 5$ esgaux chaucon a un quarré. \index{Bachet!ref.\ to}Bachet se
glorifie en ses commentaires sur Diophante d'auoir trouvé une regle en deux
cas particuliers. Je la donne generale en toute sorte de cas. Et determine
par regle si elle est possible ou non\ldots.

Voila sommairement le conte de mes recherches sur le suject des
nombres. Je ne l'ay escrit que parce que j'apprehende que le loisir
d'estendre et de mettre au long toutes ces demonstrations et ces méthodes %[**TN: 'methodes' changed to 'méthodes' for consistency]
me manquera. En tout cas cette indication seruira aux sçauants pour
trouver d'eux mesmes ce que je n'estens point, principalement si \index{Carcaui}\index{Frenicle@Frénicle!ref.\ to}M\textsuperscript{r}. de
Carcaui et Frénicle leur font part de quelques demonstrations par la %[**TN: 'Frenicle' changed to 'Frénicle' for consistency]
descente que je leur ay enuoyees sur le suject de quelques propositions
negatiues. Et peut estre la posterité me scaura gré de luy avoir fait
connoistre que les anciens n'ont pas tout sceu, et cette relation pourra
passer dans l'esprit de ceux qui viendront apres moy pour traditio
lampadis ad filios, comme parle le grand Chancelier d'Angleterre, suiuant
le sentiment et la deuise duquel j'adjousteray, multi pertransibunt et
augebitur scientia.
\end{nbQuote}\index{Numbers, theory of.!Treatment of, by Pythagoras!q3@by Fermat|)}

\nblabel{png316a}(ii)~I next proceed to mention Fermat's \emph{use in geometry of
analysis and of infinitesimals}. It would seem from his correspondence
that he had thought out the principles of analytical
geometry for himself before reading \index{Descartes!ref.\ to}Descartes's \emph{Géométrie}, and
had realised that from the equation, or, as he calls it, the
``specific property,'' of a curve all its properties could be
deduced. His extant papers on geometry deal, however, mainly
with the application of infinitesimals to the determination of the
tangents to curves, to the quadrature of curves, and to questions
of \index{Maxima and minima, determination of}maxima and minima; probably these papers are a revision of
his original manuscripts (which he destroyed), and were written
about 1663, but there is no doubt that he was in possession of
the general idea of his method for finding maxima and minima
as early as 1628 or 1629.

He obtained the \index{Subtangent}subtangent to the ellipse, \index{Cycloid}cycloid, cissoid,
conchoid, and quadratrix by making the ordinates of the curve
and a straight line the same for two points whose abscissae were
$x$ and $x-e$; but there is nothing to indicate that he was aware
that the process was general, and, though in the course of his
work he used the principle, it is probable that he never separated
it, so to speak, from the symbols of the particular problem he
was considering. The first definite statement of the method was
due to \index{Barrow!ref.\ to}Barrow,\footnote
  {See below, pp.~\pageref{xref:311}--\pageref{xref:312}.}
and was published in 1669.

\index{Conics (Analytical).!q1@and by Fermat}\index{Parabola,!quadrature of}\index{Quadrature of curves!parabolas}\index{Quadrature of curves}Fermat also obtained the areas of parabolas and hyperbolas
of any order, and determined the \index{Centres of mass}\index{Gravity,!centres of}\index{Mass, centres of}centres of mass of a few simple
laminae and of a paraboloid of revolution. As an example of
his method of solving these questions I will quote his solution of
the problem to find the area between the parabola $y^3 = px^2$, the
axis of $x$, and the line $x=a$. He says that, if the several ordinates
at the points for which $x$ is equal to $a$, $a(1-e)$, $a(1-e)^2$, $\ldots$
be drawn, then the area will be split into a number of little
rectangles whose areas are respectively
\[
  ae(pa^2)^{1/3},\, ae(1-e) \{pa^2(1-e)^2\}^{1/3},\, \ldots.
\]
The sum of these is $p^{1/3} a^{5/3} e / \{1-(1-e)^{5/3}\}$; and by a subsidiary
proposition (for he was not acquainted with the binomial
theorem) he finds the limit of this, when $e$ vanishes, to be
$\frac{3}{5} p^{1/3} a^{5/3}$. The theorems last mentioned were published only
after his death; and probably they were not written till after he
had read the works of \index{Cavalieri!ref.\ to}Cavalieri and \index{Wallis!ref.\ to}Wallis.

\index{Kepler!ref.\ to}Kepler had remarked that the values of a function immediately
adjacent to and on either side of a maximum (or minimum)
value must be equal. Fermat applied this principle to a few
examples. Thus, to find the maximum value of $x(a-x)$,
his method is essentially equivalent to taking a consecutive
value of $x$, namely $x-e$ where $e$ is very small, and putting
$x(a-x) = (x-e)(a-x+e)$. Simplifying, and ultimately putting
$e=0$, we get $x = \frac{1}{2} a$. This value of $x$ makes the given expression
a maximum.

\nblabel{png318a}(iii)~Fermat must share with \index{Pascal!ref.\ to}Pascal the honour of having
founded \index{Probabilities, theory of}\emph{the theory of probabilities}. I have already mentioned
the problem proposed to Pascal, and which he communicated
to Fermat, and have there given Pascal's solution. Fermat's
solution depends on the theory of combinations, and will be
sufficiently illustrated by the following example, the substance
of which is taken from a letter dated August~24, 1654, which
occurs in the correspondence with Pascal. Fermat discusses the
case of two players, \emph{A} and \emph{B}, where \emph{A} wants two points to win
and \emph{B} three points. Then the game will be certainly decided
in the course of four trials. Take the letters \emph{a} and \emph{b}, and write
down all the combinations that can be formed of four letters.
These combinations are 16 in number, namely, \emph{aaaa, aaab, aaba,
aabb}; \emph{abaa, abab, abba, abbb}; \emph{baaa, baab, baba, babb}; \emph{bbaa,
bbab, bbba, bbbb}. Now every combination in which \emph{a} occurs
twice or oftener represents a case favourable to \emph{A}, and every
combination in which \emph{b} occurs three times or oftener represents
a case favourable to \emph{B}. Thus, on counting them, it will be
found that there are 11 cases favourable to \emph{A}, and 5 cases
favourable to \emph{B}; and, since these cases are all equally likely, \emph{A}'s
chance of winning the game is to \emph{B}'s chance as 11 is to 5.

The only other problem on this subject which, as far as I
know, attracted the attention of Fermat was also proposed to
him by Pascal, and was as follows. A person undertakes to
throw a six with a die in eight throws; supposing him to have
made three throws without success, what portion of the stake
should he be allowed to take on condition of giving up his
fourth throw? Fermat's reasoning is as follows. The chance
of success is $1/6$, so that he should be allowed to take $1/6$ of the
stake on condition of giving up his throw. But, if we wish to
estimate the value of the fourth throw before any throw is
made, then the first throw is worth $1/6$ of the stake; the second
is worth $1/6$ of what remains, that is, $5/36$ of the stake; the
third throw is worth $1/6$ of what now remains, that is, $25/216$
of the stake; the fourth throw is worth $1/6$ of what now remains,
that is, $125/1296$ of the stake.

Fermat does not seem to have carried the matter much
further, but his correspondence with \index{Pascal!ref.\ to}Pascal shows that his views
on the fundamental principles of the subject were accurate:
those of Pascal were not altogether correct.

Fermat's reputation is quite unique in the history of science.
The problems on numbers which he had proposed long defied
all efforts to solve them, and many of them yielded only to the
skill of \index{Euler!ref.\ to}Euler. One still remains unsolved. This extraordinary
achievement has overshadowed his other work, but in fact it is
all of the highest order of excellence, and we can only regret
that he thought fit to write so little.\index{Fermat|)}

\index{Huygens|(}\nblabel{png319a}\nbparagraph{Huygens.\footnote
  {A new edition of all Huygens's works and correspondence was issued at
  the Hague in ten volumes, 1888--1905. An earlier edition of his works was
  published in six volumes, four at Leyden in 1724, and two at Amsterdam
  in 1728 (a life by \index{Sg@s'Gravesande on Huygens}\index{Gravesande, s', on Huygens}s'Gravesande is prefixed to the first volume): his scientific
  correspondence was published at the Hague in 1833.}}
\emph{Christian Huygens} was born at the Hague on
April~14, 1629, and died in the same town on June~8, 1695.
He generally wrote his name as Hugens, but I follow the usual
custom in spelling it as above: it is also sometimes written as
Huyghens. His life was uneventful, and there is little more
to record in it than a statement of his various memoirs and
researches.

In 1651 he published an essay in which he shewed the
fallacy in a system of quadratures proposed by \index{Saint-Vincent!ref.\ to}Grégoire de
Saint-Vincent, who was well versed in the geometry of the
Greeks, but had not grasped the essential points in the more
modern methods. This essay was followed by tracts on the
quadrature of the conics and the approximate rectification of
the circle.

In 1654 his attention was directed to the improvement of the
\index{Telescopes}telescope. In conjunction with his brother he devised a new
and better way of grinding and polishing lenses. As a result
of these improvements he was able during the following two
years, 1655 and 1656, to resolve numerous astronomical questions;
as, for example, the nature of Saturn's appendage. His
astronomical observations required some exact means of measuring
time, and he was thus led in 1656 to invent the pendulum \index{Clocks}clock,
as described in his tract \emph{Horologium}, 1658. The time-pieces
previously in use had been balance clocks.

In the year 1657 Huygens wrote a small work on the calculus
of \index{Probabilities, theory of}probabilities founded on the correspondence of \index{Pascal!ref.\ to}Pascal and
\index{Fermat!ref.\ to}Fermat. He spent a couple of years in England about this
time. His reputation was now so great that in 1665 \index{Louis XIV. of France, ref.\ to}Louis
XIV. offered him a pension if he would live in Paris, which
accordingly then became his place of residence.

\index{Mechanics.!z2@Development of,!x2@and by Huygens|(}In 1668 he sent to the Royal Society of London, in answer
to a problem they had proposed, a memoir in which (simultaneously
with \index{Wallis!ref.\ to}Wallis and \index{Wren!ref.\ to}Wren) he proved by experiment that
the momentum in a certain direction before the \index{Collision of bodies}collision of two
bodies is equal to the momentum in that direction after the
collision. This was one of the points in mechanics on which
\index{Descartes!ref.\ to}Descartes had been mistaken.

\index{Clocks}\nblabel{png320a}The most important of Huygens's work was his \emph{Horologium
Oscillatorium} published at Paris in 1673. The first chapter is
devoted to pendulum clocks. The second chapter contains a
complete account of the descent of heavy bodies under their own
weights in a vacuum, either vertically down or on smooth curves.
Amongst other propositions he shews that the \index{Cycloid}cycloid is \index{Tautochronous curve}tautochronous.
In the third chapter he defines \index{Evolutes}evolutes and
\index{Involutes}involutes, proves some of their more elementary properties, and
illustrates his methods by finding the evolutes of the cycloid
and the \index{Parabola,!evolute of}parabola. These are the earliest instances in which the
\index{Envelopes}envelope of a moving line was determined. In the fourth
chapter he solves the problem of the compound pendulum, and
shews that the \index{Oscillation, centre of}centres of oscillation and suspension are interchangeable.
In the fifth and last chapter he discusses again
the theory of clocks, points out that if the bob of the pendulum
were, by means of cycloidal checks, made to oscillate in a cycloid
the oscillations would be isochronous; and finishes by shewing
that the \index{Centrifugal force}centrifugal force on a body which moves round a circle
of radius $r$ with a uniform velocity $v$ varies directly as $v^{2}$
and inversely as $r$. This work contains the first attempt to
apply dynamics to bodies of finite size and not merely to
particles.

In 1675 \index{Huygens!ref.\ to}Huygens proposed to regulate the motion of watches
by the use of the balance spring, in the theory of which he had
been perhaps anticipated in a somewhat ambiguous and incomplete
statement made by Hooke in 1658. \index{Watches, invention of}Watches or portable
\index{Clocks}clocks had been invented early in the sixteenth century, and by
the end of that century were not very uncommon, but they were
clumsy and unreliable, being driven by a main spring and
regulated by a conical pulley and verge escapement; moreover,
until 1687 they had only one hand. The first watch whose
motion was regulated by a balance spring was made at Paris
under Huygens's directions, and presented by him to \index{Louis XIV. of France, ref.\ to}Louis XIV.\index{Mechanics.!z2@Development of,!x2@and by Huygens|)}

\index{Optics (geometrical).!z11@(physical)}The increasing intolerance of the Catholics led to his return
to Holland in 1681, and after the revocation of the edict of
Nantes he refused to hold any further communication with
France. He now devoted himself to the \index{Lenses, construction of}construction of lenses
of enormous focal length: of these three of focal lengths 123
feet, 180 feet, and 210 feet, were subsequently given by him to
the Royal Society of London, in whose possession they still
remain. It was about this time that he discovered the achromatic
eye-piece (for a \index{Telescopes}telescope) which is known by his name.
In 1689 he came from Holland to England in order to make
the acquaintance of \index{Newtons@Newton's \emph{Principia}!ref.\ to}Newton, whose \emph{Principia} had been published
in 1687. Huygens fully recognized the intellectual merits of
the work, but seems to have deemed any theory incomplete
which did not explain gravitation by mechanical means.

\nblabel{png321a}On his return in 1690 Huygens published his treatise on
\index{Light,!physical theories of|(}\emph{light} in which the \index{Undulatory theory (optics)|(}undulatory theory was expounded and
explained. Most of this had been written as early as 1678.
The general idea of the theory had been suggested by Robert
Hooke in 1664, but he had not investigated its consequences
in any detail. Only three ways have been suggested in which
light can be produced mechanically. Either the eye may be
supposed to send out something which, so to speak, feels the
object (as the Greeks believed); or the object perceived may
send out something which hits or affects the eye (as assumed in
the emission theory); or there may be some medium between
the eye and the object, and the object may cause some change
in the form or condition of this intervening medium and thus
affect the eye (as \index{Hooke!ref.\ to}Hooke and Huygens supposed in the \index{Wave theory (optics)}wave or
undulatory theory). According to this last theory space is filled
with an extremely rare \index{Ether, luminiferous}ether, and light is caused by a series of
waves or vibrations in this ether which are set in motion by the
pulsations of the luminous body. From this hypothesis Huygens
deduced the laws of reflexion and \index{Refraction}refraction, explained the
phenomena of double refraction, and gave a construction for
the extraordinary ray in biaxal crystals; while he found by
experiment the chief phenomena of \index{Polarization of light}polarization.\index{Undulatory theory (optics)|)}

The immense reputation and unrivalled powers of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton
led to disbelief in a theory which he rejected, and to the general
adoption of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's emission theory. Within the present
century crucial experiments have been devised which give different
results according as one or the other theory is adopted; all
these experiments agree with the results of the undulatory theory
and differ from the results of the Newtonian theory; the latter
is therefore untenable. \index{Diffraction}Until, however, the theory of \index{Interference, principle of}interference,
suggested by \index{Young, Thos.!ref.\ to}Young, was worked out by \index{Fresnel!ref.\ to}Fresnel, the
hypothesis of Huygens failed to account for all the facts, and
even now the properties which, under it, have to be attributed
to the intervening medium or ether involve difficulties of which
we still seek a solution. Hence the problem as to how the
effects of light are really produced cannot be said to be finally
solved.\index{Light,!physical theories of|)}

Besides these works Huygens took part in most of the controversies
and challenges which then played so large a part in
the mathematical world, and wrote several minor tracts. In one
of these he investigated the form and properties of the catenary.
In another he stated in general terms the rule for finding \index{Maxima and minima, determination of}maxima
and minima of which \index{Fermat!ref.\ to}Fermat had made use, and shewed that
the subtangent of an algebraical curve $f(x, y) = 0$ was equal to
$yf_{y}/f_{x}$ where $f_{y}$ is the derived function of $f(x, y)$ regarded as a
function of $y$. In some posthumous works, issued at Leyden in
1703, he further shewed how from the focal lengths of the
component lenses the magnifying power of a \index{Telescopes}telescope could be
determined; and explained some of the phenomena connected
with haloes and parhelia.

I should add that almost all his demonstrations, like those
of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton, are rigidly geometrical, and he would seem to have
made no use of the differential or fluxional calculus, though he
admitted the validity of the methods used therein. Thus, even
when first written, his works were expressed in an archaic
language, and perhaps received less attention than their intrinsic
merits deserved.

I have now traced the development of mathematics for a
period which we may take roughly as dating from 1635 to 1675,
under the influence of \index{Descartes!ref.\ to}Descartes, \index{Cavalieri!ref.\ to}Cavalieri, \index{Pascal!ref.\ to}Pascal, \index{Wallis!ref.\ to}Wallis, \index{Fermat!ref.\ to}Fermat,
and Huygens. The life of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton partly overlaps this
period; his works and influence are considered in the next
chapter.\index{Huygens|)}

\nblabel{png323a}I may dismiss \textbf{the remaining mathematicians of this time}\footnote
  {Notes on several of these mathematicians will be found in C.~Hutton's
  \emph{Mathematical Dictionary and Tracts}, 5 volumes, London, 1812--1815.}
with comparatively slight notice. The most eminent of them
are \index{Barrow!ref.\ to}\index{Torricelli!ref.\ to}\index{Mersenne!ref.\ to}\index{Roberval!ref.\ to}\emph{Bachet, Barrow, Brouncker, Collins, De~la~Hire, de~Laloubère,
\index{Frenicle@Frénicle!ref.\ to}Frénicle, James Gregory, \index{Hooke!ref.\ to}Hooke, Hudde, Nicholas Mercator,
Mersenne, Pell, Roberval, Roemer, Rolle, Saint-Vincent,
\index{Sluze, de!ref.\ to}Sluze, Torricelli, Tschirnhausen, \index{Schooten, van!ref.\ to}\index{Vans@Van Schooten!ref.\ to}van Schooten, Viviani}, and
\index{Wren!ref.\ to}\emph{Wren}. In the following notes I have arranged the above-mentioned
mathematicians so that as far as possible their chief
contributions shall come in chronological order.

\nblabel{xref:305}\index{Meziriac@Méziriac|(}\index{Demeziriac@De Méziriac|(}\index{Bachet|(}\nblabel{png323b}\nbparagraph{Bachet.} \emph{Claude Gaspard Bachet de Méziriac} was born at
Bourg in 1581, and died in 1638. He wrote the \emph{Problèmes
plaisants}, of which the first edition was issued in 1612, a
second and enlarged edition was brought out in 1624; this
contains an interesting collection of arithmetical tricks and
questions, many of which are quoted in \index{Ball, W. W. R., ref.\ to}my \index{Puzzles}\index{Recreations, mathematical}\emph{Mathematical Recreations
and Essays}. He also wrote \emph{Les éléments arithmétiques},
which exists in manuscript; and a translation of the \emph{Arithmetic}
of \index{Diophantus!ref.\ to}Diophantus. Bachet was the earliest writer who discussed
the solution of indeterminate equations by means of continued
fractions.\index{Bachet|)}\index{Demeziriac@De Méziriac|)}\index{Meziriac@Méziriac|)}

\index{Mersenne|(}\nblabel{png324a}\nbparagraph{Mersenne.} \emph{Marin Mersenne}, born in 1588 and died at Paris
in 1648, was a Franciscan friar, who made it his business to be
acquainted and correspond with the French mathematicians of
that date and many of their foreign contemporaries. In 1634
he published a translation of \index{Galileo!ref.\ to}Galileo's mechanics; in 1644 he
issued his \emph{Cogitata Physico-Mathematica}, by which he is best
known, containing an account of some experiments in physics;
he also wrote a synopsis of mathematics, which was printed in
1664.

\index{Primes|(}In the preface to the \emph{Cogitata} a statement is made about
\index{Numbers,!perfect|(}\index{Perfect numbers|(}perfect numbers, which implies that the only values of $p$ not
greater than 257 which make $N$ prime, where $N = 2^p - 1$, are 1,
2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257: all prime values
of $N$ are known as Mersenne's Numbers. Some years ago I
gave reasons for thinking that 67 was a misprint for 61. Until
1911, no error in this corrected statement was established, and
it was gradually verified for all except sixteen values of $p$. In
1911, however, it was proved that $N$ was prime when $p = 89$,
and three years later that it was prime when $p = 107$: two facts
at variance with Mersenne's statement. The prime or composite
character of N now remains unknown for only ten values of
$p$, namely, 139, 149, 157, 167, 193, 199, 227, 229, 241, and
257. We may safely say that the methods used to-day in
establishing the known results for many of the higher values of
$p$ could not have been employed by Mersenne. It would be
interesting to discover how he reached his conclusions, which are
true if $p$ does not exceed 88. Some recent writers conjecture
that his statement was the result of a guess, intelligent though
erroneous, as to the possible forms of $p$: I find it difficult to
accept this opinion, but further discussion of the problem would
be out of place here.

The theory of perfect numbers depends directly on that of
Mersenne's Numbers. \nblabel{xref:307}It is probable that all perfect numbers
are included in the formula $2^{p-1}(2^p-1)$, where $2^p-1$ is a
prime. \index{Euclid!ref.\ to}Euclid proved that any number of this form is
perfect; \index{Euler!ref.\ to}Euler shewed that the formula includes all even
perfect numbers; and there is reason to believe---though a
rigid demonstration is wanting---that an odd number cannot
be perfect. If we assume that the last of these statements is
true, then every perfect number is of the above form. Thus,
if p = 2, 3, 5, 7, 13, 17, 19, 31, 61, then, by Mersenne's rule,
the corresponding values of $2^p-1$ are prime; they are 3, 7, 31,
127, 8191, 131071, 524287, 2147483647, 2305843009213693951;
and the corresponding perfect numbers are 6, 28, 496, 8128,
33550336, 8589869056, 137438691328, 2305843008139952128,
and 2658455991569831744654692615953842176.\index{Perfect numbers|)}\index{Numbers,!perfect|)}\index{Mersenne|)}\index{Primes|)}

\index{Roberval}\nblabel{png325a}\nbparagraph{Roberval.\footnote
  {His chief works are included in the \emph{Divers Ouvrages} by Academicians,
  Paris, 1693; these were reprinted in the sixth volume of the old mémoires of
  the Academy of Sciences, Paris, 1730.}}
\emph{Gilles Personier (de) Roberval}, born at Roberval
in 1602 and died at Paris in 1675, described himself from the
place of his birth as de Roberval, a seigniorial title to which he
had no right. He discussed the nature of the \index{Tangent (geometrical)}tangents to
curves, solved some of the easier questions connected with the
\index{Cycloid}cycloid, generalized \index{Archimedes!ref.\ to}Archimedes's theorems on the spiral, wrote
on mechanics, and on the method of \index{Indivisibles, method of}indivisibles, which he rendered
more precise and logical. He was a professor in the university
of Paris, and in correspondence with nearly all the leading
mathematicians of his time.

\index{Vans@Van Schooten}\index{Schooten, van}\nbparagraph{Van Schooten.} \emph{Frans van Schooten}, to whom we owe an
edition of \index{Vieta!ref.\ to}Vieta's works, succeeded his father (who had taught
mathematics to \index{Huygens!ref.\ to}Huygens, \index{Hudde!ref.\ to}Hudde, and \index{Sluze, de!ref.\ to}\index{Desluze@De Sluze!ref.\ to}Sluze) as professor at
Leyden in 1646. He brought out in 1659 a Latin translation of
\index{Descartes!ref.\ to}Descartes's \emph{Géométrie}, and in 1657 a collection of mathematical
exercises in which he recommended the use of co-ordinates in
space of three dimensions. He died in 1661.

\index{Saint-Vincent|(}\nbparagraph{Saint-Vincent.\footnote
  {See \index{Quetelet, ref.\ to}L.~A.~J.~Quetelet's \emph{Histoire des sciences chez les Belges}, Brussels, 1866.}}
\emph{Grégoire de Saint-Vincent}, a Jesuit, born
at Bruges in 1584 and died at Ghent in 1667, discovered the
expansion of \index{Expansion,!oflog@of $\log(1+x)$}$\log (1+x)$ in ascending powers of $x$. Although a
circle-squarer he is worthy of mention for the numerous theorems
of interest which he discovered in his search after the impossible,
and \index{Montucla!ref.\ to}Montucla ingeniously remarks that ``no one ever squared
the circle with so much ability or (except for his principal object)
with so much success.'' He wrote two books on the subject, one
published in 1647 and the other in 1668, which cover some two
or three thousand closely printed pages; the fallacy in the \index{Quadrature of curves}quadrature
was pointed out by \index{Huygens!ref.\ to}Huygens. In the former work he
used indivisibles. An earlier work entitled \emph{Theoremata Mathematica},
published in 1624, contains a clear account of the method
of exhaustions, which is applied to several quadratures, notably
that of the hyperbola.\index{Saint-Vincent|)}

\index{Torricelli}\nblabel{png326a}\nbparagraph{Torricelli.\footnote
  {\index{Torricelli!ref.\ to}Torricelli's mathematical writings were published at Florence in 1644,
  under the title \emph{Opera Geometrica}; see also a memoir by \index{Loria, ref.\ to}G.~Loria, \emph{Bibliotheca
  mathematica}, series 3, vol.~i, pp.~75--89, Leipzig, 1900.}}
\emph{Evangelista Torricelli}, born at Faenza on
Oct.~15, 1608, and died at Florence in 1647, wrote on the
quadrature of the \index{Cycloid}cycloid and conics; the rectification of the
logarithmic spiral; the theory of the \index{Barometer, invention of}barometer; the value of
gravity found by observing the motion of two weights connected
by a string passing over a fixed pulley; the theory of projectiles;
and the motion of fluids.

\index{Hudde}\nbparagraph{Hudde.} \emph{Johann Hudde}, burgomaster of Amsterdam, was
born there in 1633, and died in the same town in 1704. He
wrote two tracts in 1659: one was on the reduction of equations
which have equal roots; in the other he stated what is equivalent
to the proposition that if $f(x, y) = 0$ be the algebraical
equation of a curve, then the \index{Subtangent}subtangent is $-y\dfrac{\partial f}{\partial y} \Big/ \dfrac{\partial f}{\partial x}$; but being
ignorant of the notation of the calculus his enunciation is
involved.

\index{Frenicle@Frénicle|(}\nbparagraph{Frénicle.\footnote
  {Frénicle's miscellaneous works, edited by \index{Lahire!ref.\ to}\index{Hire, De la!ref.\ to}\index{Delahire@De la Hire! ref.\ to}De~la~Hire, were published in
  the \emph{Mémoires de l'Académie}, vol.~v, 1691.}}
\emph{Bernard Frénicle de Bessy}, born in Paris circ.~1605
and died in 1670, wrote numerous papers on combinations
and on the theory of numbers, also on \index{Magic squares}magic squares. It may
be interesting to add that he challenged \index{Huygens!ref.\ to}Huygens to solve the
following system of equations in integers, $x^2+y^2=z^2$, $x^2=u^2+v^2$,
$x-y=u-v$. A solution was given by \index{Pepin@Pépin on Frénicle's problem}M.~Pépin in 1880.\index{Frenicle@Frénicle|)}

\index{Laloubère}\index{Delaloubere@De Laloubère}\nblabel{png327a}\nbparagraph{De Laloubère.} \emph{Antoine de~Laloubère}, a Jesuit, born in
Languedoc in 1600 and died at Toulouse in 1664, is chiefly
celebrated for an incorrect solution of \index{Pascal!ref.\ to}Pascal's problems on
the \index{Cycloid}cycloid, which he gave in 1660, but he has a better claim
to distinction in having been the first mathematician to study
the properties of the \index{Helix}helix.

\index{Mercator, N.}\index{Kauffmann (or Mercator)}\nbparagraph{N.~Mercator.} \emph{Nicholas Mercator} (sometimes known as
\emph{Kauffmann}) was born in Holstein about 1620, but resided most
of his life in England. He went to France in 1683, where he
designed and constructed the fountains at Versailles, but the
payment agreed on was refused unless he would turn Catholic;
he died of vexation and poverty in Paris in 1687. He wrote a
treatise on \index{Expansion,!oflog@of $\log(1+x)$}logarithms entitled \emph{Logarithmo-technica}, published in
1668, and discovered the series
\[
\log (1+x) = x - \frac{1}{2} x^2 + \frac{1}{3} x^3 - \frac{1}{4} x^4 + \ldots;
\]
he proved this by writing the equation of a hyperbola in the form
\[
y = \frac{1}{1+x} = 1 - x + x^2 - x^3 + \ldots ,
\]
to which \index{Wallis!ref.\ to}Wallis's method of quadrature could be applied. The
same series had been independently discovered by \index{Saint-Vincent!ref.\ to}Saint-Vincent.

\index{Barrow|(}\index{Barrow!ref.\ to}\nbparagraph{Barrow.\footnote
  {Barrow's mathematical works, edited by W.~Whewell, were issued at
  Cambridge in 1860. On \emph{Barrow's Geometry}, see \index{Child, J. M.}J.~M.~Child, Chicago, 1916.}}
\emph{Isaac Barrow} was born in London in 1630,
and died at Cambridge in 1677. He went to school first at
Charterhouse (where he was so troublesome that his father was
heard to pray that if it pleased God to take any of his children
he could best spare Isaac), and subsequently to Felstead. He
completed his education at Trinity College, Cambridge; after
taking his degree in 1648, he was elected to a fellowship in
1649; he then resided for a few years in college, but in 1655 he
was driven out by the persecution of the Independents. He
spent the next four years in the East of Europe, and after many
adventures returned to England in 1659. He was ordained
the next year, and appointed to the professorship of Greek at
Cambridge. In 1662 he was made professor of geometry at
Gresham College, and in 1663 was selected as the first occupier
of the Lucasian chair at Cambridge. He resigned the latter
to his pupil \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton in 1669, whose superior abilities he recognized
and frankly acknowledged. For the remainder of his
life he devoted himself to the study of divinity. He was
appointed master of Trinity College in 1672, and held the post
until his death.

He is described as ``low in stature, lean, and of a pale complexion,''
slovenly in his dress, and an inveterate smoker. He
was noted for his strength and courage, and once when travelling
in the East he saved the ship by his own prowess from capture
by pirates. A ready and caustic wit made him a favourite of
\index{Charles II. of England, ref.\ to}Charles II., and induced the courtiers to respect even if they
did not appreciate him. He wrote with a sustained and
somewhat stately eloquence, and with his blameless life and
scrupulous conscientiousness was an impressive personage of
the time.

His earliest work was a complete edition of the \index{Euclid's \emph{Elements}!ref.\ to}\emph{Elements}
of \index{Euclid!ref.\ to}Euclid, which he issued in Latin in 1655, and in English
in 1660; in 1657 he published an edition of the \emph{Data}. His
lectures, delivered in 1664, 1665, and 1666, were published in
1683 under the title \emph{Lectiones Mathematicae}; these are mostly
on the metaphysical basis for mathematical truths. His
lectures for 1667 were published in the same year, and suggest
the analysis by which \index{Archimedes!ref.\ to}Archimedes was led to his chief results.
In 1669 he issued his \emph{Lectiones Opticae et Geometricae}. It is
said in the preface that \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton revised and corrected these
lectures, adding matter of his own, but it seems probable from
\index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's remarks in the fluxional controversy that the additions
were confined to the parts which dealt with optics. This, which
is his most important work in mathematics, was republished
with a few minor alterations in 1674. In 1675 he published an
edition with numerous comments of the first four books of the
\emph{Conics} of \index{Apollonius!ref.\ to}Apollonius, and of the extant works of Archimedes
and \index{Theodosius!ref.\ to}Theodosius.

\index{Optics (geometrical).!Discussed by (among others) Euclid!z6@Barrow}In the optical lectures many problems connected with the
reflexion and \index{Refraction}refraction of light are treated with ingenuity.
The geometrical focus of a point seen by reflexion or refraction
is defined; and it is explained that the image of an object is
the locus of the geometrical foci of every point on it. \index{Barrow!ref.\ to}Barrow
also worked out a few of the easier properties of thin lenses,
and considerably simplified the Cartesian explanation of the
\index{Rainbow, explanation of}rainbow.

\begin{center}
\labelincludegraphics{329a}
\end{center}

The geometrical lectures contain some new ways of determining
the areas and \index{Tangent (geometrical)}tangents of curves. \nblabel{xref:311}The most celebrated
of these is the method given for the determination of tangents
to curves, and this is sufficiently important to require a detailed
notice, because it illustrates the way in which Barrow, \index{Hudde!ref.\ to}Hudde,
and \index{Sluze, de!ref.\ to}Sluze were working on the lines suggested by \index{Fermat!ref.\ to}Fermat
towards the methods of the differential calculus. Fermat had
observed that the tangent at a point $P$ on a curve was determined
if one other point besides $P$ on it were known; hence,
if the length of the \index{Subtangent}subtangent $MT$ could be found (thus
determining the point $T$), then the line $TP$ would be the
required tangent. Now Barrow remarked that if the abscissa
and ordinate at a point $Q$ adjacent to $P$ were drawn, he got a
small triangle $PQR$ (which he called the \index{Differential triangle, the}differential triangle,
because its sides $PR$ and $PQ$ were the differences of the
abscissae and ordinates of $P$ and $Q$), so that
\[
TM:MP=QR:RP.
\]
To find $QR:RP$ he supposed that $x,y$ were the co-ordinates of
$P$, and $x-e,y-a$ those of $Q$ (\index{Barrow!ref.\ to}Barrow actually used $p$ for $x$ and
$m$ for $y$, but I alter these to agree with the modern practice).
Substituting the co-ordinates of $Q$ in the equation of the curve,
and neglecting the squares and higher powers of $e$ and $a$ as
compared with their first powers, he obtained $e:a$. The ratio
$a/e$ was subsequently (in accordance with a suggestion made
by \index{Sluze, de!ref.\ to}Sluze) termed the \index{Coefficient, angular}\index{Angular coefficient}angular coefficient of the tangent at
the point.

Barrow applied this method to the curves (i)~$x^2(x^2+y^2)=r^2y^2$;
(ii)~$x^3+y^3=r^3$; (iii)~$x^3+y^3=rxy$, called \index{Galande, the}\emph{la galande}; (iv)
$y=(r-x)\tan \pi x/2r$, the \emph{quadratrix}; and (v)~$y = r\tan \pi x/2r$. It
will be sufficient here if I take as an illustration the simpler
case of the parabola $y^2=px$. Using the notation given
above, we have for the point $P$, $y^2=px$; and for the point
$Q$, $(y-a)^2=p(x-e)$. Subtracting we get $2ay-a^2=pe$. But,
if $a$ be an infinitesimal quantity, $a^2$ must be infinitely smaller
and therefore may be neglected when compared with the
quantities $2ay$ and $pe$. Hence $2ay=pe$, that is, $e:a=2y:p$.
Therefore $TM:y=e:a=2y:p$. Hence $TM=2y^2/p=2x$.
This is exactly the procedure of the differential calculus,
except that there we have a rule by which we can get the ratio
$a/e$ or $dy/dx$ directly without the labour of going through a
calculation similar to the above for every separate case.\index{Barrow|)}\nblabel{xref:312}

\index{Brouncker, Lord|(}\nblabel{png330a}\nbparagraph{Brouncker.} \emph{William, Viscount Brouncker}, one of the
founders of the Royal Society of London, born about 1620,
and died on April~5, 1684, was among the most brilliant
mathematicians of this time, and was in intimate relations
with \index{Wallis!ref.\ to}Wallis, \index{Fermat!ref.\ to}Fermat, and other leading mathematicians. I
mentioned above his curious reproduction of \index{Brahmagupta!ref.\ to}Brahmagupta's
solution of a certain indeterminate equation. Brouncker proved
that the area enclosed between the equilateral hyperbola $xy=1$,
the axis of $x$, and the ordinates $x=1$ and $x=2$, is equal
either to
\[
\frac{1}{1\,.\,2} + \frac{1}{3\,.\,4} + \frac{1}{5\,.\,6} + \ldots
\mbox{, or to }
1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots
\]
He also worked out other similar expressions for different
areas bounded by the hyberbola and straight lines. He wrote
on the \index{Rectification of curves|(}\index{Curves, rectification of|(}rectification of the parabola and of the \index{Cycloid}cycloid.\footnote
  {On these investigations, see his papers in the \emph{Philosophical Transactions},
  London, 1668, 1672, 1673, and 1678.}
It is
noticeable that he used \index{Infinite series, difficulties in connection with}\index{Infinite series, quadrature of curves in}infinite series to express quantities
whose values he could not otherwise determine. In answer to
a request of \index{Wallis!ref.\ to}Wallis to attempt the quadrature of the circle he
shewed that the ratio of the area of a circle to the area of the
circumscribed square, that is, the ratio of 4 to \index{P1@$\pi$,!z2@incommensurability of}$\pi$, is equal to
\[
\frac{1}{1} \cdrop{+} \frac{1^2}{2\cpad} \cdrop{+} \frac{3^2}{2\cpad} \cdrop{+} \frac{5^2}{2\cpad} \cdrop{+} \frac{7^2}{2\cpad} \cdrop{+ \ldots}
\]
\index{Continued fractions}\index{Fractions, continued}Continued fractions\footnote
  {On the history of continued fractions, see papers by \index{Gunther@Günther, S.}S.~Günther and
  \index{Favaro, A., ref.\ to}A.~Favaro in Boncompagni's \emph{Bulletino di bibliografia}, Rome, 1874, vol.~vii, pp.
  213, 451, 533; and \index{Cantor, M., ref.\ to}Cantor, vol.~ii, pp.~622, 762, 766. \index{Bombelli}Bombelli used them
  in 1572; but \index{Cataldi}Cataldi introduced the usual notation for them.}
had been employed by Bombelli in 1572,
and had been systematically used by Cataldi in his treatise
on finding the square roots of numbers, published at Bologna
in 1613. Their properties and theory were given by \index{Huygens!ref.\ to}Huygens,
1703, and \index{Euler!ref.\ to}Euler, 1744.\index{Brouncker, Lord|)}

\index{Gregory, James|(}\nblabel{png331a}\nbparagraph{James Gregory.} \emph{James Gregory}, born at Drumoak near
Aberdeen in 1638, and died at Edinburgh in October 1675, was
successively professor at St.~Andrews and Edinburgh. In 1660
he published his \emph{Optica Promota}, in which the reflecting
\index{Telescopes}telescope known by his name is described. In 1667 he issued
his \emph{Vera Circuli et Hyperbolae Quadratura} in which he shewed
how the areas of the circle and hyperbola could be obtained in
the form of infinite \index{Convergency}convergent series, and here (I believe for
the first time) we find a distinction drawn between convergent
and divergent series. This work contains a remarkable geometrical
proposition to the effect that the ratio of the area of
any arbitrary sector of a circle to that of the inscribed or
circumscribed regular polygons is not expressible by a finite
number of algebraical terms. Hence he inferred that the
\index{Infinite series, quadrature of curves in}quadrature of a circle was impossible; this was accepted by
\index{Montucla!ref.\ to}Montucla, but it is not conclusive, for it is conceivable that
some particular sector might be squared, and this particular
sector might be the whole circle. This book contains also the
earliest enunciation of the expansions in series of \index{sine x@$\operatorname{Sin} x$, series for}\index{sineinv x@$\operatorname{Sin}^{-1} x$, series for}\index{Tani@$\operatorname{Tan}^{-1} x$, series for}\index{cosineinv x@$\operatorname{Cos}^{-1}x$, series for}\index{cosine x@$\operatorname{Cos} x$, series for}\index{Expansion,!ofcosb@of $\cos x$}\index{Expansion,!ofcosc@of $\cos^{-1} x$}%
\index{Expansion,!ofsinc@of $\sin^{-1} x$}\index{Expansion,!oftan@of $\tan^{-1} x$}$\sin x$, $\cos x$,
$\sin^{-1}x$ or $\arcsin x$, and $\cos^{-1}x$ or $\arccos x$. It was reprinted
in 1668 with an appendix, \emph{Geometriae Pars}, in which Gregory
explained how the volumes of solids of revolution could be
determined. In 1671, or perhaps earlier, he established the
theorem that
\[
\theta = \tan \theta - \frac{1}{3} \tan^3 \theta + \frac{1}{5} \tan^5 \theta - \ldots ,
\]
the result being true only if $\theta$ lie between $-\frac{1}{4}\pi$ and $\frac{1}{4}\pi$. \index{P1@$\pi$,!z1@value of}This
is the theorem on which many of the subsequent calculations
of approximations to the numeral value of $\pi$ have been based.\index{Curves, rectification of|)}\index{Rectification of curves|)}\index{Gregory, James|)}

\index{Wren}\nblabel{png332a}\nbparagraph{Wren.} \emph{Sir Christopher Wren} was born at Knoyle, Wiltshire,
on October~20, 1632, and died in London on February~25,
1723. Wren's reputation as a mathematician has been
overshadowed by his fame as an architect, but he was Savilian
professor of astronomy at Oxford from 1661 to 1673, and for
some time president of the Royal Society. Together with
\index{Wallis!ref.\ to}Wallis and \index{Huygens!ref.\ to}Huygens he investigated the \index{Collision of bodies}laws of collision of
bodies; he also discovered the two systems of \index{Generating lines}generating lines
on the \index{Hyperboloid of one sheet}hyperboloid of one sheet, though it is probable that
he confined his attention to a hyperboloid of revolution.\footnote
  {See the \emph{Philosophical Transactions} London, 1669.}
Besides these he wrote papers on the resistance of fluids, and
the motion of the pendulum. He was a friend of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton
and (like Huygens, \index{Hooke!ref.\ to}Hooke, \index{Halley!ref.\ to}Halley, and others) had made
attempts to shew that the force under which the planets move
varies inversely as the square of the distance from the sun.

Wallis, \index{Brouncker, Lord!ref.\ to}Brouncker, Wren, and \index{Boyle}Boyle (the last-named being
a chemist and physicist rather than a mathematician) were the
leading philosophers who founded the \index{Royal Society of London}Royal Society of London.
The society arose from the self-styled \index{Indivisible College}``indivisible college'' in
London in 1645; most of its members moved to Oxford during
the civil war, where \index{Hooke!ref.\ to}Hooke, who was then an assistant in Boyle's
laboratory, joined in their meetings; the society was formally
constituted in London in 1660, and was incorporated on July~15,
1662. The \index{Academy,!the French}\index{French Academy}French Academy was founded in 1666, and
the \index{Berlin Academy}\index{Academy,!the Berlin}Berlin Academy in 1700. The \index{Accademia dei Lincei}Accademia dei Lincei was
founded in 1603, but was dissolved in 1630.

\index{Hooke}\nblabel{png333a}\nbparagraph{Hooke.} \emph{Robert Hooke}, born at Freshwater on July~18,
1635, and died in London on March~3, 1703, was educated at
Westminster, and Christ Church, Oxford, and in 1665 became
professor of geometry at Gresham College, a post which he
occupied till his death. He is still known by the law which
he discovered, that the \index{Tension of elastic string}\index{Elastic string, tension of}tension exerted by a stretched string
is (within certain limits) proportional to the extension, or, in
other words, that the stress is proportional to the strain. He
invented and discussed the conical \index{Pendulum, motion of}pendulum, and was the
first to state explicitly that the motions of the heavenly bodies
were merely dynamical problems. He was as jealous as he was
vain and irritable, and accused both \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton and \index{Huygens!ref.\ to}Huygens of
unfairly appropriating his results. Like Huygens, \index{Wren!ref.\ to}Wren, and
\index{Halley!ref.\ to}Halley, he made efforts to find the law of force under which the
planets move about the sun, and he believed the law to be
that of the inverse square of the distance. He, like Huygens,
discovered that the small oscillations of a coiled spiral spring
were practically isochronous, and was thus led to recommend
(possibly in 1658) the use of the balance spring in \index{Watches, invention of}watches.
He had a watch of this kind made in London in 1675; it was
finished just three months later than a similar one made in
Paris under the directions of Huygens.

\index{Collins, J.|(}\nbparagraph{Collins.} \emph{John Collins}, born near Oxford on March~5,
1625, and died in London on November~10, 1683, was a man
of great natural ability, but of slight education. Being devoted
to mathematics, he spent his spare time in correspondence with
the leading mathematicians of the time, for whom he was
always ready to do anything in his power, and he has been
described---not inaptly---as the English Mersenne. To him
we are indebted for much information on the details of the
discoveries of the period.\footnote
  {See the \emph{Commercium Epistolicum}, and \index{Rigaud, ref.\ to}S.~P.~Rigaud's \emph{Correspondence of
  Scientific Men of the Seventeenth Century}, Oxford, 1841.}\index{Collins, J.|)}

\index{Pell}\nblabel{png334a}\nbparagraph{Pell.} Another mathematician who devoted a considerable
part of his time to making known the discoveries of others, and
to correspondence with leading mathematicians, was \emph{John Pell}.
Pell was born in Sussex on March~1, 1610, and died in London
on December~10, 1685. He was educated at Trinity College,
Cambridge; he occupied in succession the mathematical chairs
at Amsterdam and Breda; he then entered the English diplomatic
service; but finally settled in 1661 in London, where he
spent the last twenty years of his life. His chief works were
an edition, with considerable new matter, of the \emph{Algebra} by
\index{Branker}Branker and \index{Rhonius}Rhonius, London, 1668; and a table of square
numbers, London, 1672.

\index{Sluze, de}\index{Desluze@De Sluze}\nbparagraph{Sluze.} \emph{René François Walther de~Sluze (Slusius)}, canon of
Liége, born on July~7, 1622, and died on March~19, 1685,
found for the \index{Subtangent}subtangent of a curve $f(x,y)=0$ an expression
which is equivalent to $-y\dfrac{\partial f}{\partial y} \Big/ \dfrac{\partial f}{\partial x}$; he wrote numerous tracts,\footnote
  {Some of his papers were published by \index{Lepage@Le Paige}Le Paige in vol.~xvii of
  Boncompagni's \emph{Bulletino di bibliografia}, Rome, 1884.}
and in particular discussed at some length spirals and points of
inflexion.

\index{Viviani}\nbparagraph{Viviani.}\nblabel{png334b} \emph{Vincenzo Viviani}, a pupil of \index{Galileo!ref.\ to}Galileo and \index{Torricelli!ref.\ to}Torricelli,
born at Florence on April~5, 1622, and died there on September~22,
1703, brought out in 1659 a restoration of the lost book of
\index{Apollonius!ref.\ to}Apollonius on conic sections, and in 1701 a restoration of the
work of \index{Aristaeus!ref.\ to}Aristaeus. He explained in 1677 how an \index{Angle,!trisection of}\index{Trisection of angle}angle could
be trisected by the aid of the equilateral hyperbola or the
conchoid. In 1692 he proposed the problem to construct four
windows in a hemispherical vault so that the remainder of the
surface can be accurately determined; a celebrated problem, of
which analytical solutions were given by \index{Wallis!ref.\ to}Wallis, \index{Leibnitz!ref.\ to}Leibnitz, \index{Gregory, David!ref.\ to}David
Gregory, and \index{Bernoulli, James!ref.\ to}James Bernoulli.

\index{Tschirnhausen}\nblabel{png335a}\nbparagraph{Tschirnhausen.} \emph{Ehrenfried Walther von Tschirnhausen} was
born at Kislingswalde on April~10, 1631, and died at Dresden
on October~11, 1708. In 1682 he worked out the theory of
\index{Caustics are rectifiable}caustics by reflexion, or, as they were usually called, \index{Catacaustics}catacaustics,
and shewed that they were \index{Rectification of curves}\index{Curves, rectification of}rectifiable. This was the second
case in which the \index{Envelopes}envelope of a moving line was determined.
He constructed burning mirrors of great power. The transformation
by which he removed certain intermediate terms from
a given algebraical equation is well known; it was published in
the \emph{Acta Eruditorum} for 1683.

\index{Hire, De la}\index{Lahire}\index{Delahire@De la Hire}\nbparagraph{De la Hire.} \emph{Philippe De~la~Hire} (or \emph{Lahire}), born in Paris
on March~18, 1640, and died there on April~21, 1719, wrote on
graphical methods, 1673; on the conic sections, 1685; a treatise
on \index{Epicycloids}epicycloids, 1694; one on roulettes, 1702; and, lastly,
another on conchoids, 1708. His works on conic sections and
epicycloids were founded on the teaching of \index{Desargues!ref.\ to}Desargues, whose
favourite pupil he was. He also translated the essay of
\index{Moschopulus!ref.\ to}Moschopulus on \index{Magic squares}magic squares, and collected many of the
theorems on them which were previously known; this was
published in 1705.

\index{Roemer}\nbparagraph{Roemer.} \emph{Olof Roemer}, born at Aarhuus on September~25,
1644, and died at Copenhagen on September~19, 1710, was the
first to measure the \index{Light,!velocity of}velocity of light; this was done in 1675 by
means of the eclipses of Jupiter's satellites. \index{Diffraction}He brought the
transit and mural circle into common use, the altazimuth having
been previously generally employed, and it was on his recommendation
that astronomical observations of stars were subsequently
made in general on the meridian. He was also the first
to introduce micrometers and reading microscopes into an observatory.
He also deduced from the properties of epicycloids the
form of the teeth in toothed-wheels best fitted to secure a uniform
motion.

\index{Rolle}\nbparagraph{Rolle.} \emph{Michel Rolle}, born at Ambert on April~21, 1652,
and died in Paris on November~8, 1719, wrote an algebra in
1689, which contains the theorem on the position of the \index{Roots of equations,!position of}\index{Equations, differential!position of roots}roots
of an equation which is known by his name. He published in
1696 a treatise on the \index{Equations, differential!indeterminate}solutions of equations, whether determinate
or indeterminate, and he produced several other minor
works. He taught that the differential calculus, which, as we
shall see later, had been introduced towards the close of the
seventeenth century, was nothing but a collection of ingenious
fallacies.

\nbChapter{XVI}{The Life and Works of Newton}{}{}
  {Newton's life and works are discussed in \emph{The Memoirs of Newton}, by D.~Brewster,
  2 volumes, Edinburgh, second edition, 1860. An edition of most
  of Newton's works was published by S.~Horsley in 5 volumes, London, 1779--1785;
  and a bibliography of them was issued by G.~J.~Gray, Cambridge,
  second edition, 1907; see also the catalogue of the Portsmouth Collection of
  Newton's papers, Cambridge, 1888. My \emph{Essay on the Genesis, Contents, and
  History of Newton's Principia}, London, 1893, may be also consulted.}
\index{Geometry.!z2@Classical synthetic geometry, discussed or used by nearly all the mathematicians considered in the first period, chapters \textsc{ii-v}!also by Newton and his School, chapters \textsc{xvi, xviii}}\index{Astronomy.!z19@Physical astronomy created by Newton, chapter \textsc{xvi}}\index{Ball, W. W. R., ref.\ to}\index{Gray on Newton's writings}\index{Horsley on Newton}\index{Brewster, D., ref.\ to}\nblabel{png337a}\index{Newtons@Newton's \emph{Principia}!ref.\ to}\index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)|(}

\nblabel{png337b}\First{The} mathematicians considered in the \hyperref[chapter:XV]{last chapter} commenced
the creation of those processes which distinguish modern mathematics.
The extraordinary abilities of Newton enabled him
within a few years to perfect the more elementary of those
processes, and to distinctly advance every branch of mathematical
science then studied, as well as to create some new
subjects. Newton was the contemporary and friend of \index{Wallis!ref.\ to}Wallis,
\index{Huygens!ref.\ to}Huygens, and others of those mentioned in the \hyperref[chapter:XV]{last chapter}, but
though most of his mathematical work was done between the
years 1665 and 1686, the bulk of it was not printed---at any
rate in book-form---till some years later.

I propose to discuss the works of Newton more fully than
those of other mathematicians, partly because of the intrinsic
importance of his discoveries, and partly because this book is
mainly intended for English readers, and the development of
mathematics in Great Britain was for a century entirely in the
hands of the Newtonian school.

\nblabel{png338a}\emph{Isaac Newton} was born in Lincolnshire, near Grantham, on
December~25, 1642, and died at Kensington, London, on March~20,
1727. He was educated at Trinity College, Cambridge, and
lived there from 1661 till 1696, during which time he produced
the bulk of his work in mathematics; in 1696 he was appointed
to a valuable Government office, and moved to London, where
he resided till his death.

His father, who had died shortly before Newton was born,
was a yeoman farmer, and it was intended that Newton should
carry on the paternal farm. He was sent to school at Grantham,
where his learning and mechanical proficiency excited some
attention. In 1656 he returned home to learn the business of a
farmer, but spent most of his time solving problems, making
experiments, or devising mechanical models; his mother noticing
this, sensibly resolved to find some more congenial occupation
for him, and his uncle, having been himself educated at Trinity
College, Cambridge, recommended that he should be sent there.

In 1661 Newton accordingly entered as a student at Cambridge,
where for the first time he found himself among
surroundings which were likely to develop his powers. He
seems, however, to have had but little interest for general society
or for any pursuits save science and mathematics. Luckily he
kept a diary, and we can thus form a fair idea of the course of
education of the most advanced students at an English university
at that time. He had not read any mathematics before
coming into residence, but was acquainted with \index{Sanderson's \emph{Logic}}Sanderson's
\emph{Logic}, which was then frequently read as preliminary to mathematics.
At the beginning of his first October term he happened
to stroll down to Stourbridge Fair, and there picked up a book
on astrology, but could not understand it on account of the
geometry and trigonometry. He therefore bought a Euclid, and
was surprised to find how obvious the propositions seemed. He
thereupon read \index{Oughtred!ref.\ to}Oughtred's \emph{Clavis} and \index{Descartes!ref.\ to}Descartes's \emph{Géométrie}, the
latter of which he managed to master by himself, though with
some difficulty. The interest he felt in the subject led him to
take up mathematics rather than chemistry as a serious study.
His subsequent mathematical reading as an undergraduate was
founded on \index{Kepler!ref.\ to}Kepler's \emph{Optics}, the works of \index{Vieta!ref.\ to}Vieta, van Schooten's
\emph{Miscellanies}, \index{Descartes!ref.\ to}Descartes's \emph{Géométrie}, and \index{Wallis!ref.\ to}Wallis's \emph{Arithmetica
Infinitorum}: he also attended \index{Barrow!ref.\ to}Barrow's lectures. At a later
time, on reading Euclid more carefully, he formed a high
opinion of it as an instrument of education, and he used to
express his regret that he had not applied himself to geometry
before proceeding to algebraic analysis.

\nblabel{png339a}There is a manuscript of his, dated May~28, 1665, written in
the same year as that in which he took his B.A. degree, which
is the earliest documentary proof of his invention of \index{Fluxions}fluxions.
It was about the same time that he discovered the binomial
theorem.\footnote
  {See below, pp.~\pageref{xref:327}, \pageref{xref:341}.}

On account of the plague the college was sent down during
parts of the year 1665 and 1666, and for several months at this
time Newton lived at home. This period was crowded with
brilliant discoveries. He thought out the fundamental principles
of his \index{Gravity,!law of|(}theory of gravitation, namely, that every particle of
matter attracts every other particle, and he suspected that the
\index{Attraction, theories of|(}attraction varied as the product of their masses and inversely as
the square of the distance between them. He also worked out
the fluxional calculus tolerably completely: thus in a manuscript
dated November~13, 1665, he used fluxions to find the tangent
and the radius of curvature at any point on a curve, and in
October 1666 he applied them to several problems in the theory
of equations. Newton communicated these results to his friends
and pupils from and after 1669, but they were not published in
print till many years later. It was also whilst staying at home
at this time that he devised some instruments for \index{Lenses, construction of}grinding
lenses to particular forms other than spherical, and perhaps he
decomposed solar light into different \index{Colours, theory of}colours.

\nblabel{png339b}
\begin{center}
\labelincludegraphics{340a}
\end{center}

Leaving out details and taking round numbers only, his
reasoning at this time on the theory of gravitation seems to
have been as follows. He suspected that the force which
retained the moon in its orbit about the earth was the same as
terrestrial gravity, and to verify this hypothesis he proceeded
thus. He knew that, if a stone were allowed to fall near the
surface of the earth, the attraction of the earth (that is, the
weight of the stone) caused it to move through 16 feet in
one second. The moon's orbit relative to the earth is nearly
a circle; and as a rough approximation, taking it to be so, he
knew the distance of the moon, and therefore the length of its
path; he also knew the time the moon took to go once round
it, namely, a month. Hence he could easily find its velocity at
any point such as $M$. He could therefore find the distance
$MT$ through which it would move in the next second if it
were not pulled by the earth's attraction. At the end of that
second it was however at $M'$, and therefore the earth $E$ must
have pulled it through the distance $TM'$ in one second (assuming
the direction of the earth's pull to be constant). Now he and
several physicists of the time had conjectured from \index{Kepler's laws}Kepler's third
law that the attraction of the earth on a body would be found
to decrease as the body was removed farther away from the
earth inversely as the square of the distance from the centre
of the earth;\footnote
  {An argument leading to this result is given below on page~\pageref{xref:332}.}
if this were the actual law and if gravity were
the sole force which retained the moon in its orbit, then $TM'$
should be to 16 feet inversely as the square of the distance
of the moon from the centre of the earth to the square of
the radius of the earth. In 1679, when he repeated the
investigation, $TM'$ was found to have the value which was
required by the hypothesis, and the verification was complete;
but in 1666 his estimate of the distance of the moon was
inaccurate, and when he made the calculation he found that
$TM'$ was about one-eighth less than it ought to have been on
his hypothesis.

This discrepancy does not seem to have shaken his faith in
the belief that gravity extended as far as the moon and varied
inversely as the square of the distance; but, from \index{Whiston!ref.\ to}Whiston's
notes of a conversation with Newton, it would seem that
Newton inferred that some other force---probably \index{Cartesian vortices}\index{Vortices, Cartesian}Descartes's
vortices---acted on the moon as well as gravity. This statement
is confirmed by \index{Pemberton, ref.\ to}Pemberton's account of the investigation. It
seems, moreover, that Newton already believed firmly in the
principle of universal gravitation, that is, that every particle
of matter attracts every other particle, and suspected that the
attraction varied as the product of their masses and inversely
as the square of the distance between them; but it is certain
that he did not then know what the attraction of a spherical
mass on any external point would be, and did not think it
likely that a particle would be attracted by the earth as if the
latter were concentrated into a single particle at its centre.\index{Attraction, theories of|)}\index{Gravity,!law of|)}

\nblabel{png341a}On his return to Cambridge in 1667 Newton was elected
to a fellowship at his college, and permanently took up his
residence there. In the early part of 1669, or perhaps in
1668, he revised \index{Barrow!ref.\ to}Barrow's lectures for him. The end of the
fourteenth lecture is known to have been written by Newton,
but how much of the rest is due to his suggestions cannot now
be determined. As soon as this was finished he was asked by
Barrow and \index{Collins, J.!ref.\ to}Collins to edit and add notes to a translation of
\index{Kinckhuysen, ref.\ to}Kinckhuysen's \emph{Algebra}; he consented to do this, but on condition
that his name should not appear in the matter. In 1670 he also
began a systematic exposition of his analysis by infinite series,
the object of which was to express the ordinate of a curve
in an infinite algebraical series every term of which can be
integrated by \index{Wallis!ref.\ to}Wallis's rule; his results on this subject had been
communicated to \index{Barrow!ref.\ to}Barrow, \index{Collins, J.!ref.\ to}Collins, and others in 1669. This
was never finished: the fragment was published in 1711, but
the substance of it had been printed as an appendix to the
\index{Optics (geometrical).!Discussed by (among others) Euclid!z7@Newton|(}\emph{Optics} in 1704. These works were only the fruit of Newton's
leisure, most of his time during these two years being given up
to optical researches.

\nblabel{png342a}In October, 1669, Barrow resigned the Lucasian chair in
favour of Newton. During his tenure of the professorship,
it was Newton's practice to lecture publicly once a week, for
from half-an-hour to an hour at a time, in one term of each
year, probably dictating his lectures as rapidly as they could
be taken down; and in the week following the lecture to
devote four hours to appointments which he gave to students
who wished to come to his rooms to discuss the results of the
previous lecture. He never repeated a course, which usually
consisted of nine or ten lectures, and generally the lectures of
one course began from the point at which the preceding course
had ended. The manuscripts of his lectures for seventeen out
of the first eighteen years of his tenure are extant.

\nblabel{png342b}When first appointed Newton chose optics for the subject
of his lectures and researches, and before the end of 1669 he
had worked out the details of his \index{Colours, theory of}discovery of the decomposition
of a ray of white light into rays of different colours
by means of a prism. The complete explanation of the \index{Rainbow, explanation of}theory
of the rainbow followed from this discovery. These discoveries
formed the subject-matter of the lectures which he delivered
as Lucasian professor in the years 1669, 1670, and 1671. The
chief new results were embodied in a paper communicated
to the Royal Society in February, 1672, and subsequently
published in the \emph{Philosophical Transactions}. The manuscript
of his original lectures was printed in 1729 under the title
\emph{Lectiones Opticae}. This work is divided into two books, the
first of which contains four sections and the second five. The
first section of the first book deals with the decomposition
of solar light by a prism in consequence of the unequal refrangibility
of the rays that compose it, and a description
of his experiments is added. The second section contains an
account of the method which Newton invented for the determining
the coefficients of \index{Refraction}refraction of different bodies. This
is done by making a ray pass through a prism of the material
so that the deviation is a minimum; and he proves that, if the
angle of the prism be $i$ and the deviation of the ray be $\delta$, the
refractive index will be $\sin{\frac{1}{2}(i+\delta)} \cosec \frac{1}{2}i$. The third section
is on refractions at plane surfaces; he here shews that if a
ray pass through a prism with minimum deviation, the angle
of incidence is equal to the angle of emergence; most of this
section is devoted to geometrical solutions of different problems.
The fourth section contains a discussion of refractions at curved
surfaces. The second book treats of his \index{Colours, theory of}\index{Rainbow, explanation of}theory of colours and
of the rainbow.

\index{Optics (geometrical).!z11@(physical)}By a curious chapter of accidents Newton failed to correct
the chromatic aberration of two colours by means of a couple
of prisms. He therefore abandoned the hope of making a
refracting \index{Telescopes}telescope which should be achromatic, and instead
designed a reflecting telescope, probably on the model of a
small one which he had made in 1668. The form he used is
that still known by his name; the idea of it was naturally
suggested by \index{Gregory, James!ref.\ to}Gregory's telescope. In 1672 he invented a
reflecting \index{Microscope}microscope, and some years later he invented the
\index{Sextant, invention of}sextant which was rediscovered by \index{Hadley, ref.\ to}J.~Hadley in 1731.\index{Optics (geometrical).!Discussed by (among others) Euclid!z7@Newton|)}

His professorial lectures from 1673 to 1683 were on
algebra and the theory of equations, and are described below;
but much of his time during these years was occupied with
other investigations, and I may remark that throughout his
life Newton must have devoted at least as much attention to
chemistry and theology as to mathematics, though his conclusions
are not of sufficient interest to require mention here. His theory
of colours and his deductions from his optical experiments were
at first attacked with considerable vehemence. The correspondence
which this entailed on Newton occupied nearly all his
leisure in the years 1672 to 1675, and proved extremely distasteful
to him. Writing on December~9, 1675, he says, \index{Light,!physical theories of}``I was so
persecuted with discussions arising out of my theory of light,
that I blamed my own imprudence for parting with so substantial
a blessing as my quiet to run after a shadow.'' Again, on
November~18, 1676, he observes, ``I see I have made myself a
slave to philosophy; but, if I get rid of \index{Linus of Liége}Mr.~Linus's business, I
will resolutely bid adieu to it eternally, excepting what I do for
my private satisfaction, or leave to come out after me; for I see
a man must either resolve to put out nothing new, or to become
a slave to defend it.'' The unreasonable dislike to have his
conclusions doubted or to be involved in any correspondence
about them was a prominent trait in Newton's character.

\nblabel{png344a}Newton was deeply interested in the question as to how the
effects of light were really produced, and by the end of 1675 he
had worked out the \index{Corpuscular theory of light}corpuscular or \index{Emission theory of light}emission theory, and had
shewn how it would account for all the various phenomena of
geometrical optics, such as reflexion, refraction, colours, diffraction,
\&c. To do this, however, he was obliged to add a somewhat
artificial rider, that the corpuscles had alternating fits of
easy reflexion and easy refraction communicated to them by an
\index{Ether, luminiferous}ether which filled space. The theory is now known to be
untenable, but it should be noted that Newton enunciated it as
a hypothesis from which certain results would follow: it would
seem that he believed the wave theory to be intrinsically more
probable, but it was the difficulty of explaining diffraction on
that theory that led him to suggest another hypothesis.

Newton's corpuscular theory was expounded in memoirs communicated
to the Royal Society in December 1675, which are
substantially reproduced in his \emph{Optics}, published in 1704. In
the latter work he dealt in detail with his theory of fits of easy
reflexion and transmission, and the colours of thin plates, to
which he added an explanation of the colours of thick plates
[bk.~\textsc{ii}, part~4] and observations on the inflexion of light [bk.~\textsc{iii}].

Two letters written by Newton in the year 1676 are sufficiently
interesting to justify an allusion to them.\footnote
  {See \index{Wallis!ref.\ to}J.~Wallis, \emph{Opera}, vol.~iii, Oxford, 1699, p.~622 \emph{et seq.}}
\nblabel{png345a}\index{Leibnitz!ref.\ to}Leibnitz, in 1674, in
a correspondence with \index{Oldenburg}Oldenburg, wrote saying that he possessed
\index{Quadrature of curves}\index{Infinite series, quadrature of curves in|(}``general analytical methods depending on infinite series.''
Oldenburg, in reply, told him that Newton and \index{Gregory, James!ref.\ to}Gregory had
used such series in their work. In answer to a request for
information, Newton wrote on June~13, 1676, giving a brief
account of his method. \nblabel{xref:327}\index{Interpolation, method of}\index{Expansion,!ofcosc@of $\cos^{-1} x$}%
\index{Expansion,!ofsinc@of $\sin^{-1} x$}He here enunciated the binomial
theorem, which he stated, in effect, in the form that if A, B, C, D, \dots
denote the successive terms in the \index{Expansion,!ofbin@of binomial}expansion of $(\mathrm{P}+\mathrm{PQ})^{m/n}$,
then
\[
(\mathrm{P}+\mathrm{PQ})^{m/n} = \mathrm{A} + \dfrac{m}{n}\mathrm{AQ} + \dfrac{m-n}{2n}\mathrm{BQ} + \dfrac{m-2n}{3n}\mathrm{CQ} + \dfrac{m-3n}{4n}\mathrm{DQ} + \ldots
\]
where $\mathrm{A}=\mathrm{P}^{m/n}$. He gave examples of its use. He also gave the
expansion of \index{sine x@$\operatorname{Sin} x$, series for}\index{sineinv x@$\operatorname{Sin}^{-1} x$, series for}$\sin^{-1}x$, from which he deduced that of $\sin x$: this
seems to be the earliest known instance of a \index{Reversion of series}\index{Series!reversion of}reversion of series.
He also inserted an expression for the rectification of an elliptic
arc in an infinite series.

Leibnitz wrote on August~27 asking for fuller details; and
Newton, on October~24, 1676, sent, through Oldenburg, an
account of the way in which he had been led to some of his
results. The main results may be briefly summarized. He
begins by saying that altogether he had used three methods
for expansion in series. His first was arrived at from the
study of the method of \index{Interpolation, method of}interpolation. Thus, by considering
the series of expressions for $(1-x^{2})^{0/2}$, $(1-x^{2})^{2/2}$, $(1-x^{2})^{4/2}$, $\ldots$ ,
he deduced by interpolations a rule connecting the successive
coefficients in the expansions of $(1-x^{2})^{1/2}$, $(1-x^{2})^{3/2}$, $\ldots$; and
then by analogy obtained the expression for the general term
in the expansion of a binomial. He then tested his result in
various ways; for instance in the case of $(1-x^{2})^{1/2}$, by extracting
the square root of $1-x^{2}$, \emph{more arithmetico}, and by
forming the square of the expansion of $(1-x^{2})^{1/2}$, which reduced
to $1-x^{2}$. He also used the series to determine the areas of
the circle and the hyperbola in infinite series, and found that
the results were the same as those he had arrived at by other
means.

Having established this result, he then discarded the method
of interpolation, and employed his binomial theorem to express
(when possible) the ordinate of a curve in an infinite series
in ascending powers of the abscissa, and thus by \index{Wallis!ref.\ to}Wallis's
method he obtained expressions in infinite series for the areas
and arcs of curves in the manner described in the appendix to
his \emph{Optics} and in his \emph{De Analysi per Equationes Numero Terminorum
Infinitas}. He states that he had employed this second
method before the plague in 1665--66, and goes on to say that
he was then obliged to leave Cambridge, and subsequently
(presumably on his return to Cambridge) he ceased to pursue
these ideas, as he found that \index{Kauffmann (or Mercator)}\index{Mercator, N.!ref.\ to}Nicholas Mercator had employed
some of them in his \emph{Logarithmo-technica}, published in 1668;
and he supposed that the remainder had been or would be found
out before he himself was likely to publish his discoveries.\index{Infinite series, quadrature of curves in|)}

Newton next explains that he had also a third method, of
which (he says) he had about 1669 sent an account to \index{Barrow!ref.\ to}Barrow
and \index{Collins, J.!ref.\ to}Collins, illustrated by applications to areas, \index{Rectification of curves}\index{Curves, rectification of}rectification,
cubature, \&c. This was the method of \index{Fluxions}fluxions; but Newton
gives no description of it here, though he adds some illustrations
of its use. The first illustration is on the quadrature of the
curve represented by the equation
\[
y = ax^{m} (b + cx^{n})^{p},
\]
which he says can be effected as a sum of $(m+1)/n$ terms if
$(m+1)/n$ be a positive integer, and which he thinks cannot
otherwise be effected except by an infinite series.\footnote
  {This is not so, the integration is possible if $p+(m+1)/n$ be an integer.}
He also gives
a list of other forms which are immediately integrable, of which
the chief are
\[
\dfrac{x^{mn-1}}{a+bx^{n}+cx^{2n}},\ \dfrac{x^{(m+1/2)n-1}}{a+bx^{n}+cx^{2n}},\ x^{mn-1}(a+bx^{n}+cx^{2n})^{\pm1/2},
\]
\[
x^{mn-1}(a+bx^{n})^{\pm1/2}(c+dx^{n})^{-1},\ x^{mn-n-1}(a+bx^{n})(c+dx^{n})^{-1/2};
\]
where $m$ is a positive integer and $n$ is any number whatever.
Lastly, he points out that the area of any curve can be easily
determined approximately by the method of interpolation
described below in discussing his \index{Differentials}\emph{Methodus Differentialis}.

At the end of his letter Newton alludes to the solution of the
``inverse problem of tangents,'' a subject on which \index{Leibnitz!ref.\ to}Leibnitz had
asked for information. He gives formulae for \index{Reversion of series}\index{Series!reversion of}reversing any
series, but says that besides these formulae he has two methods
for solving such questions, which for the present he will not
describe except by an anagram which, being read, is as follows,
``Una methodus consistit in extractione \index{Fluents}fluentis quantitatis ex
aequatione simul involvente fluxionem ejus: altera tantum in
assumptione seriei pro quantitate qualibet incognita ex qua
caetera commode derivari possunt, et in collatione terminorum
homologorum aequationis resultantis, ad eruendos terminos
assumptae seriei.''

He implies in this letter that he is worried by the questions
he is asked and the controversies raised about every new matter
which he produces, which shew his rashness in publishing ``quod
umbram captando eatenus perdideram quietem meam, rem prorsus
substantialem.''

Leibnitz, in his answer, dated June~21, 1677, explains his
method of drawing tangents to curves, which he says proceeds
``not by fluxions of lines, but by the differences of numbers'';
and he introduces his notation of $dx$ and $dy$ for the infinitesimal
differences between the co-ordinates of two consecutive
points on a curve. He also gives a solution of the problem to
find a curve whose \index{Subtangent!constant}subtangent is constant, which shews that he
could integrate.

In 1679 \index{Hooke!ref.\ to}Hooke, at the request of the Royal Society, wrote
to Newton expressing a hope that he would make further communications
to the Society, and informing him of various facts
then recently discovered. Newton replied saying that he had
abandoned the study of philosophy, but he added that the
earth's diurnal motion might be proved by the experiment of
observing the deviation from the perpendicular of a stone
dropped from a height to the ground---an experiment which
was subsequently made by the Society and succeeded. \index{Hooke!ref.\ to}Hooke
in his letter mentioned \index{Picard, J.}Picard's geodetical researches; in these
Picard used a value of the radius of the earth which is substantially
correct. \nblabel{png348a}This led Newton to repeat, with Picard's data,
his calculations of 1666 on the lunar orbit, \index{Gravity,!law of}and he thus verified
his supposition that gravity extended as far as the moon and
varied inversely as the square of the distance. He then proceeded
to consider the general theory of motion of a particle
under a centripetal force, that is, one directed to a fixed point,
and showed that the vector would sweep over equal areas in equal
times. He also proved that, if a particle \index{Elliptic orbits of planets}describe an ellipse under
a centripetal force to a focus, the law must be that of the inverse
square of the distance from the focus, and conversely, that the
orbit of a particle projected under the influence of such a force
would be a conic (or, it may be, he thought only an ellipse).
Obeying his rule to publish nothing which could land him in a
scientific controversy these results were locked up in his note-books,
and it was only a specific question addressed to him five
years later that led to their publication.

\nblabel{png348b}The \emph{Universal Arithmetic}, which is on algebra, \index{Equations, differential!theory of|(}theory of
equations, and miscellaneous problems, contains the substance
of Newton's lectures during the years 1673 to 1683. His
manuscript of it is still extant; \index{Whiston}\index{Whiston!ref.\ to}Whiston\footnote
  {\emph{William Whiston}, born in Leicestershire on December~9, 1667, educated
  at Clare College, Cambridge, of which society he was a fellow, and died in
  London on August~22, 1752, wrote several works on astronomy. He acted as
  Newton's deputy in the Lucasian chair from 1699, and in 1703 succeeded him
  as professor, but he was expelled in 1711, mainly for theological reasons. He
  was succeeded by Nicholas \index{Saunderson of Cambridge}Saunderson, the blind mathematician, who was
  born in Yorkshire in 1682, and died at Christ's College, Cambridge, on
  April~19, 1739.}
extracted a somewhat
reluctant permission from Newton to print it, and it was
published in 1707. Amongst several new theorems on various
points in algebra and the theory of equations Newton here
enunciates the following important results. He explains that
the equation whose roots are the solution of a given problem
will have as many roots as there are different possible cases;
and he considers how it happens that the equation to which
a problem leads may contain \index{Equations, differential!position of roots|(}\index{Roots of equations,!position of|(}roots which do not satisfy the
original question. He extends \index{Descartes!rule of signs of}Descartes's rule of signs to give
limits to the number of imaginary roots. He uses the \index{Continuity, principle of}principle
of continuity to explain how two real and unequal roots may
become imaginary in passing through equality, and illustrates
this by geometrical considerations; thence he shews that
imaginary roots must occur in pairs. Newton also here gives
rules to find a superior limit to the positive roots of a numerical
equation, and to determine the approximate values of the
numerical roots. He further enunciates the theorem known by
his name for finding the sum of the $n$th powers of the roots of
an equation, and laid the \index{Roots of equations,!symmetrical functions of}\index{Symmetrical functions of roots of an equation}foundation of the theory of symmetrical
functions of the roots of an equation.

\index{Algebra.!z8@Developed by (amongst others) Descartes!x2@Newton}The most interesting theorem contained in the work is his
attempt to find a rule (analogous to that of \index{Descartes!ref.\ to}Descartes for real
roots) by which the number of imaginary roots of an equation
can be determined. He knew that the result which he obtained
was not universally true, but he gave no proof and did not
explain what were the exceptions to the rule. His theorem is as
follows. Suppose the equation to be of the $n$th degree arranged
in descending powers of $x$ (the coefficient of $x^n$ being positive),
and suppose the $n + 1$ fractions
\[
1,\ \frac{n}{n-1}\;\frac{2}{1},\ \frac{n-1}{n-2}\;\frac{3}{2},\ \ldots,\ \frac{n-p+1}{n-p}\;\frac{p+1}{p},\ \ldots,\ \frac{2}{1}\;\frac{n}{n-1},\ 1
\]
to be formed and written below the corresponding terms of the
equation, then, if the square of any term when multiplied by the
corresponding fraction be greater than the product of the terms
on each side of it, put a plus sign above it: otherwise put a
minus sign above it, and put a plus sign above the first and last
terms. Now consider any two consecutive terms in the original
equation, and the two symbols written above them. Then we
may have any one of the four following cases: ($\alpha$)~the terms of
the same sign and the symbols of the same sign; ($\beta$)~the terms
of the same sign and the symbols of opposite signs; ($\gamma$)~the
terms of opposite signs and the symbols of the same sign; ($\delta$)~the
terms of opposite signs and the symbols of opposite signs.
Then it has been shewn that the number of negative roots will
not exceed the number of cases~($\alpha$), and the number of positive
roots will not exceed the number of cases~($\gamma$); and therefore the
number of imaginary roots is not less than the number of cases
($\beta$)~and~($\delta$). In other words the number of changes of signs in
the row of symbols written above the equation is an inferior
limit to the number of imaginary roots. Newton, however,
asserted that ``you may almost know how many roots are
impossible'' by counting the changes of sign in the series of
symbols formed as above. That is to say, he thought that in
general the actual number of positive, negative, and imaginary
roots could be got by the rule and not merely superior or
inferior limits to these numbers. But though he knew that the
rule was not universal he could not find (or at any rate did
not state) what were the exceptions to it: this problem was
subsequently discussed by \index{Campbell}Campbell, \index{Maclaurin!ref.\ to}Maclaurin, \index{Euler!ref.\ to}Euler, and
other writers; at last in 1865 \index{Sylvester!ref.\ to}Sylvester succeeded in proving
the general result.\footnote
  {See the \emph{Proceedings of the London Mathematical Society}, 1865, vol.~i
  no.~2.}\index{Equations, differential!theory of|)}\index{Roots of equations,!position of|)}\index{Equations, differential!position of roots|)}

\nblabel{xref:332}In August, 1684, \index{Halley!ref.\ to}Halley came to Cambridge in order to consult
Newton about the \index{Gravity,!law of|(}law of gravitation. \index{Hooke!ref.\ to}Hooke, \index{Huygens!ref.\ to}Huygens,
Halley, and \index{Wren!ref.\ to}Wren had all conjectured that the \index{Attraction, theories of}force of the
attraction of the sun or earth on an external particle varied
inversely as the square of the distance. These writers seem
independently to have shewn that, if \index{Kepler!ref.\ to}Kepler's conclusions were
rigorously true, as to which they were not quite certain, the
law of attraction must be that of the inverse square. Probably
their argument was as follows. If $v$ be the velocity of a planet,
$r$ the radius of its orbit taken as a circle, and $T$ its periodic
time, $v=2\pi r/T$. But, if $f$ be the acceleration to the centre of
the circle, we have $f = v^{2}/r$. Therefore, substituting the above
value of $v$, $f = 4\pi^{2}r/T^{2}$. Now, by \index{Kepler's laws}Kepler's third law, $T^{2}$ varies
as $r^{3}$; hence $f$ varies inversely as $r^{2}$. They could not, however,
deduce from the law the orbits of the planets. Halley explained
that their investigations were stopped by their inability to solve
this problem, and asked Newton if he could find out what the
orbit of a planet would be if the \index{Attraction, theories of|(}law of attraction were that of
the inverse square. \index{Elliptic orbits of planets}Newton immediately replied that it was an
ellipse, and promised to send or write out afresh the demonstration
of it which he had found in 1679. This was sent in
November, 1684.

\nblabel{png351a}Instigated by \index{Halley!ref.\ to}Halley, Newton now returned to the problem
of gravitation; and before the autumn of 1684, he had worked
out the substance of propositions 1--19, 21, 30, 32--35 in the
first book of the \index{Newtons@Newton's \emph{Principia}|(}\emph{Principia}. These, together with notes on the
laws of motion and various lemmas, were read for his lectures
in the Michaelmas Term, 1684.

In November Halley received Newton's promised communication,
which probably consisted of the substance of propositions
1, 11, and either proposition 17 or the first corollary of
proposition 13; thereupon Halley again went to Cambridge,
where he saw ``a curious treatise, \emph{De Motu}, drawn up since
August.'' Most likely this contained Newton's manuscript
notes of the lectures above alluded to: these notes are now
in the university library, and are headed ``\emph{De Motu Corporum}.''
Halley begged that the results might be published,
and finally secured a promise that they should be sent to the
Royal Society: they were accordingly communicated to the
Society not later than February, 1685, in the paper \emph{De Motu},
which contains the substance of the following propositions in
the \emph{Principia}, book~\textsc{i}, props.~1, 4, 6, 7, 10, 11, 15, 17, 32;
book~\textsc{ii}, props.~2, 3, 4.

It seems also to have been due to the influence and tact of
Halley at this visit in November, 1684, that Newton undertook
to attack the whole problem of gravitation, and practically
pledged himself to publish his results: these are contained in
the \emph{Principia}. As yet Newton had not determined the attraction
of a spherical body on an external point, nor had he
calculated the details of the planetary motions even if the
members of the solar system could be regarded as points. The
first problem was solved in 1685, probably either in January
or February. \nblabel{png352a}``No sooner,'' to quote from \index{Glaisher}Dr.~Glaisher's
address on the bicentenary of the publication of the \emph{Principia},
``had Newton proved this superb theorem---and we know from
his own words that he had no expectation of so beautiful a
result till it emerged from his mathematical investigation---than
all the mechanism of the universe at once lay spread before
him. When he discovered the theorems that form the first
three sections of book \textsc{i}, when he gave them in his lectures of
1684, he was unaware that the sun and earth exerted their
attractions as if they were but points. How different must
these propositions have seemed to Newton's eyes when he
realized that these results, which he had believed to be only
approximately true when applied to the solar system, were
really exact! Hitherto they had been true only in so far as he
could regard the sun as a point compared to the distance of
the planets, or the earth as a point compared to the distance
of the moon---a distance amounting to only about sixty times
the earth's radius---but now they were mathematically true, excepting
only for the slight deviation from a perfectly spherical
form of the sun, earth, and planets. We can imagine the effect
of this sudden transition from approximation to exactitude in
stimulating Newton's mind to still greater efforts. It was now
in his power to apply mathematical analysis with absolute
precision to the actual problems of astronomy.''

\index{Mechanics.!z3@Treated dynamically by Newton}Of the three fundamental principles applied in the \emph{Principia}
we may say that the idea that every particle attracts every
other particle in the universe was formed at least as early as
1666; the law of equable description of areas, its consequences,
and the fact that if the law of attraction were that of the
inverse square the orbit of a particle about a centre of force
would be a conic were proved in 1679; and, lastly, the discovery
that a sphere, whose density at any point depends only on the
distance from the centre, attracts an external point as if the
whole mass were collected at its centre was made in 1685.
It was this last discovery that enabled him to apply the first
two principles to the phenomena of bodies of finite size.\index{Attraction, theories of|)}\index{Gravity,!law of|)}

\nblabel{png353a}The draft of the first book of the \emph{Principia} was finished
before the summer of 1685, but the corrections and additions
took some time, and the book was not presented to the Royal
Society until April~28, 1686. This book is given up to the
consideration of the motion of particles or bodies in free space
either in known orbits, or under the action of known forces,
or under their mutual attraction; and in particular to indicating
how the effects of \index{Disturbing forces}disturbing forces may be calculated.
In it also Newton generalizes the law of attraction into a
statement that every particle of matter in the universe attracts
every other particle with a force which varies directly as the
product of their masses, and inversely as the square of the
distance between them; and he thence deduces the law of
attraction for spherical shells of constant density. The book
is prefaced by an introduction on the science of dynamics,
which defines the limits of mathematical investigation. His
object, he says, is to apply mathematics to the phenomena
of nature; among these phenomena motion is one of the
most important; now motion is the effect of force, and,
though he does not know what is the nature or origin of
force, still many of its effects can be measured; and it is
these that form the subject-matter of the work.

The second book of the \emph{Principia} was completed by the
summer of 1686. This book treats of motion in a resisting
medium, and of hydrostatics and hydrodynamics, with special
applications to waves, tides, and acoustics. He concludes it
by shewing that the \index{Vortices, Cartesian}\index{Cartesian vortices}Cartesian theory of vortices was inconsistent
both with the known facts and with the laws of
motion.

The next nine or ten months were devoted to the third
book. Probably for this originally he had no materials ready.
He commences by discussing when and how far it is justifiable
to construct hypotheses or theories to account for
known phenomena. He proceeds to apply the theorems
obtained in the first book to the chief phenomena of the
solar system, and to determine the masses and distances of
the planets and (whenever sufficient data existed) of their
satellites. In particular the motion of the moon, the various
inequalities therein, and the theory of the tides are worked
out in detail. He also investigates the theory of comets,
shews that they belong to the solar system, explains how
from three observations the orbit can be determined, and
illustrates his results by considering certain special comets.
The third book as we have it is but little more than a sketch
of what Newton had finally proposed to himself to accomplish;
his original scheme is among the ``Portsmouth papers,'' and
his notes shew that he continued to work at it for some years
after the publication of the first edition of the \emph{Principia}: the
most interesting of his memoranda are those in which by
means of fluxions he has carried his results beyond the point
at which he was able to translate them into geometry.\footnote
  {For a fuller account of the \emph{Principia} see \index{Ball, W. W. R., ref.\ to}my \emph{Essay on the Genesis,
  Contents, and History of Newton's Principia}, London, 1893.}

The demonstrations throughout the work are geometrical,
but to readers of ordinary ability are rendered unnecessarily
difficult by the absence of illustrations and explanations, and
by the fact that no clue is given to the method by which
Newton arrived at his results. The reason why it was presented
in a geometrical form appears to have been that the
infinitesimal calculus was then unknown, and, had Newton
used it to demonstrate results which were in themselves
opposed to the prevalent philosophy of the time, the controversy
as to the truth of his results would have been hampered
by a dispute concerning the validity of the methods used
in proving them. He therefore cast the whole reasoning
into a geometrical shape which, if somewhat longer, can at
any rate be made intelligible to all mathematical students,
So closely did he follow the lines of Greek geometry that he
constantly used \index{Graphical methods}graphical methods, and represented forces,
velocities, and other magnitudes in the Euclidean way by
straight lines (\emph{ex.~gr.}\ book~\textsc{i}, lemma~10), and not by a certain
number of units. The latter and modern method had been
introduced by \index{Wallis!ref.\ to}Wallis, and must have been familiar to Newton.
The effect of his confining himself rigorously to classical
geometry is that the \emph{Principia} is written in a language which
is archaic, even if not unfamiliar.

The adoption of geometrical methods in the \emph{Principia} for
purposes of demonstration does not indicate a preference on
Newton's part for geometry over analysis as an instrument
of research, for it is known now that Newton used the \index{Fluxions}fluxional
calculus in the first instance in finding some of the theorems,
especially those towards the end of book~\textsc{i} and in book~\textsc{ii};
and in fact one of the most important uses of that calculus is
stated in book~\textsc{ii}, lemma~2. But it is only just to remark
that, at the time of its publication and for nearly a century
afterwards, the differential and fluxional calculus were not fully
developed, and did not possess the same superiority over the
method he adopted which they do now; and it is a matter for
astonishment that when Newton did employ the calculus he
was able to use it to so good an effect.

\nblabel{png355a}The printing of the work was slow, and it was not finally
published till the summer of 1687. The cost was borne by
\index{Halley!ref.\ to}Halley, who also corrected the proofs, and even put his own
researches on one side to press the printing forward. The
conciseness, absence of illustrations, and synthetical character
of the book restricted the numbers of those who were able to
appreciate its value; and, though nearly all competent critics
admitted the validity of the conclusions, some little time
elapsed before it affected the current beliefs of educated men.
I should be inclined to say (but on this point opinions differ
widely) that within ten years of its publication it was generally
accepted in Britain as giving a correct account of the laws of
the universe; it was similarly accepted within about twenty
years on the continent, except in France, where the \index{Vortices, Cartesian}\index{Cartesian vortices}Cartesian
hypothesis held its ground until \index{Voltaire on Newton}Voltaire in 1733 took up the
advocacy of the Newtonian theory.

\nblabel{png356a}The manuscript of the \emph{Principia} was finished by 1686.
Newton devoted the remainder of that year to his paper on
physical optics, the greater part of which is given up to the
subject of diffraction.\index{Newtons@Newton's \emph{Principia}|)}

In 1687 \index{James II. of England, ref.\ to}James II. having tried to force the university to
admit as a master of arts a Roman Catholic priest who refused
to take the oaths of supremacy and allegiance, Newton took a
prominent part in resisting the illegal interference of the king,
and was one of the deputation sent to London to protect the
rights of the university. The active part taken by Newton in
this affair led to his being in 1689 elected member for the
university. This parliament only lasted thirteen months, and
on its dissolution he gave up his seat. He was subsequently
returned in 1701, but he never took any prominent part in
politics.

On his coming back to Cambridge in 1690 he resumed his
mathematical studies and correspondence, but probably did not
lecture. The two letters to \index{Wallis!ref.\ to}Wallis, in which he explained his
method of \index{Fluxions}fluxions and \index{Fluents}fluents, were written in 1692 and published
in 1693. Towards the close of 1692 and throughout the
two following years, Newton had a long illness, suffering from
insomnia and general nervous irritability. Perhaps he never
quite regained his elasticity of mind, and, though after his
recovery he shewed the same power in solving any question
propounded to him, he ceased thenceforward to do original
work on his own initiative, and it was somewhat difficult to
stir him to activity in new subjects.

In 1694 Newton began to collect data connected with the
irregularities of the moon's motion with the view of revising the
part of the \emph{Principia} which dealt with that subject. To render
the observations more accurate, he forwarded to \index{Flamsteed}Flamsteed\footnote
  {\emph{John Flamsteed}, born at Derby in 1646 and died at Greenwich in 1719,
  was one of the most distinguished astronomers of this age, and the first
  astronomer-royal. Besides much valuable work in astronomy, he invented
  the system (published in 1680) of drawing maps by projecting the surface of
  the sphere on an enveloping cone, which can then be unwrapped. His life
  by \index{Baily, R. F., on Flamsteed}R.~F.~Baily was published in London in 1835, but various statements in
  it should be read side by side with those in \index{Brewster, D., ref.\ to}Brewster's life of Newton.
  Flamsteed was succeeded as astronomer-royal by \index{Halley!ref.\ to}Edmund Halley (see below,
  pp.~\pageref{xref:379}--\pageref{xref:380}).}
a table of corrections for \index{Refraction}refraction which he had previously made.
This was not published till 1721, when \index{Halley!ref.\ to}Halley communicated it
to the Royal Society. The original calculations of Newton and
the papers connected with them are in the Portsmouth collection,
and shew that Newton obtained it by finding the path of a ray,
by means of quadratures, in a manner equivalent to the solution
of a differential equation. As an illustration of Newton's
genius, I may mention that even as late as 1754 \index{Euler!ref.\ to}Euler failed to
solve the same problem. In 1782 \index{Laplace!ref.\ to}Laplace gave a rule for constructing
such a table, and his results agree substantially with those of Newton.

\nblabel{png357a}I do not suppose that Newton would in any case have produced
much more original work after his illness; but his
appointment in 1696 as warden, and his promotion in 1699
to the mastership of the Mint, at a salary of £1500 a year,
brought his scientific investigations to an end, though it was
only after this that many of his previous investigations were
published in the form of books. In 1696 he moved to London,
in 1701 he resigned the Lucasian chair, and in 1703 he was
elected president of the Royal Society.

\nblabel{png357b}In 1704 Newton published his \emph{Optics}, which contains the
results of the papers already mentioned. To the first edition
of this book were appended two minor works which have no
special connection with optics; one being on \index{Cubic curves, Newton on|(}cubic curves, the
other on the quadrature of curves and on \index{Fluxions}fluxions. Both of
them were manuscripts with which his friends and pupils
were familiar, but they were here published \emph{urbi et orbi} for the
first time.

The first of these appendices is entitled \emph{Enumeratio Linearum
Tertii Ordinis};\footnote
  {On this work and its bibliography, see \index{Ball, W. W. R., ref.\ to}my memoir in the \emph{Transactions
  of the London Mathematical Society}, 1891, vol.~xxii, pp.~104--143.}
the object seems to be to illustrate the use of
analytical geometry, and as the application to conics was well
known, Newton selected the theory of cubics.

\nblabel{xref:340}\nblabel{png358a}He begins with some general theorems, and \index{Curves, classification of}classifies curves
according as their equations are algebraical or transcendental;
the former being cut by a straight line in a number of points
(real or imaginary) equal to the degree of the curve, the latter
being cut by a straight line in an infinite number of points.
Newton then shews that many of the most important properties
of conics have their analogues in the theory of \index{Curves of the third degree|(}cubics, and he
discusses the theory of \index{Asymptotes, theory of}asymptotes and curvilinear diameters.

After these general theorems, he commences his detailed
examination of cubics by pointing out that a cubic must have
at least one real point at infinity. If the asymptote or tangent
at this point be at a finite distance, it may be taken for the
axis of $y$. This asymptote will cut the curve in three points
altogether, of which at least two are at infinity. If the third
point be at a finite distance, then (by one of his general theorems
on asymptotes) the equation can be written in the form
\[
xy^{2} + hy = ax^{3} + bx^{2} + cx + d,
\]
where the axes of $x$ and $y$ are the asymptotes of the hyperbola
which is the locus of the middle points of all chords drawn
parallel to the axis of $y$; while, if the third point in which this
asymptote cuts the curve be also at infinity, the equation can be
written in the form
\[
xy = ax^{3} + bx^{2} + cx + d.
\]

Next he takes the case where the tangent at the real point
at infinity is not at a finite distance. A line parallel to the
direction in which the curve goes to infinity may be taken as
the axis of $y$. Any such line will cut the curve in three points
altogether, of which one is by hypothesis at infinity, and one is
necessarily at a finite distance. He then shews that if the
remaining point in which this line cuts the curve be at a finite
distance, the equation can be written in the form
\[
y^{2} = ax^{3} + bx^{2} + cx + d;
\]
while if it be at an infinite distance, the equation can be
written in the form
\[
y = ax^{3} + bx^{2} + cx + d.
\]

Any cubic is therefore reducible to one of four characteristic
forms. \index{Multiple points}Each of these forms is then discussed in detail, and the
possibility of the existence of double points, isolated ovals, \&c.,
is worked out. The final result is that in all there are seventy-eight
possible forms which a cubic may take. Of these Newton
enumerated only seventy-two; four of the remainder were
mentioned by \index{Stirling}Stirling in 1717, one by \index{Nicole!ref.\ to}Nicole in 1731, and one
by \index{Bernoulli, Nicholas!ref.\ to}Nicholas Bernoulli about the same time.

In the course of the work Newton states the remarkable
theorem that, just as the shadow of a circle (cast by a luminous
point on a plane) gives rise to all the conics, so the shadows of
the curves represented by the equation $y^{2} = ax^{3} + bx^{2} + cx + d$
give rise to all the cubics. This remained an unsolved puzzle
until 1731, when Nicole and \index{Clairaut!ref.\ to}Clairaut gave demonstrations of
it; a better proof is that given by \index{Murdoch}Murdoch in 1740, which
depends on the classification of these curves into five species
according as to whether their points of intersection with the axis
of $x$ are real and unequal, real and two of them equal (two
cases), real and all equal, or two imaginary and one real.\index{Cubic curves, Newton on|)}\index{Curves of the third degree|)}

In this tract Newton also discusses double points in the
plane and at infinity, the description of curves satisfying given
conditions, and the graphical solution of problems by the use of
curves.

\nblabel{png359a}The second appendix to the \emph{Optics} is entitled \emph{De Quadratura
Curvarum}. Most of it had been communicated to \index{Barrow!ref.\ to}Barrow in
1668 or 1669, and probably was familiar to Newton's pupils
and friends from that time onwards. It consists of two parts.

\nblabel{xref:341}The bulk of the first part is a statement of Newton's method
of effecting the quadrature and \index{Rectification of curves}\index{Curves, rectification of|(}rectification of curves by means
of \index{Quadrature of curves|(}\index{Infinite series, quadrature of curves in|(}infinite series; it is noticeable as containing the earliest use
in print of literal \index{Exponents}\index{Indices}indices, and a printed statement of the
binomial theorem, but these novelties are introduced only
incidentally. The main object is to give rules for developing a
function of $x$ in a series in ascending powers of $x$, so as to
enable mathematicians to effect the quadrature of any curve
in which the ordinate $y$ can be expressed as an explicit
algebraical function of the abscissa $x$. \index{Wallis!ref.\ to}Wallis had shewn how
this quadrature could be found when $y$ was given as a sum of a
number of multiples of powers of $x$, and Newton's rules of
\index{Expansions@Expansion in series}expansion here established rendered possible the similar quadrature
of any curve whose ordinate can be expressed as the sum
of an \index{Infinite series, difficulties in connection with}infinite number of such terms. In this way he effects the
quadrature of the curves
\[
y=\frac{a^2}{b+x},\quad y=(a^2\pm x^2)^\frac{1}{2},\quad y=(x-x^2)^\frac{1}{2},\quad y=\left(\frac{1+a x^2}{1-b x^2}\right)^\frac{1}{2},
\]
but naturally the results are expressed as infinite series. He
then proceeds to curves whose ordinate is given as an implicit
function of the abscissa; and he gives a method by which $y$ can
be expressed as an infinite series in ascending powers of $x$,
but the application of the rule to any curve demands in general
such complicated numerical calculations as to render it of little
value. He concludes this part by shewing that the \index{Rectification of curves}rectification
of a curve can be effected in a somewhat similar way. His
process is equivalent to finding the integral with regard to $x$
of $(1+{\dot{y}}^2)^\frac{1}{2}$ in the form of an infinite series. I should add
that Newton indicates the importance of determining whether
the series are \index{Convergency}convergent---an observation far in advance of
his time---but he knew of no general test for the purpose;
and in fact it was not until \index{Gauss!ref.\ to}Gauss and \index{Cauchy!ref.\ to}Cauchy took up the
question that the necessity of such limitations was commonly
recognized.\index{Curves, rectification of|)}

The part of the appendix which I have just described is
practically the same as Newton's manuscript \emph{De Analysi per
Equationes Numero Terminorum Infinitas}, which was subsequently
printed in 1711. It is said that this was originally
intended to form an appendix to \index{Kinckhuysen, ref.\ to}Kinckhuysen's \emph{Algebra},
which, as I have already said, he at one time intended to edit.
The substance of it was communicated to \index{Barrow!ref.\ to}Barrow, and by him
to \index{Collins, J.!ref.\ to}Collins, in letters of July~31 and August~12, 1669; and a
summary of part of it was included in the letter of October~24,
1676, sent to \index{Leibnitz!ref.\ to}Leibnitz.

It should be read in connection with Newton's \emph{Methodus
Differentialis}, also published in 1711. Some additional
theorems are there given, and he discusses his \index{Interpolation, method of}method of
interpolation, which had been briefly described in the letter
of October~24, 1676. The principle is this. If $y = \phi(x)$ be a
function of $x$, and if, when $x$ is successively put equal to
$a_{1}, a_{2}, \ldots$ , the values of $y$ be known and be $b_{1}, b_{2}, \ldots$ , then a
parabola whose equation is $y = p + qx + rx^{2}+ \ldots$ can be drawn
through the points $(a_{1}, b_{1}), (a_{2}, b_{2}), \ldots$ , and the ordinate of this
parabola may be taken as an approximation to the ordinate of
the curve. The degree of the parabola will of course be one
less than the number of given points. Newton points out
that in this way the areas of any curves can be approximately
determined.\index{Infinite series, quadrature of curves in|)}\index{Quadrature of curves|)}

\nblabel{png361a}The second part of this appendix to the \emph{Optics} contains a
description of Newton's method of \index{Fluxions}fluxions. This is best considered
in connection with Newton's manuscript on the same
subject which was published by \index{Colson on Newton's fluxions}John Colson in 1736, and of
which it is a summary.

\index{Fluxional calculus|(}The invention of the \index{Calculus, infinitesimal|(}infinitesimal calculus was one of the
great intellectual achievements of the seventeenth century. This
method of analysis, expressed in the notation of fluxions and
\index{Fluents}fluents, was used by Newton in or before 1666, but no account
of it was published until 1693, though its general outline was
known by his friends and pupils long anterior to that year, and
no complete exposition of his methods was given before 1736.

The idea of a fluxion or \index{Differential coefficient}differential coefficient, as treated at
this time, is simple. When two quantities---\emph{e.g.} the radius of a
sphere and its volume---are so related that a change in one
causes a change in the other, the one is said to be a function of
the other. The ratio of the rates at which they change is
termed the differential coefficient or fluxion of the one with
regard to the other, and the process by which this ratio is
determined is known as differentiation. Knowing the differential
coefficient and one set of corresponding values of the two
quantities, it is possible by summation to determine the relation
between them, as \index{Cavalieri!ref.\ to}Cavalieri and others had shewn; but often the
process is difficult. If, however, we can reverse the process of
differentiation we can obtain this result directly. This process
of reversal is termed integration. It was at once seen that
problems connected with the quadrature of curves, and the
determination of volumes (which were soluble by summation, as
had been shewn by the employment of indivisibles), were
reducible to integration. In mechanics also, by integration,
velocities could be deduced from known accelerations, and
distances traversed from known velocities. In short, wherever
things change according to known laws, here was a possible
method of finding the relation between them. It is true that,
when we try to express observed phenomena in the language of
the calculus, we usually obtain an equation involving the
variables, and their differential coefficients---and possibly the
solution may be beyond our powers. Even so, the method is
often fruitful, and its use marked a real advance in thought and
power.

I proceed to describe somewhat fully Newton's methods as
described by \index{Colson on Newton's fluxions}Colson. Newton assumed that all geometrical
magnitudes might be conceived as generated by continuous
motion; thus a line may be considered as generated by the
motion of a point, a surface by that of a line, a solid by that of
a surface, a plane angle by the rotation of a line, and so on.
The quantity thus generated was defined by him as the \index{Fluents}fluent
or flowing quantity. The velocity of the moving magnitude
was defined as the \index{Fluxions}fluxion of the \index{Fluents}fluent. This seems to be the
earliest definite recognition of the idea of a continuous function,
though it had been foreshadowed in some of Napier's papers.

Newton's treatment of the subject is as follows. There are
two kinds of problems. The object of the first is to find the
fluxion of a given quantity, or more generally ``the relation of
the fluents being given, to find the relation of their fluxions.''
This is equivalent to differentiation. The object of the second
or inverse method of fluxions is from the fluxion or some
relations involving it to determine the \index{Fluents}fluent, or more generally
``an equation being proposed exhibiting the relation of the
\index{Fluxions}fluxions of quantities, to find the relations of those quantities,
or fluents, to one another.''\footnoteibid{}{fnibid:colsona}
  {\index{Colson on Newton's fluxions}Colson's edition of Newton's manuscript,}{pp.~21,~22.}
  %[**TN: page numbers in footnote changed to arabic for consistency with following footnotes]
This is equivalent either to
integration which Newton termed the method of quadrature,
or to the solution of a differential equation which was called
by Newton the inverse method of tangents. The methods for
solving these problems are discussed at considerable length.

Newton then went on to apply these results to questions
connected with the \index{Maxima and minima, determination of}maxima and minima of quantities, the
method of drawing tangents to curves, and the curvature of
curves (namely, the determination of the centre of curvature,
the radius of curvature, and the rate at which the radius of
curvature increases). He next considered the quadrature of
curves, and the \index{Rectification of curves}\index{Curves, rectification of}rectification of curves.\footnoteibid{fnibid:colsona}{fnibid:colsonb}
  {\index{Colson on Newton's fluxions}Colson's edition of Newton's manuscript,}{pp.~22,~23.}
  %[**TN: page numbers in footnote changed to arabic for consistency with following footnotes]
In finding the maximum
and minimum of functions of one variable we regard the
change of sign of the difference between two consecutive values
of the function as the true criterion; but his argument is that
when a quantity increasing has attained its maximum it can
have no further increment, or when decreasing it has attained
its minimum it can have no further decrement; consequently
the fluxion must be equal to nothing.

It has been remarked that neither Newton nor \index{Leibnitz!ref.\ to}Leibnitz
produced a calculus, that is, a classified collection of rules; and
that the problems they discussed were treated from first principles.
That, no doubt, is the usual sequence in the history of
such discoveries, though the fact is frequently forgotten by
subsequent writers. In this case I think the statement, so far
as Newton's treatment of the differential or fluxional part of
the calculus is concerned, is incorrect, as the foregoing account
sufficiently shews.

If a flowing quantity or \index{Fluents}fluent were represented by $x$,
Newton denoted its fluxion by $\dot{x}$, the fluxion of $\dot{x}$ or second
\index{Fluxions}fluxion of $x$ by $\ddot{x}$, and so on. Similarly the \index{Fluents}fluent of $x$ was
denoted by \fbox{$\mathstrut\,x\,$}, or sometimes by $x'$ or $[x]$. The infinitely small
part by which a fluent such as $x$ increased in a small interval of
time measured by $o$ was called the \index{Moments in theory of fluxions}moment of the fluent; and
its value was shewn\footnoteibid{fnibid:colsonb}{fnibid:colsonc}
  {\index{Colson on Newton's fluxions}Colson's edition of Newton's manuscript,}{p.~24.}
to be $\dot{x}o$. Newton adds the important
remark that thus we may in any problem neglect the terms
multiplied by the second and higher powers of $o$, and we can
always find an equation between the co-ordinates $x$, $y$ of a
point on a curve and their fluxions $\dot{x}$, $\dot{y}$. It is an application of
this principle which constitutes one of the chief values of the
calculus; for if we desire to find the effect produced by
several causes on a system, then, if we can find the effect produced
by each cause when acting alone in a very small time,
the total effect produced in that time will be equal to the sum
of the separate effects. I should here note the fact that \index{Vince, ref.\ to}Vince
and other English writers in the eighteenth century used $\dot{x}$ to
denote the increment of $x$ and not the velocity with which it
increased; that is, $\dot{x}$ in their writings stands for what Newton
would have expressed by $\dot{x}o$ and what \index{Leibnitz!ref.\ to}Leibnitz would have
written as $dx$.

I need not discuss in detail the manner in which Newton
treated the problems above mentioned. I will only add that,
in spite of the form of his definition, the introduction into
geometry of the idea of time was evaded by supposing that
some quantity (\emph{ex.~gr.}\ the abscissa of a point on a curve)
increased equably; and the required results then depend on
the rate at which other quantities (\emph{ex.~gr.}\ the ordinate or
radius of curvature) increase relatively to the one so chosen.\footnoteibid{fnibid:colsonc}{fnibid:colsond}
  {\index{Colson on Newton's fluxions}Colson's edition of Newton's manuscript,}{p.~20}
The \index{Fluents}fluent so chosen is what we now call the independent
variable; its fluxion was termed the ``principal fluxion''; and,
of course, if it were denoted by $x$, then $\dot{x}$ was constant, and
consequently $\ddot{x}$ = 0.

There is no question that Newton used a method of fluxions
in 1666, and it is practically certain that accounts of it were
communicated in manuscript to friends and pupils from and
after 1669. The manuscript, from which most of the above
summary has been taken, is believed to have been written
between 1671 and 1677, and to have been in circulation at
Cambridge from that time onwards, though it is probable that
parts were rewritten from time to time. It was unfortunate that
it was not published at once. Strangers at a distance naturally
judged of the method by the letter to \index{Wallis!ref.\ to}Wallis in 1692, or by the
\emph{Tractatus de Quadratura Curvarum}, and were not aware that
it had been so completely developed at an earlier date. This
was the cause of numerous misunderstandings. At the same
time it must be added that all mathematical analysis was leading
up to the ideas and methods of the infinitesimal calculus. Foreshadowings
of the principles and even of the language of that
calculus can be found in the writings of \index{Napier of Merchiston!ref.\ to}Napier, \index{Kepler!ref.\ to}Kepler, \index{Cavalieri!ref.\ to}Cavalieri,
\index{Pascal!ref.\ to}Pascal, \index{Fermat!ref.\ to}Fermat, Wallis, and \index{Barrow!ref.\ to}Barrow. It was Newton's
good luck to come at a time when everything was ripe for the
discovery, and his ability enabled him to construct almost at
once a complete calculus.

\nblabel{png365a}The infinitesimal calculus can also be expressed in the notation
of the differential calculus: a notation which was invented by
\index{Leibnitz!ref.\ to}Leibnitz probably in 1675, certainly by 1677, and was published
in 1684, some nine years before the earliest printed account of
Newton's method of \index{Fluxions}fluxions. \index{Fluxional calculus!controversy}But the question whether the
general idea of the calculus expressed in that notation was
obtained by Leibnitz from Newton, or whether it was discovered
independently, gave rise to a long and bitter controversy. The
leading facts are given in the \hyperref[chapter:XVII]{next chapter}.\index{Calculus, infinitesimal|)}

The remaining events of Newton's life require little or no
comment. In 1705 he was knighted. From this time onwards
he devoted much of his leisure to theology, and wrote at great
length on prophecies and predictions, subjects which had always
been of interest to him. His \emph{Universal Arithmetic} was published
by Whiston in 1707, and his \emph{Analysis by Infinite Series}
in 1711; but Newton had nothing to do with the preparation
of either of these for the press. His evidence before the House
of Commons in 1714 on the \index{Longitude}determination of longitude at sea
marks an important epoch in the history of navigation.

\index{Fluxional calculus!controversy}The dispute with \index{Leibnitz!ref.\ to}Leibnitz as to whether he had derived the
ideas of the differential calculus from Newton or invented it
independently originated about 1708, and occupied much
of Newton's time, especially between the years 1709 and
1716.\index{Fluxional calculus|)}

In 1709 Newton was persuaded to allow \index{Cotes!ref.\ to}Cotes to prepare
the long-talked-of second edition of the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia}; it was issued
in March 1713. A third edition was published in 1726 under
the direction of \index{Pemberton, ref.\ to}Henry Pemberton. In 1725 Newton's health
began to fail. \nblabel{png366a}He died on March~20, 1727, and eight days
later was buried in Westminster Abbey.

\nblabel{png366b}His chief works, taking them in their order of publication,
are the \index{Newtons@Newton's \emph{Principia}}\emph{Principia}, published in 1687; the \emph{Optics} (with appendices
on \emph{cubic curves}, the \emph{quadrature and rectification of curves
by the use of infinite series}, and the \emph{method of fluxions}), published
in 1704; the \emph{Universal Arithmetic}, published in 1707;
the \emph{Analysis per Series}, \index{Fluxions}\emph{Fluxiones}, \&c., and the \emph{Methodus Differentialis},
published in 1711; the \emph{Lectiones Opticae}, published in
1729; the \emph{Method of Fluxions}, \&c.\ (that is, \emph{Newton's manuscript
on fluxions}), translated by \index{Colson on Newton's fluxions}J.~Colson and published in 1736; and
the \emph{Geometria Analytica}, printed in 1779 in the first volume of
Horsley's edition of Newton's works.

In appearance Newton was short, and towards the close of
his life rather stout, but well set, with a square lower jaw,
brown eyes, a broad forehead, and rather sharp features. His
hair turned grey before he was thirty, and remained thick and
white as silver till his death.

As to his manners, he dressed slovenly, was rather languid,
and was often so absorbed in his own thoughts as to be anything
but a lively companion. Many anecdotes of his extreme
absence of mind when engaged in any investigation have been
preserved. Thus once when riding home from Grantham he
dismounted to lead his horse up a steep hill; when he turned at
the top to remount, he found that he had the bridle in his hand,
while his horse had slipped it and gone away. Again, on the
few occasions when he sacrificed his time to entertain his friends,
if he left them to get more wine or for any similar reason, he
would as often as not be found after the lapse of some time
working out a problem, oblivious alike of his expectant guests
and of his errand. He took no exercise, indulged in no amusements,
and worked incessantly, often spending eighteen or nineteen
hours out of the twenty-four in writing.

\nblabel{png367a}In character he was religious and conscientious, with an
exceptionally high standard of morality, having, as \index{Burnet on Newton}Bishop
Burnet said, ``the whitest soul'' he ever knew. Newton was
always perfectly straightforward and honest; but in his controversies
with \index{Leibnitz!ref.\ to}Leibnitz, \index{Hooke!ref.\ to}Hooke, and others, though scrupulously
just, he was not generous; and it would seem that he frequently
took offence at a chance expression when none was intended.
He modestly attributed his discoveries largely to the admirable
work done by his predecessors; and once explained that, if he
had seen farther than other men, it was only because he had
stood on the shoulders of giants. He summed up his own
estimate of his work in the sentence, ``I do not know what I
may appear to the world; but to myself I seem to have been
only like a boy, playing on the sea-shore, and diverting myself,
in now and then finding a smoother pebble, or a prettier shell
than ordinary, whilst the great ocean of truth lay all undiscovered
before me.'' He was morbidly sensitive to being
involved in any discussions. I believe that, with the exception
of his papers on optics, every one of his works was published
only under pressure from his friends and against his own wishes.
There are several instances of his communicating papers and
results on condition that his name should not be published:
thus when in 1669 he had, at \index{Collins, J.!ref.\ to}Collins's request, solved some
problems on harmonic series and on annuities which had previously
baffled investigation, he only gave permission that his
results should be published ``so it be,'' as he says, ``without my
name to it; for I see not what there is desirable in public
esteem, were I able to acquire and maintain it: it would perhaps
increase my acquaintance, the thing which I chiefly study
to decline.''

Perhaps the most wonderful single illustration of his powers
was the composition in seven months of the first book of the
\index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia}, and the expression of the numerous and complex
results in classical geometrical form. As other illustrations of
his ability I may mention his solutions of the problem of \index{Pappus!ref.\ to}Pappus,
of \index{Bernoulli, John!ref.\ to}John Bernoulli's challenge, and of the question of orthogonal
\index{Trajectories}trajectories. The problem of Pappus, here alluded to, is to find
the locus of a point such that the rectangle under its distances
from two given straight lines shall be in a given ratio to the
rectangle under its distances from two other given straight lines.
Many geometricians from the time of \index{Apollonius!ref.\ to}Apollonius had tried to
find a geometrical solution and had failed, but what had proved
insuperable to his predecessors seems to have presented little
difficulty to Newton who gave an elegant demonstration that
the locus was a conic. Geometry, said \index{Lagrange!ref.\ to}Lagrange when recommending
the study of analysis to his pupils, is a strong bow,
but it is one which only a Newton can fully utilize. As another
example I may mention that in 1696 \index{Bernoulli, John!ref.\ to}John Bernoulli challenged
mathematicians (i)~to determine the \index{Brachistochrone, the}brachistochrone, and (ii)
to find a curve such that if any line drawn from a fixed point $O$
cut it in $P$ and $Q$ then $OP^{n} + OQ^{n}$ would be constant. \index{Leibnitz!ref.\ to}Leibnitz
solved the first of these questions after an interval of rather
more than six months, and then suggested they should be sent
as a challenge to Newton and others. Newton received the
problems on Jan.~29, 1697, and the next day gave the complete
solutions of both, at the same time generalising the second
question. An almost exactly similar case occurred in 1716
when Newton was asked to find the orthogonal trajectory of a
family of curves. In five hours Newton solved the problem in
the form in which it was propounded to him, and laid down the
principles for finding trajectories.

It is almost impossible to describe the effect of Newton's
writings without being suspected of exaggeration. But, if the
state of mathematical knowledge in 1669 or at the death of
\index{Pascal!ref.\ to}Pascal or \index{Fermat!ref.\ to}Fermat be compared with what was known in 1700
it will be seen how immense was the advance. In fact we
may say that it took mathematicians half a century or more
before they were able to assimilate the work produced in those
years.

\nblabel{png369a}In pure geometry Newton did not establish any new methods,
but no modern writer has shewn the same power in using those
of classical geometry. In algebra and the theory of equations
he introduced the system of literal indices, established the
binomial theorem, and created no inconsiderable part of the
theory of equations: one rule which he enunciated in this
subject remained till a few years ago an unsolved riddle which
had overtaxed the resources of succeeding mathematicians. In
analytical geometry, he introduced the modern classification of
curves into algebraical and transcendental; and established
many of the fundamental properties of asymptotes, multiple
points, and isolated loops, illustrated by a discussion of cubic
curves. The \index{Fluxions}fluxional or infinitesimal calculus was invented by
Newton in or before the year 1666, and circulated in manuscript
amongst his friends in and after the year 1669, though no
account of the method was printed till 1693. The fact that the
results are nowadays expressed in a different notation has led
to Newton's investigations on this subject being somewhat
overlooked.

Newton, further, was the first to place dynamics on a
satisfactory basis, and from dynamics he deduced the theory of
statics: this was in the introduction to the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia} published
in 1687. The theory of attractions, the application of the
principles of mechanics to the solar system, the creation of
physical astronomy, and the establishment of the law of universal
gravitation are due to him, and were first published in the
same work, but of the nature of gravity he confessed his
ignorance, though he found inconceivable the idea of action at
a distance. The particular questions connected with the motion
of the earth and moon were worked out as fully as was then
possible. \index{Hydrodynamics. Developed by!z1@Newton}The theory of hydrodynamics was created in the
second book of the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia}, and he added considerably to the
theory of \index{Hydrostatics. Developed by!z5@Newton}hydrostatics which may be said to have been first
discussed in modern times by \index{Pascal!ref.\ to}Pascal. The theory of the propagation
of waves, and in particular the application to determine
the velocity of sound, is due to Newton and was published in
1687. In geometrical optics, he explained amongst other things
the decomposition of light and the theory of the rainbow; he
invented the reflecting \index{Telescopes}telescope known by his name, and the
sextant. In physical optics, he suggested and elaborated the
emission theory of light.

The above list does not exhaust the subjects he investigated,
but it will serve to illustrate how marked was his influence on
the history of mathematics. On his writings and on their
effects, it will be enough to quote the remarks of two or three
of those who were subsequently concerned with the subject-matter
of the \emph{Principia}. \index{Lagrange!ref.\ to}Lagrange described the \emph{Principia} as
the greatest production of the human mind, and said he felt
dazed at such an illustration of what man's intellect might be
capable. In describing the effect of his own writings and those
of \index{Laplace!ref.\ to}Laplace it was a favourite remark of his that Newton was
not only the greatest genius that had ever existed, but he was
also the most fortunate, for as there is but one universe, it can
happen but to one man in the world's history to be the interpreter
of its laws. Laplace, who is in general very sparing of
his praise, makes of Newton the one exception, and the words
in which he enumerates the causes which ``will always assure
to the \emph{Principia} a pre-eminence above all the other productions
of human genius'' have been often quoted. Not less remarkable
is the homage rendered by \index{Gauss!ref.\ to}Gauss; for other great mathematicians
or philosophers he used the epithets magnus, or clarus, or
clarissimus: for Newton alone he kept the prefix summus.
Finally \index{Biot!ref.\ to}Biot, who had made a special study of Newton's works,
sums up his remarks by saying, ``comme géomètre et comme
expérimentateur Newton est sans égal; par la réunion de ces
deux genres de génies à leur plus haut degré, il est sans
exemple.''\index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)|)}

\nbChapter{XVII}{Leibnitz and the Mathematicians of the First Half of the Eighteenth Century}{Leibnitz}{}
  {See Cantor, vol.~iii; other authorities for the mathematicians of the
  period are mentioned in the footnotes.}
\index{Mechanics.!z4@Subsequently extended by (among others) D'Alembert, Maclaurin, Euler, Lagrange, Laplace, and Poisson, chapters \textsc{xvii, xviii}}\index{Cantor, M., ref.\ to}\nblabel{png371a}

\nblabel{png371b}\First{I have} briefly traced in the \hyperref[chapter:XVI]{last chapter} the nature and extent
of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's contributions to science. Modern analysis is,
however, derived directly from the works of \index{Leibnitz!ref.\ to}Leibnitz and the
elder Bernoullis; and it is immaterial to us whether the fundamental
ideas of it were obtained by them from Newton, or
discovered independently. The English mathematicians of
the years considered in this chapter continued to use the
language and notation of Newton; they are thus somewhat
distinct from their continental contemporaries, and I have therefore
grouped them together in a section by themselves.


\section*{Leibnitz and the Bernoullis.\nblabel{png371c}}

\index{Leibnitz|(}\nblabel{png371d}\nbparagraph{Leibnitz.\footnote
  {See the life of Leibnitz by \index{Guhrauer on Leibnitz}G.~E.~Guhrauer, two volumes and a supplement,
  Breslau, 1842 and 1846.  Leibnitz's mathematical papers have been
  collected and edited by \index{Gerhardt, ref.\ to}C.~J.~Gerhardt in seven volumes, Berlin and Halle,
  1849--63.}}
\emph{Gottfried Wilhelm Leibnitz} (or \emph{Leibniz}) was born
at Leipzig on June~21 (O.S.), 1646, and died at Hanover on
November~14, 1716. His father died before he was six, and the
teaching at the school to which he was then sent was inefficient,
but his industry triumphed over all difficulties; by the
time he was twelve he had taught himself to read Latin easily,
and had begun Greek; and before he was twenty he had
mastered the ordinary text-books on mathematics, philosophy,
theology, and law. Refused the degree of doctor of laws at
Leipzig by those who were jealous of his youth and learning, he
moved to Nuremberg. An essay which he there wrote on the
study of law was dedicated to the Elector of Mainz, and led to
his appointment by the elector on a commission for the revision
of some statutes, from which he was subsequently promoted to
the diplomatic service. In the latter capacity he supported
(unsuccessfully) the claims of the German candidate for the
crown of Poland. The violent seizure of various small places in
Alsace in 1670 excited universal alarm in Germany as to the
designs of \index{Louis XIV. of France, ref.\ to}Louis XIV.; and \index{Leibnitz!ref.\ to}Leibnitz drew up a scheme by which
it was proposed to offer German co-operation, if France liked to
take Egypt, and use the possession of that country as a basis for
attack against Holland in Asia, provided France would agree to
leave Germany undisturbed. This bears a curious resemblance
to the similar plan by which \index{Napoleon I.}Napoleon I. proposed to attack
England. In 1672 Leibnitz went to Paris on the invitation of
the French government to explain the details of the scheme, but
nothing came of it.

At Paris he met \index{Huygens!ref.\ to}Huygens who was then residing there,
and their conversation led Leibnitz to study geometry, which
he described as opening a new world to him; though as a
matter of fact he had previously written some tracts on various
minor points in mathematics, the most important being a paper
on combinations written in 1668, and a description of a new
\index{Arithmetical!machine}\index{Calculating machine}calculating machine. In January, 1673, he was sent on a
political mission to London, where he stopped some months and
made the acquaintance of \index{Oldenburg}Oldenburg, \index{Collins, J.!ref.\ to}Collins, and others; it
was at this time that he communicated the memoir to the Royal
Society in which he was found to have been forestalled by
\index{Mouton}Mouton.

In 1673 the Elector of Mainz died, and in the following year
\index{Leibnitz!ref.\ to}Leibnitz entered the service of the Brunswick family; in 1676
he again visited London, and then moved to Hanover, where,
till his death, he occupied the well-paid post of librarian in the
ducal library. His pen was thenceforth employed in all the
political matters which affected the Hanoverian family, and his
services were recognized by honours and distinctions of various
kinds; his memoranda on the various political, historical, and
theological questions which concerned the dynasty during the
forty years from 1673 to 1713 form a valuable contribution to
the history of that time.

Leibnitz's appointment in the Hanoverian service gave him
more time for his favourite pursuits. He used to assert that as
the first-fruit of his increased leisure, he invented the differential
and integral calculus in 1674, but the earliest traces of the use
of it in his extant note-books do not occur till 1675, and it was
not till 1677 that we find it developed into a consistent system;
it was not published till 1684. Most of his mathematical
papers were produced within the ten years from 1682 to 1692,
and many of them in a journal, called the \emph{Acta Eruditorum},
founded by himself and Otto Mencke in 1682, which had a
wide circulation on the continent.

\nblabel{png373a}Leibnitz occupies at least as large a place in the history of
philosophy as he does in the history of mathematics. Most of
his philosophical writings were composed in the last twenty or
twenty-five years of his life; and the point as to whether his
views were original or whether they were appropriated from
\index{Spinoza and Leibnitz}Spinoza, whom he visited in 1676, is still in question among
philosophers, though the evidence seems to point to the originality
of Leibnitz. As to Leibnitz's system of philosophy it will
be enough to say that he regarded the ultimate elements of the
universe as individual percipient beings whom he called monads.
According to him the monads are centres of force, and substance
is force, while space, matter, and motion are merely phenomenal;
finally, the existence of God is inferred from the existing
harmony among the monads. His services to literature were
almost as considerable as those to philosophy; in particular, I
may single out his overthrow of the then prevalent belief that
Hebrew was the primeval language of the human race.

In 1700 the \index{Academy,!the Berlin}\index{Berlin Academy}academy of Berlin was created on his advice, and
he drew up the first body of statutes for it. On the accession
in 1714 of his master, \index{George I. of England, ref.\ to}George I., to the throne of England,
\index{Leibnitz!ref.\ to}Leibnitz was thrown aside as a useless tool; he was forbidden
to come to England; and the last two years of his life were
spent in neglect and dishonour. He died at Hanover in 1716.
He was overfond of money and personal distinctions; was
unscrupulous, as perhaps might be expected of a professional
diplomatist of that time; but possessed singularly attractive
manners, and all who once came under the charm of his personal
presence remained sincerely attached to him. His mathematical
reputation was largely augmented by the eminent position that
he occupied in diplomacy, philosophy, and literature; and the
power thence derived was considerably increased by his influence
in the management of the \emph{Acta Eruditorum}.

\index{Calculus, infinitesimal|(}\index{Fluxional calculus!controversy|(}\nblabel{png374a}The last years of his life---from 1709 to 1716--were embittered
by the long controversy with \index{Keill}John Keill, \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton, and
others, as to whether he had discovered the differential calculus
independently of Newton's previous investigations, or whether
he had derived the fundamental idea from Newton, and merely
invented another notation for it. The controversy\footnote
  {The case in favour of the independent invention by Leibnitz is stated in
  \index{Gerhardt, ref.\ to}Gerhardt's \emph{Leibnizens mathematische Schriften}; and in the third volume of
  \index{Cantor, M., ref.\ to}M.~Cantor's \emph{Geschichte der Mathematik}. The arguments on the other side
  are given in \index{Sloman on calculus}H.~Sloman's \emph{Leibnitzens Anspruch auf die Erfindung der
  Differenzialrechnung}, Leipzig, 1857, of which an English translation, with
  additions by Dr.~Sloman, was published at Cambridge in 1860. A summary
  of the evidence will be found in \index{Gibson on origin of calculus}G.~A.~Gibson's memoir, \emph{Proceedings of the
  Edinburgh Mathematical Society}, vol.~xiv, 1896, pp.~148--174. The history
  of the invention of the calculus is given in an article on it in the ninth edition
  of the \emph{Encyclopaedia Britannica}, and in \index{Mansion on the calculus}P.~Mansion's \emph{Esquisse de l'histoire
  du calcul infinitésimal}, Gand, 1887.}
occupies a
place in the scientific history of the early years of the eighteenth
century quite disproportionate to its true importance, but it so
materially affected the history of mathematics in western
Europe, that I feel obliged to give the leading facts, though I
am reluctant to take up so much space with questions of a
personal character.

The ideas of the infinitesimal calculus can be expressed
either in the notation of \index{Fluxions}fluxions or in that of differentials.
The former was used by \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton in 1666, but no distinct
account of it was printed till 1693. The earliest use of the
latter in the note-books of \index{Leibnitz!ref.\ to}Leibnitz may be probably referred to
1675, it was employed in the letter sent to Newton in 1677, and
an account of it was printed in the memoir of 1684 described
below. There is no question that the differential notation is due
to Leibnitz, and the sole question is as to whether the general
idea of the calculus was taken from Newton or discovered
independently.

The case in favour of the independent invention by Leibnitz
rests on the ground that he published a description of his
method some years before Newton printed anything on fluxions,
that he always alluded to the discovery as being his own invention,
and that for some years this statement was unchallenged;
while of course there must be a strong presumption that he
acted in good faith. To rebut this case it is necessary to shew
(i)~that he saw some of Newton's papers on the subject in or
before 1675, or at least 1677, and (ii)~that he thence derived the
fundamental ideas of the calculus. The fact that his claim was
unchallenged for some years is, in the particular circumstances
of the case, immaterial.

That Leibnitz saw some of Newton's manuscripts was always
intrinsically probable; but when, in 1849, \index{Gerhardt, ref.\ to}C.~J.~Gerhardt\footnote
  {Gerhardt, \emph{Leibnizens mathematische Schriften}, vol.~i, p.~7.}
examined Leibnitz's papers he found among them a manuscript
copy, the existence of which had been previously unsuspected,
in Leibnitz's handwriting, of extracts from Newton's \emph{De Analysi
per Equationes Numero Terminorum Infinitas} (which was
printed in the \emph{De Quadratura Curvarum} in 1704), together
with notes on their expression in the differential notation.
The question of the date at which these extracts were made is
therefore all-important. \index{Tschirnhausen!ref.\ to}Tschirnhausen seems to have possessed
a copy of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's \emph{De Analysi} in 1675, and as in that year
he and \index{Leibnitz!ref.\ to}Leibnitz were engaged together on a piece of work,
it is not impossible that these extracts were made then. It
is also possible that they may have been made in 1676, for
Leibnitz discussed the question of analysis by infinite series
with \index{Collins, J.!ref.\ to}Collins and \index{Oldenburg}Oldenburg in that year, and it is \emph{a priori}
probable that they would have then shewn him the manuscript
of Newton on that subject, a copy of which was possessed by
one or both of them. On the other hand it may be supposed
that Leibnitz made the extracts from the printed copy in or
after 1704. Leibnitz shortly before his death admitted in a
letter to \index{Conti}Conti that in 1676 Collins had shewn him some
Newtonian papers, but implied that they were of little or no
value,---presumably he referred to Newton's letters of June 13
and Oct.~24, 1676, and to the letter of Dec.~10, 1672, on the
method of tangents, extracts from which accompanied\footnote
  {\index{Gerhardt, ref.\ to}Gerhardt, vol.~i, p.~91.}
the
letter of June~13,---but it is remarkable that, on the receipt of
these letters, Leibnitz should have made no further inquiries,
unless he was already aware from other sources of the method
followed by Newton.

Whether Leibnitz made no use of the manuscript from
which he had copied extracts, or whether he had previously
invented the calculus, are questions on which at this distance
of time no direct evidence is available. It is, however, worth
noting that the unpublished Portsmouth Papers shew that
when, in 1711, Newton went carefully into the whole dispute,
he picked out this manuscript as the one which had probably
somehow fallen into the hands of Leibnitz.\footnote
  {\emph{Catalogue of Portsmouth Papers}, pp.~xvi,~xvii,~7,~8.}
At that time
there was no direct evidence that Leibnitz had seen this
manuscript before it was printed in 1704, and accordingly
Newton's conjecture was not published; but Gerhardt's discovery
of the copy made by Leibnitz tends to confirm the
accuracy of Newton's judgment in the matter. It is said by
those who question Leibnitz's good faith that to a man of his
ability the manuscript, especially if supplemented by the letter
of Dec.~10, 1672, would supply sufficient hints to give him a
clue to the methods of the calculus, though as the \index{Fluxions}fluxional
notation is not employed in it anyone who used it would have
to invent a notation; but this is denied by others.

There was at first no reason to suspect the good faith of
\index{Leibnitz!ref.\ to}Leibnitz; and it was not until the appearance in 1704 of an
anonymous review of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's tract on quadrature, in which
it was implied that Newton had borrowed the idea of the
fluxional calculus from Leibnitz, that any responsible mathematician\footnote
  {In 1699 \index{Duillier}Duillier had accused Leibnitz of plagiarism from Newton, but
  Duillier was not a person of much importance.}
questioned the statement that Leibnitz had invented
the calculus independently of Newton. It is universally
admitted that there was no justification or authority for the
statements made in this review, which was rightly attributed
to Leibnitz. But the subsequent discussion led to a critical
examination of the whole question, and doubt was expressed as
to whether Leibnitz had not derived the fundamental idea from
Newton. The case against Leibnitz as it appeared to Newton's
friends was summed up in the \index{Commercium Epistolicum@\emph{Commercium Epistolicum}}\emph{Commercium Epistolicum} issued
in 1712, and detailed references are given for all the facts
mentioned.

No such summary (with facts, dates, and references) of the
case for Leibnitz was issued by his friends; but \index{Bernoulli, John!ref.\ to}John Bernoulli
attempted to indirectly weaken the evidence by attacking the
personal character of Newton: this was in a letter dated June~7,
1713. The charges were false, and, when pressed for an
explanation of them, Bernoulli most solemnly denied having
written the letter. In accepting the denial Newton added in a
private letter to him the following remarks, which are interesting
as giving Newton's account of why he was at last induced to
take any part in the controversy. ``I have never,'' said he,
``grasped at fame among foreign nations, but I am very
desirous to preserve my character for honesty, which the
author of that epistle, as if by the authority of a great
judge, had endeavoured to wrest from me. Now that I am
old, I have little pleasure in mathematical studies, and I have
never tried to propagate my opinions over the world, but have
rather taken care not to involve myself in disputes on account
of them.''

\index{Leibnitz!ref.\ to}Leibnitz's defence or explanation of his silence is given in
the following letter, dated April~9, 1716, from him to \index{Conti}Conti.
``Pour répondre de point en point à l'ouvrage publié contre
moi, il falloit un autre ouvrage aussi grand pour le moins que
celui-là: il falloit entrer dans un grand détail de quantité de
minutiés passées il y a trente à quarante ans, dont je ne me
souvenois guère: il me falloit chercher mes vieilles lettres,
dont plusieurs se sont perdues, outre que le plus souvent je
n'ai point gardé les minutes des miennes: et les autres sont
ensevelies dans un grand tas de papiers, que je ne pouvois
débrouiller qu'avec du temps et de la patience; mais je n'en
avois guère le loisir, étant chargé présentement d'occupations
d'une toute autre nature.''

The death of Leibnitz in 1716 only put a temporary stop
to the controversy which was bitterly debated for many years
later. The question is one of difficulty; the evidence is conflicting
and circumstantial; and every one must judge for
himself which opinion seems most reasonable. Essentially it
is a case of Leibnitz's word against a number of suspicious
details pointing against him. His unacknowledged possession
of a copy of part of one of Newton's manuscripts may be
explicable; but the fact that on more than one occasion he
deliberately altered or added to important documents (\emph{ex.~gr.}\ the
letter of June~7, 1713, in the \emph{Charta Volans}, and that of
April~8, 1716, in the \emph{Acta Eruditorum}), before publishing
them, and, what is worse, that a material date in one of his
manuscripts has been falsified\footnote
  {\index{Cantor, M., ref.\ to}Cantor, who advocates Leibnitz's claims, thinks that the falsification
  must be taken to be Leibnitz's act: see Cantor, vol.~iii, p.~176.}
(1675 being altered to 1673),
makes his own testimony on the subject of little value. It
must be recollected that what he is alleged to have received was
rather a number of suggestions than an account of the calculus;
and it is possible that as he did not publish his results of
1677 until 1684, and that as the notation and subsequent
development of it were all of his own invention, he may have
been led, thirty years later, to minimize any assistance which
he had obtained originally, and finally to consider that it was
immaterial. During the eighteenth century the prevalent
opinion was against \index{Leibnitz!ref.\ to}Leibnitz, but to-day the majority of writers
incline to think it more likely that the inventions were
independent.

If we must confine ourselves to one system of notation then
there can be no doubt that that which was invented by Leibnitz
is better fitted for most of the purposes to which the infinitesimal
calculus is applied than that of \index{Fluxions}fluxions, and for some
(such as the calculus of variations) it is indeed almost essential.
It should be remembered, however, that at the beginning of the
eighteenth century the methods of the infinitesimal calculus had
not been systematized, and either notation was equally good.
The development of that calculus was the main work of the
mathematicians of the first half of the eighteenth century. The
differential form was adopted by continental mathematicians.
The application of it by \index{Euler!ref.\ to}Euler, \index{Lagrange!ref.\ to}Lagrange, and \index{Laplace!ref.\ to}Laplace to the
principles of mechanics laid down in the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia} was the great
achievement of the last half of that century, and finally demonstrated
the superiority of the differential to the fluxional calculus.
The translation of the \emph{Principia} into the language of modern
analysis, and the filling in of the details of the Newtonian theory
by the aid of that analysis, were effected by Laplace.

The controversy with Leibnitz was regarded in England as
an attempt by foreigners to defraud \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton of the credit of
his invention, and the question was complicated on both sides
by national jealousies. It was therefore natural, though it was
unfortunate, that in England the geometrical and fluxional
methods as used by Newton were alone studied and employed.
For more than a century the English school was thus out of
touch with continental mathematicians. The consequence was
that, in spite of the brilliant band of scholars formed by \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton,
the improvements in the methods of analysis gradually effected
on the continent were almost unknown in Britain. It was not
until 1820 that the value of analytical methods was fully recognized
in England, and that Newton's countrymen again took any
large share in the development of mathematics.\index{Fluxional calculus!controversy|)}

Leaving now this long controversy I come to the discussion
of the mathematical papers produced by \index{Leibnitz!ref.\ to}Leibnitz, all the more
important of which were published in the \emph{Acta Eruditorum}.
They are mainly concerned with applications of the infinitesimal
calculus and with various questions on mechanics.

\nblabel{png380a}The only papers of first-rate importance which he produced
are those on the differential calculus. The earliest of these was
one published in the \emph{Acta Eruditorum} for October, 1684, in
which he enunciated a general method for finding \index{Maxima and minima, determination of}maxima and
minima, and for drawing tangents to curves. One inverse
problem, namely, to find the curve whose \index{Subtangent!constant}subtangent is constant,
was also discussed. The notation is the same as that with
which we are familiar, and the differential coefficients of $x^{n}$ and
of products and quotients are determined. In 1686 he wrote
a paper on the principles of the new calculus. In both of these
papers the \index{Continuity, principle of}principle of continuity is explicitly assumed, while
his treatment of the subject is based on the use of infinitesimals
and not on that of the limiting value of ratios. In answer to
some objections which were raised in 1694 by \index{Nieuwentyt}Bernard Nieuwentyt,
who asserted that $dy/dx$ stood for an unmeaning quantity like
$0/0$, Leibnitz explained, in the same way as \index{Barrow!ref.\ to}Barrow had
previously done, that the value of $dy/dx$ in geometry could be
expressed as the ratio of two finite quantities. I think that
Leibnitz's statement of the objects and methods of the infinitesimal
calculus as contained in these papers, which are the three
most important memoirs on it that he produced, is somewhat
obscure, and his attempt to place the subject on a metaphysical
basis did not tend to clearness; but the fact that all the results
of modern mathematics are expressed in the language invented
by \index{Leibnitz!ref.\ to}Leibnitz has proved the best monument of his work. Like
\index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton, he treated integration not only as a summation, but as
the inverse of differentiation.\index{Calculus, infinitesimal|)}

In 1686 and 1692 he wrote papers on \index{Osculating circle}osculating curves.
These, however, contain some bad blunders, as, for example, the
assertion that an osculating circle will necessarily cut a curve
in four consecutive points: this error was pointed out by \index{Bernoulli, John!ref.\ to}John
Bernoulli, but in his article of 1692 Leibnitz defended his
original assertion, and insisted that a circle could never cross a
curve where it touched it.

In 1692 Leibnitz wrote a memoir in which he laid the
foundation of the theory of \index{Envelopes}envelopes. This was further developed
in another paper in 1694, in which he introduced for the
first time the terms \index{Coord@Co-ordinates}``co-ordinates'' and ``axes of co-ordinates.''

\nblabel{png381a}Leibnitz also published a good many papers on mechanical
subjects; but some of them contain mistakes which shew that
he did not understand the principles of the subject. Thus, in
1685, he wrote a memoir to find the pressure exerted by a
sphere of weight $W$ placed between two inclined planes of complementary
inclinations, placed so that the lines of greatest
slope are perpendicular to the line of the intersection of the
planes. He asserted that the pressure on each plane must
consist of two components, ``unum quo decliviter descendere
tendit, alterum quo planum declive premit.'' He further said
that for metaphysical reasons the sum of the two pressures must
be equal to $W$. Hence, if $R$ and $R'$ be the required pressures,
and $\alpha$ and $\frac{1}{2}\pi-\alpha$ the inclinations of the planes, he finds that
\[
R=\frac{1}{2}W(1-\sin \alpha+\cos \alpha) \quad\text{and}\quad R'=\frac{1}{2}W(1-\cos \alpha+\sin \alpha).
\]
The true values are $R=W\cos \alpha$ and $R'=W\sin \alpha$. Nevertheless
some of his papers on mechanics are valuable. Of these the
most important were two, in 1689 and 1694, in which he solved
the problem of finding an \index{Isochronous curve}isochronous curve; one, in 1697, on
the \index{Curve of quickest descent}curve of quickest descent (this was the problem sent as a
challenge to Newton); and two, in 1691 and 1692, in which
he stated the intrinsic equation of the curve assumed by a
flexible rope suspended from two points, that is, the \index{Catenary}catenary,
but gave no proof. This last problem had been originally
proposed by \index{Galileo!ref.\ to}Galileo.

In 1689, that is, two years after the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia} had been
published, he wrote on the \index{Planetary motions}movements of the planets which he
stated were produced by a motion of the ether. Not only
were the equations of motion which he obtained wrong, but his
deductions from them were not even in accordance with his own
axioms. In another memoir in 1706, that is, nearly twenty
years after the \emph{Principia} had been written, he admitted that
he had made some mistakes in his former paper, but adhered
to his previous conclusions, and summed the matter up by
saying ``it is certain that gravitation generates a new force at
each instant to the centre, but the centrifugal force also generates
another away from the centre\ldots. The centrifugal force may
be considered in two aspects according as the movement is
treated as along the tangent to the curve or as along the arc
of the circle itself.'' It seems clear from this paper that he did
not really understand the principles of dynamics, and it is hardly
necessary to consider his work on the subject in further detail.
Much of it is vitiated by a constant confusion between momentum
and kinetic energy: when the force is ``passive'' he uses the
first, which he calls the \index{Vis mortua}\emph{vis mortua}, as the measure of a force;
when the force is ``active'' he uses the latter, the double of
which he calls the \index{Vis viva}\emph{vis viva}.

\index{vers x@$\operatorname{Vers} x$, series for}\index{Expansion,!ofe@of $e^{x}$}\index{Expansion,!ofvers@of $\vers x$}%
\index{sine x@$\operatorname{Sin} x$, series for}\index{Expansion,!oftan@of $\tan^{-1} x$}\index{Tani@$\operatorname{Tan}^{-1} x$, series for}%
\index{Expansion,!oflog@of $\log(1+x)$}The series quoted by \index{Leibnitz!ref.\ to}Leibnitz comprise those for $e^x$,
$\log(1+x)$, $\sin x$, $\vers x$, and $\tan^{-1}x$; all of these had been
previously published, and he rarely, if ever, added any demonstrations.
Leibnitz (like \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton) recognised the importance
of James \index{Gregory, James!ref.\ to}Gregory's remarks on the necessity of examining
whether \index{Infinite series, difficulties in connection with}infinite series are \index{Convergency}convergent or divergent, and proposed
a test to distinguish series whose terms are alternately positive
and negative. In 1693 he explained the method of \index{Expansions@Expansion in series}expansion
by \index{Indeterminate coefficients}indeterminate coefficients, though his applications were not
free from error.

To sum the matter up briefly, it seems to me that Leibnitz's
work exhibits great skill in analysis, but much of it is unfinished,
and when he leaves his symbols and attempts to interpret
his results he frequently commits blunders. No doubt
the demands of politics, philosophy, and literature on his time
may have prevented him from elaborating any problem completely
or writing a systematic exposition of his views, though
they are no excuse for the mistakes of principle which occur in
his papers. Some of his memoirs contain suggestions of
methods which have now become valuable means of analysis,
such as the use of \index{Determinants}determinants and of \index{Indeterminate coefficients}indeterminate coefficients;
but when a writer of manifold interests like \index{Leibnitz!ref.\ to}Leibnitz
throws out innumerable suggestions, some of them are likely to
turn out valuable; and to enumerate these (which he did not
work out) without reckoning the others (which are wrong) gives
a false impression of the value of his work. But in spite of
this, his title to fame rests on a sure basis, for by his advocacy
of the differential calculus his name is inseparably connected
with one of the chief instruments of analysis, as that of
\index{Descartes!ref.\ to}Descartes---another philosopher---is similarly connected with
analytical geometry.

\nblabel{png383a}Leibnitz was but one amongst several continental writers
whose papers in the \emph{Acta Eruditorum} familiarised mathematicians
with the use of the differential calculus. Among the
most important of these were \index{Bernoulli, James!ref.\ to}James and \index{Bernoulli, John!ref.\ to}John Bernoulli, both
of whom were warm friends and admirers of Leibnitz, and to
their devoted advocacy his reputation is largely due. Not only
did they take a prominent part in nearly every mathematical
question then discussed, but nearly all the leading mathematicians
on the continent during the first half of the eighteenth
century came directly or indirectly under the influence of one
or both of them.\index{Leibnitz|)}

The Bernoullis\footnote
  {See the account in the \emph{Allgemeine deutsche Biographie}, vol.~ii, Leipzig,
  1875, pp.~470--483.}
(or as they are sometimes, and perhaps
more correctly, called, the Bernouillis) were a family of Dutch
origin, who were driven from Holland by the Spanish persecutions,
and finally settled at Bâle in Switzerland. The first
member of the family who attained distinction in mathematics
was James.

\index{Bernoulli, James|(}\index{Bernoulli, James!ref.\ to}\nblabel{png384a}\nbparagraph{James Bernoulli.\footnote
  {See the \emph{éloge} by \index{Defontenelle@De Fontenelle, ref.\ to}\index{Fontenelle, de, ref.\ to}B.~de~Fontenelle,
  Paris, 1766; also \index{Montucla!ref.\ to}Montucla's \emph{Histoire},
  vol.~ii. A collected edition of the works of James Bernoulli was published
  in two volumes at Geneva in 1744, and an account of his life is prefixed to
  the first volume.}}
\emph{Jacob} or \emph{James Bernoulli} was born at
Bâle on December~27, 1654; in 1687 he was appointed to a
chair of mathematics in the university there; and occupied it
until his death on August~16, 1705.

He was one of the earliest to realize how powerful as an
instrument of analysis was the \index{Calculus, infinitesimal}infinitesimal calculus, and he
applied it to several problems, but he did not himself invent
any new processes. His great influence was uniformly and
successfully exerted in favour of the use of the differential calculus,
and his lessons on it, which were written in the form of
two essays in 1691 and are published in the second volume of
his works, shew how completely he had even then grasped the
principles of the new analysis. These lectures, which contain
the earliest use of the term integral, were the first published
attempt to construct an integral calculus; for \index{Leibnitz!ref.\ to}Leibnitz had
treated each problem by itself, and had not laid down any
general rules on the subject.

The most important discoveries of James Bernoulli were
his solution of the problem to find an \index{Isochronous curve}isochronous curve; his
proof that the construction for the \index{Catenary}catenary which had been
given by Leibnitz was correct, and his extension of this to
strings of variable density and under a central force; his
determination of the form taken by an elastic rod fixed at one
end and acted on by a given force at the other, the \index{Elastica}\emph{elastica};
also of a flexible rectangular sheet with two sides fixed horizontally
and filled with a heavy liquid, the \index{Lintearia}\emph{lintearia}; and, lastly,
of a sail filled with wind, the \index{Velaria}\emph{velaria}. In 1696 he offered a
reward for the general solution of \index{Isoperimetrical problem}isoperimetrical figures, that
is, of figures of a given species and given perimeter which shall
include a maximum area: his own solution, published in 1701,
is correct as far as it goes. In 1698 he published an essay on
the differential calculus and its applications to geometry. He
here investigated the chief properties of the \index{Equiangular spiral}\index{Spiral, the equiangular}equiangular spiral,
and especially noticed the manner in which various curves
deduced from it reproduced the original curve: struck by this
fact he begged that, in imitation of \index{Archimedes!ref.\ to}Archimedes, an equiangular
spiral should be engraved on his tombstone with the inscription
\emph{eadem numero mutata resurgo}. He also brought out in 1695
an edition of \index{Descartes!ref.\ to}Descartes's \emph{Géométrie}. In his \emph{Ars Conjectandi},
published in 1713, he established the fundamental principles of
the calculus of \index{Probabilities, theory of}probabilities; in the course of the work he
defined the numbers known by his name\footnote
  {A bibliography of \index{Bernoulli's numbers}Bernoulli's Numbers was given by \index{Ely on Bernoulli's numbers}G.~S.~Ely, in the
  American \emph{Journal of Mathematics}, 1882, vol.~v, pp.~228--235.}
and explained their
use, he also gave some theorems on finite differences. His
higher lectures were mostly on the theory of series; these were
published by \index{Bernoulli, Nicholas!ref.\ to}Nicholas Bernoulli in 1713.\index{Bernoulli, James|)}

\index{Bernoulli, John|(}\index{Bernoulli, John!ref.\ to}\nblabel{png385a}\nbparagraph{John Bernoulli.\footnote
  {\index{D'Alembert!ref.\ to}D'Alembert wrote a eulogistic \emph{éloge} on the work and influence of John
  Bernoulli, but he explicitly refused to deal with his private life or quarrels;
  see also \index{Montucla!ref.\ to}Montucla's \emph{Histoire}, vol.~ii. A collected edition of the works of
  John Bernoulli was published at Geneva in four volumes in 1742, and his
  correspondence with \index{Leibnitz!ref.\ to}Leibnitz was published in two volumes at the same
  place in 1745.}}
\emph{John Bernoulli}, the brother of \index{Bernoulli, James!ref.\ to}James
Bernoulli, was born at Bâle on August~7, 1667, and died there
on January~1, 1748. He occupied the chair of mathematics
at Groningen from 1695 to 1705; and at Bâle, where he
succeeded his brother, from 1705 to 1748. To all who did not
acknowledge his merits in a manner commensurate with his
own view of them he behaved most unjustly: as an illustration
of his character it may be mentioned that he attempted to substitute
for an incorrect solution of his own on the problem of
\index{Isoperimetrical problem}isoperimetrical curves another stolen from his brother James,
while he expelled his son Daniel from his house for obtaining
a prize from the French Academy which he had expected to
receive himself. He was, however, the most successful teacher
of his age, and had the faculty of inspiring his pupils with
almost as passionate a zeal for mathematics as he felt himself.
The general adoption on the continent of the differential rather
than the \index{Fluxions}fluxional notation was largely due to his influence.

Leaving out of account his innumerable controversies, the
chief discoveries of \index{Bernoulli, John!ref.\ to}John Bernoulli were the \index{Exponential calculus}exponential calculus,
the treatment of \index{Trigonometry.!z9@Development of, by John Bernoulli}trigonometry as a branch of analysis,
the conditions for a \index{Geodesics}geodesic, the determination of orthogonal
\index{Trajectories}trajectories, the solution of the \index{Brachistochrone, the}brachistochrone, the statement
that a ray of light pursues such a path that $\sum\mu ds$ is a minimum,
and the enunciation of the principle of virtual work. I believe
that he was the first to denote the accelerating effect of \index{Gravity,!symbol for}gravity
by an algebraical sign $g$, and he thus arrived at the formula
$v^2 = 2gh$: the same result would have been previously expressed
by the proportion $v_1^2 : v_2^2 = h_1 : h_2$. The \index{Functions, notation for}notation $\phi x$ to indicate
a function\footnote
  {On the meaning assigned at first to the word \emph{function} see a note by
  \index{Cantor, M., ref.\ to}M.~Cantor, \emph{L'Intermédiaire des mathématiciens}, January 1896, vol.~iii, pp.~22--23.}
of $x$ was introduced by him in 1718, and displaced
the notation $X$ or $\xi$ proposed by him in 1698; but the general
adoption of symbols like $f$, $F$, $\phi$, $\psi$, \dots to represent functions,
seems to be mainly due to \index{Euler!ref.\ to}Euler and \index{Lagrange!ref.\ to}Lagrange.\index{Bernoulli, John|)}

\index{Bernoullis, the younger|(}\nblabel{png386a}Several members of the same family, but of a younger
generation, enriched mathematics by their teaching and
writings. The most important of these were the three sons of
John; namely, \index{Bernoulli, Nicholas}Nicholas, \index{Bernoulli, Daniel!ref.\ to}Daniel, and \index{Bernoulli, John II.}John the younger; and
the two sons of John the younger, who bore the names of
John and James. To make the account complete I add here
their respective dates. \index{Bernoulli, Nicholas!ref.\ to}\emph{Nicholas Bernoulli}, the eldest of the
three sons of John, was born on Jan.~27, 1695, and was
drowned at Petrograd, where he was professor, on July~26,
1726. \emph{Daniel Bernoulli}, the second son of John, was born on
Feb.~9, 1700, and died on March~17, 1782; he was professor
first at Petrograd and afterwards at Bâle, and shares with
Euler the unique distinction of having gained the prize proposed
annually by the French Academy no less than ten times: I refer
to him again a few pages later. \index{Bernoulli, John!ref.\ to}\emph{John Bernoulli}, the younger,
a brother of Nicholas and Daniel, was born on May~18, 1710,
and died in 1790; he also was a professor at Bâle. He left
two sons, \index{Bernoulli, John III.}\emph{John} and \index{Bernoulli, James II.}\emph{James}: of these, the former, who was born
on Dec.~4, 1744, and died on July~10, 1807, was astronomer-royal
and director of mathematical studies at Berlin; while
the latter, who was born on Oct.~17, 1759, and died in
July 1789, was successively professor at Bâle, Verona, and
Petrograd.\index{Bernoullis, the younger|)}


\section*{The development of analysis on the continent.\nblabel{png387a}}

Leaving for a moment the English mathematicians of the
first half of the eighteenth century we come next to a number
of continental writers who barely escape mediocrity, and to
whom it will be necessary to devote but few words. Their
writings mark the steps by which analytical geometry and the
differential and integral calculus were perfected and made
familiar to mathematicians. Nearly all of them were pupils
of one or other of the two elder Bernoullis, and they were so
nearly contemporaries that it is difficult to arrange them
chronologically. The most eminent of them are \emph{Cramer, de
Gua, de Montmort, Fagnano, l'Hospital, Nicole, Parent,
Riccati, Saurin}, and \emph{Varignon}.

\index{Lh@L'Hospital|(}\index{Hospital, l'|(}\index{Delhospital@De l'Hospital|(}\nblabel{png387b}\nbparagraph{L'Hospital.} \emph{Guillaume Fran\c{c}ois Antoine l'Hospital, Marquis
de St.-Mesme}, born at Paris in 1661, and died there on
Feb.~2, 1704, was among the earliest pupils of \index{Bernoulli, John!ref.\ to}John Bernoulli,
who, in 1691, spent some months at l'Hospital's house in
Paris for the purpose of teaching him the new calculus. It
seems strange, but it is substantially true, that a knowledge of
the \index{Calculus, infinitesimal|(}infinitesimal calculus and the power of using it was then
confined to \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton, \index{Leibnitz!ref.\ to}Leibnitz, and the two elder Bernoullis---and
it will be noticed that they were the only mathematicians
who solved the more difficult problems then proposed as challenges.
There was at that time no text-book on the subject,
and the credit of putting together the first treatise which
explained the principles and use of the method is due to
l'Hospital; it was published in 1696 under the title \emph{Analyse des
infiniment petits}. This contains a partial investigation of
the \index{Limiting values}limiting value of the ratio of functions which for a certain
value of the variable take the \index{Indeterminate forms}indeterminate form $0:0$, a
problem solved by \index{Bernoulli, John!ref.\ to}John Bernoulli in 1704. This work had
a wide circulation; it brought the differential notation into
general use in France, and helped to make it known in
Europe. A supplement, containing a similar treatment of
the integral calculus, together with additions to the differential
calculus which had been made in the following half century,
was published at Paris, 1754--56, by \index{Bougainville, De}\index{De Bougainville}L.~A. de~Bougainville.

L'Hospital took part in most of the challenges issued
by \index{Leibnitz!ref.\ to}Leibnitz, the Bernoullis, and other continental mathematicians
of the time; in particular he gave a solution of
the \index{Brachistochrone, the}brachistochrone, and investigated the form of the \index{Resistance, solid of least}\index{Solid of least resistance}solid
of least resistance of which \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton in the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia} had
stated the result. He also wrote a treatise on analytical
conics, which was published in 1707, and for nearly a century
was deemed a standard work on the subject.\index{Delhospital@De l'Hospital|)}\index{Hospital, l'|)}\index{Lh@L'Hospital|)}

\index{Varignon}\nblabel{png388a}\nbparagraph{Varignon.\footnote
  {See the \emph{éloge} by B.~de~Fontenelle, Paris, 1766.}}
\emph{Pierre Varignon}, born at Caen in 1654, and
died in Paris on Dec.~22, 1722, was an intimate friend of
Newton, Leibnitz, and the Bernoullis, and, after l'Hospital, was
the earliest and most powerful advocate in France of the use of
the differential calculus. \index{Expansions@Expansion in series}\index{Infinite series, difficulties in connection with}He realized the necessity of obtaining
a test for examining the \index{Convergency}convergency of series, but the
analytical difficulties were beyond his powers. He simplified
the proofs of many of the leading propositions in mechanics,
and in 1687 recast the treatment of the subject, basing it on
the composition of \index{Triangle of forces}\index{Forces!parallelogram of}\index{Parallelogram of forces}\index{Forces!triangle of}forces. His works were published at Paris
in 1725.

\index{Montmort, de}\index{Demontmort@De Montmort}\nbparagraph{De Montmort.\nbnamespace Nicole.} \emph{Pierre Raymond de~Montmort},
born at Paris on Oct.~27, 1678, and died there on Oct.~7,
1719, was interested in the subject of \index{Differences, finite}finite differences. He
determined in 1713 the sum of $n$ terms of a finite series of
the form
\[
na+\frac{n(n-1)}{1\cdot2}\Delta a + \frac{n(n-1)(n-2)}{1\cdot2\cdot3}\Delta^{2}a+\ldots;
\]
{\nbstretchyspace a theorem which seems to have been independently rediscovered
by \index{Goldbach}Chr.~Goldbach in 1718.} \index{Nicole}\emph{François Nicole}, who
was born at Paris on Dec.~23, 1683, and died there on
Jan.~18, 1758, published his \emph{Traité du calcul des differences
finies} in 1717; it contains rules both for forming differences
and for effecting the summation of given series. Besides this,
in 1706 he wrote a work on roulettes, especially spherical
\index{Epicycloids}epicycloids; and in 1729 and 1731 he published memoirs on
\index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's essay on curves of the third degree.

\index{Saurin}\index{Parent}\index{Gua, de}\index{Degua@De Gua}\nblabel{png389a}\nbparagraph{Parent.\nbnamespace Saurin.\nbnamespace De~Gua.} \emph{Antoine Parent}, born at
Paris on Sept.~16, 1666, and died there on Sept.~26, 1716,
wrote in 1700 on analytical geometry of three dimensions.
His works were collected and published in three volumes at
Paris in 1713. \emph{Joseph Saurin}, born at Courtaison in 1659,
and died at Paris on Dec.~29, 1737, was the first to show how
the tangents at the \index{Multiple points}multiple points of curves could be determined
by analysis. \index{Malves, de}\index{Demalves@De Malves}\emph{Jean Paul de~Gua de~Malves} was born at
Carcassonne in 1713, and died at Paris on June~2, 1785. He
published in 1740 a work on analytical geometry in which he
applied it, without the aid of the differential calculus, to find
the tangents, asymptotes, and various singular points of an
algebraical curve; and he further shewed how singular points
and isolated loops were affected by conical projection. He
gave the proof of \index{Descartes!rule of signs of}Descartes's rule of signs which is to be
found in most modern works. It is not clear whether Descartes
ever proved it strictly, and Newton seems to have regarded it
as obvious.

\index{Cramer, G.}\nbparagraph{Cramer.} \emph{Gabriel Cramer}, born at Geneva in 1704, and
died at Bagnols in 1752, was professor at Geneva. The work
by which he is best known is his treatise on algebraic
curves\footnote
  {See Cantor, chapter cxvi.}
published in 1750, which, as far as it goes, is fairly
complete; it contains the earliest demonstration that a curve
of the $n$th degree is in general determined if $\frac{1}{2}n(n + 3)$ points
on it be given. This work is still sometimes read. Besides
this, he edited the works of the two elder Bernoullis; and
wrote on the physical cause of the spheroidal shape of the
planets and the motion of their apses, 1730, and on \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's
treatment of cubic curves, 1746.

\index{Riccati}\nblabel{png390a}\nbparagraph{Riccati.} \emph{Jacopo Francesco, Count Riccati}, born at Venice
on May~28, 1676, and died at Trèves on April~15, 1754, did
a great deal to disseminate a knowledge of the Newtonian
philosophy in Italy. Besides the equation known by his
name, certain cases of which he succeeded in integrating, he
discussed the question of the possibility of lowering the order
of a given \index{Equations, differential}\index{Differential equations}differential equation. His works were published at
Trèves in four volumes in 1758. He had two sons who wrote
on several minor points connected with the integral calculus
and differential equations, and applied the calculus to several
mechanical questions: these were \emph{Vincenzo}, who was born in
1707 and died in 1775, and \emph{Giordano}, who was born in 1709
and died in 1790.

\index{Toschi}\index{Fagnano}\nbparagraph{Fagnano.} \emph{Giulio Carlo, Count Fagnano}, and \emph{Marquis de
Toschi}, born at Sinigaglia on Dec.~6, 1682, and died on Sept.~26,
1766, may be said to have been the first writer who directed
attention to the theory of elliptic functions. Failing to \index{Ellipse, area of!rectification of}rectify
the ellipse or hyperbola, Fagnano attempted to determine arcs
whose difference should be rectifiable. He also pointed out
the remarkable analogy existing between the integrals which
represent the arc of a circle and the arc of a lemniscate.
Finally he proved the formula
\[
\pi = 2i \log\{(1-i)/(1+i)\},
\]
where $i$ stands for $\sqrt{-1}$. His works were collected and
published in two volumes at Pesaro in 1750.

It was inevitable that some mathematicians should object to
methods of analysis founded on the infinitesimal calculus. The
most prominent of these were \emph{Viviani, De~la~Hire}, and \emph{Rolle},
whose names were mentioned at the \hyperref[png334b]{close of chapter~\textsc{xv}}.

So far no one of the school of \index{Leibnitz!ref.\ to}Leibnitz and the two elder
Bernoullis had shewn any exceptional ability, but by the action
of a number of second-rate writers the methods and language
of analytical geometry and the differential calculus had become
well known by about 1740. The close of this school is marked
by the appearance of \index{Clairaut!ref.\ to}\emph{Clairaut}, \index{D'Alembert!ref.\ to}\emph{D'Alembert}, and \index{Bernoulli, Daniel!ref.\ to}\emph{Daniel Bernoulli}.
Their lives overlap the period considered in the \hyperref[chapter:XVIII]{next chapter},
but, though it is difficult to draw a sharp dividing line which
shall separate by a definite date the mathematicians there considered
from those whose writings are discussed in this chapter,
I think that on the whole the works of these three writers are
best treated here.\index{Calculus, infinitesimal|)}

\index{Clairaut|(}\nblabel{png391a}\nbparagraph{Clairaut.} \emph{Alexis Claude Clairaut} was born at Paris on
May~13, 1713, and died there on May~17, 1765. He belongs
to the small group of children who, though of exceptional
precocity, survive and maintain their powers when grown up.
As early as the age of twelve he wrote a memoir on four
geometrical curves; but his first important work was a treatise
on \index{Tortuous curves}tortuous curves, published when he was eighteen---a work
which procured for him admission to the French Academy. In
1731 he gave a demonstration of the fact noted by \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton
that all curves of the third order were projections of one of five
parabolas.

\index{Degree, length of}\index{Astronomy.!z20@Developed by (amongst others) Clairaut}\index{Earth,!dimensions of}In 1741 Clairaut went on a scientific expedition to measure
the length of a meridian degree on the earth's surface, and on
his return in 1743 he published his \emph{Théorie de la figure de la
terre}. This is founded on a paper by \index{Maclaurin!ref.\ to}Maclaurin, wherein it had
been shewn that a mass of homogeneous fluid set in rotation
about a line through its centre of mass would, under the mutual
\index{Attraction, theories of}attraction of its particles, take the form of a spheroid. This
work of Clairaut treated of heterogeneous spheroids and contains
the proof of his formula for the accelerating effect of \index{Gravity,!law of}gravity in
a place of latitude $l$, namely,
\[
g = G \left\{1 + \left(\tfrac{5}{2} m - \epsilon\right)\sin^{2}l\right\},
\]
where $G$ is the value of equatorial gravity, $m$ the ratio of the
centrifugal force to gravity at the equator, and $\epsilon$ the ellipticity
of a meridian section of the earth. In 1849 Stokes\footnote
  {See \emph{Cambridge Philosophical Transactions}, vol.~viii, pp.~672--695.}
shewed
that the same result was true whatever was the interior constitution
or density of the earth, provided the surface was a
spheroid of equilibrium of small ellipticity.

Impressed by the power of geometry as shewn in the writings
of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton and \index{Maclaurin!ref.\ to}Maclaurin, Clairaut abandoned analysis, and his
next work, the \emph{Théorie de la lune}, published in 1752, is strictly
Newtonian in character. This contains the explanation of the
motion of the \index{Apse, motion of lunar}apse which had previously puzzled astronomers,
and which Clairaut had at first deemed so inexplicable that he
was on the point of publishing a new hypothesis as to the law
of attraction when it occurred to him to carry the approximation
to the third order, and he thereupon found that the result was
in accordance with the observations. This was followed in 1754
by some lunar tables. Clairaut subsequently wrote various
papers on the orbit of the moon, and on the motion of \index{Comets}comets
as affected by the perturbation of the planets, particularly on
the path of \index{Halley!ref.\ to}Halley's comet.

His growing popularity in society hindered his scientific
work: ``engagé,'' says \index{Bossut on Clairaut}Bossut, ``à des soupers, à des veilles,
entraîné par un goût vif pour les femmes, voulant allier le
plaisir à ses travaux ordinaires, il perdit le repos, la santé,
enfin la vie à l'âge de cinquante-deux ans.''\index{Clairaut|)}

\index{D'Alembert|(}\nblabel{png392a}\nbparagraph{D'Alembert.\footnote
  {\index{Bertrand}Bertrand, \index{Condorcet!ref.\ to}Condorcet, and \index{Bastien on D'Alembert}J.~Bastien have left sketches of D'Alembert's
  life. His literary works have been published, but there is no complete edition
  of his scientific writings. Some papers and letters, discovered comparatively
  recently, were published by \index{Henry C., ref.\ to}C.~Henry at Paris in 1887.}}
\emph{Jean-le-Rond D'Alembert} was born at Paris
on November~16, 1717, and died there on October~29, 1783.
He was the illegitimate child of the chevalier \index{Destouches, ref.\ to}Destouches.
Being abandoned by his mother on the steps of the little church
of St.~Jean-le-Rond, which then nestled under the great porch
of Notre-Dame, he was taken to the parish commissary, who,
following the usual practice in such cases, gave him the Christian
name of Jean-le-Rond; I do not know by what authority he
subsequently assumed the right to prefix \emph{de} to his name. He
was boarded out by the parish with the wife of a glazier in a
small way of business who lived near the cathedral, and here he
found a real home, though a humble one. His father appears
to have looked after him, and paid for his going to a school
where he obtained a fair mathematical education.

An essay written by him in 1738 on the integral calculus,
and another in 1740 on ``ducks and drakes'' or ricochets,
attracted some attention, and in the same year he was elected
a member of the French Academy; this was probably due to
the influence of his father. It is to his credit that he absolutely
refused to leave his adopted mother, with whom he continued
to live until her death in 1757. It cannot be said that she
sympathised with his success, for at the height of his fame she
remonstrated with him for wasting his talents on such work:
``Vous ne serez jamais qu'un philosophe,'' said she, ``et qu'est-ce
qu'un philosophe? c'est un fou qui se tourmente pendant sa vie,
pour qu'on parle de lui lorsqu'il n'y sera plus.''

Nearly all his mathematical works were produced during
the years 1743 to 1754. The first of these was his \emph{Traité de
dynamique}, published in 1743, in which he enunciates the principle
known by his name, namely, that the ``internal forces of
inertia'' (that is, the forces which resist acceleration) must be
equal and opposite to the forces which produce the acceleration.
This may be inferred from \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's second reading of his third
law of motion, but the full consequences had not been realized
previously. The application of this principle enables us to obtain
the \index{Equations, differential|(}\index{Differential equations|(}differential equations of motion of any rigid system.

In 1744 D'Alembert published his \emph{Traité de l'équilibre et
du mouvement des fluides}, in which he applies his principle to
fluids; this led to partial differential equations which he was
then unable to solve. In 1745 he developed that part of the
subject which dealt with the motion of air in his \emph{Théorie générale
des vents}, and this again led him to partial differential equations.
A second edition of this in 1746 was dedicated to \index{Frederick the Great of Prussia, ref.\ to}Frederick the
Great of Prussia, and procured an invitation to Berlin and the
offer of a pension; he declined the former, but subsequently,
after some pressing, pocketed his pride and the latter. \nblabel{png394a}In 1747
he applied the differential calculus to the problem of a \index{Vibrating string|(}\index{String, vibrating, theory of|(}vibrating
string, and again arrived at a partial differential equation.

His analysis had three times brought him to an equation
of the form
\[
  \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2},
\]
and he now succeeded in shewing that it was satisfied by
\[
  u = \phi(x+t) + \psi(x-t),
\]
where $\phi$ and $\psi$ are arbitrary functions. It may be interesting
to give his solution which was published in the transactions
of the Berlin Academy for 1747. He begins by saying that, if
$\dfrac{\partial u}{\partial x}$ be denoted by $p$ and $\dfrac{\partial u}{\partial t}$ by $q$, then
\[
  du = pdx+qdt.
\]
But, by the given equation, $\dfrac{\partial q}{\partial t} = \dfrac{\partial p}{\partial x}$, and therefore $pdt+qdx$ is
also an exact differential: denote it by $dv$.
\begin{DPgather*}
\lintertext{\rlap{Therefore}} dv = pdt+qdx.\\
\lintertext{\rlap{Hence}}  du+dv = (pdx+qdt) + (pdt+qdx) = (p+q)(dx+dt),\\
\lintertext{\rlap{and}}    du-dv = (pdx+qdt) - (pdt+qdx) = (p-q)(dx-dt).
\end{DPgather*}
Thus $u+v$ must be a function of $x+t$, and $u-v$ must be a
function of $x-t$. We may therefore put
\begin{DPgather*}
                 u+v = 2\phi(x+t),\\
\lintertext{and} u-v = 2\psi(x-t).\\
\lintertext{Hence} u = \phi(x+t) + \psi(x-t).
\end{DPgather*}

D'Alembert added that the conditions of the physical problem
of a vibrating string demand that, when $x=0$, $u$ should vanish
for all values of $t$. Hence identically
\[
  \phi(t) + \psi(-t) = 0.
\]
Assuming that both functions can be expanded in integral
powers of $t$, this requires that they should contain only odd
powers. Hence
\[
  \psi(-t) = -\phi(t) = \phi(-t).
\]
Therefore
\[
  u = \phi(x+t) + \phi(x-t).
\]

\index{Euler!ref.\ to}Euler now took the matter up and shewed that the equation
of the form of the string was $\dfrac{\partial^2 u}{\partial t^2} = a^2\dfrac{\partial^2 u}{\partial x^2}$, and that the general
integral was $u = \phi(x+at) + \psi(x-at)$, where $\phi$ and $\psi$ are
arbitrary functions.\index{Differential equations|)}\index{Equations, differential|)}\index{String, vibrating, theory of|)}\index{Vibrating string|)}

The chief remaining contributions of D'Alembert to mathematics
were on physical astronomy, especially on the precession
of the equinoxes, and on variations in the obliquity of the
ecliptic. These were collected in his \emph{Système du monde}, published
in three volumes in 1754.

During the latter part of his life he was mainly occupied
with the great French encyclopaedia. For this he wrote the
introduction, and numerous philosophical and mathematical
articles; the best are those on geometry and on probabilities.
His style is brilliant, but not polished, and faithfully reflects
his character, which was bold, honest, and frank. He defended
a severe criticism which he had offered on some mediocre work
by the remark, ``j'aime mieux être incivil qu'ennuyé''; and
with his dislike of sycophants and bores it is not surprising that
during his life he had more enemies than friends.\index{D'Alembert|)}

\index{Bernoulli, Daniel|(}\nblabel{png395a}\nbparagraph{Daniel Bernoulli.\footnote
  {The only account of Daniel Bernoulli's life with which I am acquainted is
  the \emph{éloge} by his friend \index{Condorcet}\index{Decondorcet@De Condorcet}Condorcet. \emph{Marie Jean Antoine Nicolas Caritat}, \emph{Marquis
  de~Condorcet}, was born in Picardy on Sept.~17, 1743, and fell a victim to the
  republican terrorists on March~28, 1794. He was secretary to the Academy,
  and is the author of numerous \emph{éloges}. He is perhaps more celebrated for his
  studies in philosophy, literature, and politics than in mathematics, but his
  mathematical treatment of probabilities, and his discussion of differential
  equations and finite differences, shew an ability which might have put him in
  the first rank had he concentrated his attention on mathematics. He sacrificed
  himself in a vain effort to guide the revolutionary torrent into a constitutional
  channel.}}
\emph{Daniel Bernoulli}, whose name I mentioned
above, and who was by far the ablest of the younger Bernoullis,
was a contemporary and intimate friend of \index{Euler!ref.\ to}Euler, whose works
are mentioned in the \hyperref[chapter:XVIII]{next chapter}. Daniel Bernoulli was born
on Feb.~9, 1700, and died at Bâle, where he was professor of
natural philosophy, on March~17, 1782. He went to Petrograd
in 1724 as professor of mathematics, but the roughness of the
social life was distasteful to him, and he was not sorry when
a temporary illness in 1733 allowed him to plead his health as
an excuse for leaving. He then returned to Bâle, and held
successively chairs of medicine, metaphysics, and natural philosophy
there.

His earliest mathematical work was the \emph{Exercitationes}, published
in 1724, which contains a solution of the differential
equation proposed by \index{Riccati!ref.\ to}Riccati. Two years later he pointed out
for the first time the frequent desirability of resolving a compound
motion into motions of translation and motions of rotation.
His chief work is his \index{Hydrodynamics. Developed by!z2@D'Alembert}\emph{Hydrodynamica}, published in 1738;
it resembles \index{Lagrange!ref.\ to}Lagrange's \emph{Mécanique analytique} in being arranged
so that all the results are consequences of a single principle,
namely, in this case, the \index{Conservation of energy}\index{Energy, conservation of}conservation of energy. This was
followed by a memoir on the \index{Tides, theory of}theory of the tides, to which, conjointly
with memoirs by Euler and \index{Maclaurin!ref.\ to}Maclaurin, a prize was
awarded by the French Academy: these three memoirs contain
all that was done on this subject between the publication of
\index{Newtons@Newton's \emph{Principia}!ref.\ to}Newton's \emph{Principia} and the investigations of \index{Laplace!ref.\ to}Laplace. Bernoulli
also wrote a large number of papers on various mechanical
questions, especially on problems connected with \index{Vibrating string}\index{String, vibrating, theory of}vibrating strings,
and the solutions given by \index{Taylor (Brook)!ref.\ to}Taylor and by \index{D'Alembert!ref.\ to}D'Alembert. He is
the earliest writer who attempted to formulate a kinetic theory
of gases, and he applied the idea to explain the law associated
with the names of \index{Boyle}Boyle and \index{Mariotte}Mariotte.\index{Bernoulli, Daniel|)}


\section*{The English mathematicians of the eighteenth century.\nblabel{png396a}}

I have reserved a notice of the English mathematicians who
succeeded \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton, in order that the members of the English
school may be all treated together. It was almost a matter of
course that the English should at first have adopted the notation
of Newton in the \index{Calculus, infinitesimal}infinitesimal calculus in preference to that of
\index{Leibnitz!ref.\ to}Leibnitz, and consequently the English school would in any case
have developed on somewhat different lines to that on the continent,
where a knowledge of the infinitesimal calculus was derived
solely from Leibnitz and the Bernoullis. But this separation
into two distinct schools became very marked owing to the
action of Leibnitz and \index{Bernoulli, John!ref.\ to}John Bernoulli, which was naturally
resented by Newton's friends; and so for forty or fifty years, to
the disadvantage of both sides, the quarrel raged. The leading
members of the English school were \index{Cotes!ref.\ to}\emph{Cotes, Demoivre, Ditton,
\index{Gregory, David}David Gregory, \index{Halley!ref.\ to}Halley, \index{Maclaurin!ref.\ to}Maclaurin, Simpson}, and \emph{Taylor}. I
may, however, again remind my readers that as we approach
modern times the number of capable mathematicians in Britain,
France, Germany, and Italy becomes very considerable, but that
in a popular sketch like this book it is only the leading men
whom I propose to mention.

To David Gregory, Halley, and Ditton I need devote but few
words.

\nblabel{png397a}\nbparagraph{David Gregory.} \emph{David Gregory}, the nephew of the James
Gregory mentioned \hyperref[png331a]{above}, born at Aberdeen on June~24, 1661,
and died at Maidenhead on Oct.~10, 1708, was appointed
professor at Edinburgh in 1684, and in 1691 was on Newton's
recommendation elected Savilian professor at Oxford. His
chief works are one on geometry, issued in 1684; one on optics,
published in 1695, which contains [p.~98] the earliest suggestion
of the possibility of making an achromatic combination of lenses;
and one on the Newtonian geometry, physics, and astronomy,
issued in 1702.

\nblabel{xref:379}\index{Halley}\nbparagraph{Halley.} \emph{Edmund Halley}, born in London in 1656, and
died at Greenwich in 1742, was educated at St.~Paul's School,
London, and Queen's College, Oxford, in 1703 succeeded \index{Wallis!ref.\ to}Wallis
as Savilian professor, and subsequently in 1720 was appointed
astronomer-royal in succession to \index{Flamsteed!ref.\ to}Flamsteed, whose \emph{Historia
Coelestis Britannica} he edited; the first and imperfect edition
was issued in 1712. Halley's name will be recollected for the
generous manner in which he secured the immediate publication
of \index{Newtons@Newton's \emph{Principia}!ref.\ to}Newton's \emph{Principia} in 1687. Most of his original work was
on astronomy and allied subjects, and lies outside the limits of
this book; it may be, however, said that the work is of excellent
quality, and both Lalande and \index{Mairan}Mairan speak of it in the highest
terms. \index{Halley!ref.\ to}Halley conjecturally restored the eighth and lost book
of the conics of \index{Apollonius!ref.\ to}Apollonius, and in 1710 brought out a magnificent
edition of the whole work; he also edited the works of
\index{Serenus!ref.\ to}Serenus, those of \index{Menelaus!ref.\ to}Menelaus, and some of the minor works of
Apollonius. He was in his turn succeeded at Greenwich as
astronomer-royal by \index{Bradley}Bradley.\footnote
  {\emph{James Bradley}, born in Gloucestershire in 1692, and died in 1762, was
  the most distinguished astronomer of the first half of the eighteenth century.
  Among his more important discoveries were the explanation of \index{Aberration (astronomical)}astronomical
  aberration (1729), the cause of \index{Nutation}nutation (1748), and his empirical formula
  for corrections for \index{Refraction}refraction. It is perhaps not too much to say that he was
  the first astronomer who made the art of observing part of a methodical
  science.}\nblabel{xref:380}

\index{Ditton, H.}\nblabel{png398a}\nbparagraph{Ditton.} \emph{Humphry Ditton} was born at Salisbury on May~29,
1675, and died in London in 1715 at Christ's Hospital, where
he was mathematical master. He does not seem to have paid
much attention to mathematics until he came to London about
1705, and his early death was a distinct loss to English science.
He published in 1706 a text book on \index{Fluxions}fluxions; this and another
similar work by \index{Jones, Wm.}William Jones, which was issued in 1711,
occupied in England much the same place that \index{Hospital, l'!ref.\ to}\index{Delhospital@De l'Hospital!ref.\ to}\index{Lh@L'Hospital!ref.\ to}l'Hospital's
treatise did in France. In 1709 Ditton issued an algebra, and
in 1712 a treatise on perspective. He also wrote numerous
papers in the \emph{Philosophical Transactions}. He was the earliest
writer to attempt to explain the phenomenon of \index{Capillarity}capillarity
on mathematical principles; and he invented a method for
finding the \index{Longitude}longitude, which has been since used on various
occasions.

\index{Taylor (Brook)|(}\nblabel{png398b}\nbparagraph{Taylor.\footnote
  {An account of his life by \index{Young, Sir Wm., on Taylor}Sir William Young is prefixed to the \emph{Contemplatio
  Philosophica}. This was printed at London in 1793 for private
  circulation and is now extremely rare.}}
\emph{Brook Taylor,} born at Edmonton on August~18,
1685, and died in London on December~29, 1731, was educated
at St.~John's College, Cambridge, and was among the most
enthusiastic of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's admirers. From the year 1712 onwards
he wrote numerous papers in the \emph{Philosophical Transactions},
in which, among other things, he discussed the motion of
projectiles, the \index{Oscillation, centre of}centre of oscillation, and the forms taken by
liquids when raised by \index{Capillarity}capillarity. In 1719 he resigned the
secretaryship of the Royal Society and abandoned the study
of mathematics. His earliest work, and that by which he is
generally known, is his \emph{Methodus Incrementorum Directa et
Inversa}, published in London in 1715. \nblabel{png399a}This contains [prop.~7]
a proof of the well-known \index{Expansion,!offh@of $f(x + h)$}theorem
\[
f(x + h) = f(x) + hf'(x) + \frac{h^{2}}{|\underline{2}}f''(x) + \dots ,
\]
by which a function of a single variable can be \index{Expansions@Expansion in series}expanded in
powers of it. He does not consider the convergency of the
series, and the proof which involves numerous assumptions is
not worth reproducing. The work also includes several
theorems on \index{Interpolation, method of}interpolation. \index{Taylors@Taylor's theorem}Taylor was the earliest writer to
deal with theorems on the change of the independent variable;
he was perhaps the first to realize the possibility of a \index{Operations, calculus of}\index{Calculus of operations}calculus
of operation, and just as he denotes the \emph{n}th differential coefficient
of $y$ by $y_{n}$, so he uses $y_{-1}$ to represent the integral of $y$;
lastly, he is usually recognized as the creator of the theory of
\index{Finite differences}\index{Differences, finite}finite differences.

The applications of the calculus to various questions given in
the \emph{Methodus} have hardly received that attention they deserve.
The most important of them is the theory of the transverse
\index{String, vibrating, theory of}\index{Vibrating string}vibrations of strings, a problem which had baffled previous
investigators. In this investigation Taylor shews that the
number of half-vibrations executed in a second is
\[
\pi \sqrt{DP/LN},
\]
where $L$ is the length of the string, $N$ its weight, $P$ the weight
which stretches it, and $D$ the length of a seconds pendulum.
This is correct, but in arriving at it he assumes that every
point of the string will pass through its position of equilibrium
at the same instant, a restriction which \index{D'Alembert!ref.\ to}D'Alembert
subsequently shewed to be unnecessary. Taylor also found the
form which the string assumes at any instant.

\nblabel{png400a}The \emph{Methodus} also contains the earliest determination of
the differential equation of the path of a ray of light when
traversing a heterogeneous medium; and, assuming that the
density of the air depends only on its distance from the
earth's surface, Taylor obtained by means of quadratures the
approximate form of the curve. The form of the \index{Catenary}catenary and
the determination of the centres of oscillation and percussion
are also discussed.

A treatise on \index{Perspective}perspective by Taylor, published in 1719,
contains the earliest general enunciation of the principle of
\index{Vanishing points}vanishing points; though the idea of vanishing points for
horizontal and parallel lines in a picture hung in a vertical
plane had been enunciated by \index{Ubaldi}Guido Ubaldi in his \emph{Perspectivae
Libri}, Pisa, 1600, and by \index{Stevinus!ref.\ to}Stevinus in his \emph{Sciagraphia}, Leyden,
1608.\index{Taylor (Brook)|)}

\index{Cotes|(}\nblabel{png400b}\nbparagraph{Cotes.} \emph{Roger Cotes} was born near Leicester on July~10,
1682, and died at Cambridge on June~5, 1716. He was
educated at Trinity College, Cambridge, of which society he
was a fellow, and in 1706 was elected to the newly-created
Plumian chair of astronomy in the university of Cambridge.
From 1709 to 1713 his time was mainly occupied in editing
the second edition of the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia}. The remark of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton
that if only Cotes had lived ``we might have known something''
indicates the opinion of his abilities held by most of
his contemporaries.

Cotes's writings were collected and published in 1722
under the titles \emph{Harmonia Mensurarum} and \emph{Opera Miscellanea}.
His lectures on hydrostatics were published in 1738.
A large part of the \emph{Harmonia Mensurarum} is given up
to the decomposition and integration of rational algebraical
expressions. That part which deals with the theory of partial
fractions was left unfinished, but was completed by \index{Demoivre!ref.\ to}\index{Moivre, de!ref.\ to}Demoivre.
Cotes's theorem in trigonometry, which depends on forming the
quadratic factors of $x^{n} - 1$, is well known. The proposition that
``if from a fixed point $O$ a line be drawn cutting a curve in
$Q_{1}$, $Q_{2}$, $\dots$, $Q_{n}$, and a point $P$ be taken on the line so that the
reciprocal of $OP$ is the arithmetic mean of the reciprocals of
$OQ_{1}$, $OQ_{2}$, $\dots$, $OQ_{n}$, then the locus of $P$ will be a straight line''
is also due to Cotes. The title of the book was derived from
the latter theorem. The \emph{Opera Miscellanea} contains a paper
on the method for determining the most \index{Probabilities, theory of}probable result from
a number of observations. This was the earliest attempt to
frame a \index{Errors, theory of}theory of errors. It also contains essays on \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's
\emph{Methodus Differentialis}, on the construction of tables by the
method of differences, on the descent of a body under gravity,
on the cycloidal \index{Pendulum, motion of}pendulum, and on projectiles.\index{Cotes|)}

\index{Moivre, de|(}\index{Demoivre|(}\nblabel{png401a}\nbparagraph{Demoivre.} \emph{Abraham Demoivre} (more correctly written
as \emph{de\allowbreak~Moivre}) was born at Vitry on May~26, 1667, and died in
London on November~27, 1754. His parents came to England
when he was a boy, and his education and friends were alike
English. His interest in the higher mathematics is said to
have originated in his coming by chance across a copy of
\index{Newtons@Newton's \emph{Principia}!ref.\ to}Newton's \emph{Principia}. From the \emph{éloge} on him delivered in 1754
before the French Academy it would seem that his work
as a teacher of mathematics had led him to the house of the
\index{Devonshire, Earl of, ref.\ to}Earl of Devonshire at the instant when Newton, who had
asked permission to present a copy of his work to the earl,
was coming out. Taking up the book, and charmed by the far-reaching
conclusions and the apparent simplicity of the reasoning,
Demoivre thought nothing would be easier than to master the
subject, but to his surprise found that to follow the argument
overtaxed his powers. He, however, bought a copy, and as he
had but little leisure he tore out the pages in order to carry one
or two of them loose in his pocket so that he could study them
in the intervals of his work as a teacher. Subsequently he
joined the Royal Society, and became intimately connected with
Newton, \index{Halley!ref.\ to}Halley, and other mathematicians of the English
school. The manner of his death has a certain interest for
psychologists. Shortly before it he declared that it was necessary
for him to sleep some ten minutes or a quarter of an hour
longer each day than the preceding one. The day after he had thus
reached a total of something over twenty-three hours he slept
up to the limit of twenty-four hours, and then died in his sleep.

He is best known for having, together with \index{Lambert!ref.\ to}Lambert,
created that part of \index{Trigonometry.!z9@Development of, by John Bernoulli!x1@Demoivre}trigonometry which deals with imaginary
quantities. Two theorems on this part of the subject are still
connected with his name, namely, that which asserts that
$\cos nx + i \sin nx$ is one of the values of $(\cos x + i \sin x)^{n}$, and
that which gives the various quadratic factors of $x^{2n} - 2px^{n} + 1$.
His chief works, other than numerous papers in the \emph{Philosophical
Transactions}, were \emph{The Doctrine of Chances}, published
in 1718, and the \emph{Miscellanea Analytica}, published in 1730. In
the former the theory of \index{Recurring series}recurring series was first given, and
the theory of partial fractions which \index{Cotes!ref.\ to}Cotes's premature death
had left unfinished was completed, while the \index{Probabilities, theory of}rule for finding
the probability of a compound event was enunciated. The
latter book, besides the trigonometrical propositions mentioned
above, contains some theorems in astronomy, but they are treated
as problems in analysis.\index{Demoivre|)}\index{Moivre, de|)}

\index{Maclaurin|(}\nblabel{png402a}\nbparagraph{Maclaurin.\footnote
  {A sketch of Maclaurin's life is prefixed to his posthumous account of
  Newton's discoveries, London, 1748.}}\index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}
\emph{Colin Maclaurin}, who was born at Kilmodan
in Argyllshire in February 1698, and died at York on June~14,
1746, was educated at the university of Glasgow; in 1717
he was elected, at the early age of nineteen, professor of
mathematics at Aberdeen; and in 1725 he was appointed the
deputy of the mathematical professor at Edinburgh, and ultimately
succeeded him. There was some difficulty in securing a
stipend for a deputy, and Newton privately wrote offering to
bear the cost so as to enable the university to secure the services
of Maclaurin. Maclaurin took an active part in opposing the
advance of the \index{Pretender, the Young, ref.\ to}Young Pretender in 1745; on the approach of
the Highlanders he fled to York, but the exposure in the
trenches at Edinburgh and the privations he endured in his
escape proved fatal to him.

His chief works are his \emph{Geometria Organica}, London, 1720;
his \emph{De Linearum Geometricarum Proprietatibus}, London, 1720;
his \emph{Treatise on Fluxions}, Edinburgh, 1742; his \emph{Algebra},
London, 1748; and his \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}\emph{Account of Newton's Discoveries}, London,
1748.

\index{Geometry.!z8@Modern synthetic geometry, originated with Desargues!x4@Maclaurin}\nblabel{png403a}The first section of the first part of the \emph{Geometria Organica}
is on \index{Conic Sections (Geometrical).!Discussed by most of the Greek geometricians after Menaechmus!x6@and Maclaurin}conics; the second on nodal cubics; the third on other
cubics and on quartics; and the fourth section is on general
properties of curves. Newton had shewn that, if two angles
bounded by straight lines turn round their respective summits
so that the point of intersection of two of these lines moves
along a straight line, the other point of intersection will
describe a conic; and, if the first point move along a conic, the
second will describe a quartic. Maclaurin gave an analytical
discussion of the general theorem, and shewed how by this
method various curves could be practically traced. This work
contains an elaborate discussion on curves and their \index{Pedals}pedals,
a branch of geometry which he had created in two papers
published in the \emph{Philosophical Transactions} for 1718 and
1719.

The second part of the work is divided into three sections
and an appendix. The first section contains a proof of \index{Cotes!ref.\ to}Cotes's
theorem above alluded to; and also the analogous theorem
(discovered by himself) that, if a straight line $OP_{1}P_{2}\dots$ drawn
through a fixed point $O$ cut a curve of the $n$th degree in $n$
points $P_{1}$, $P_{2}$,~$\dots$, and if the tangents at $P_{1}$, $P_{2}$,~$\dots$ cut a fixed
line $Ox$ in points $A_{1}$, $A_{2}$,~$\dots$, then the sum of the reciprocals
of the distances $OA_{1}$, $OA_{2}$,~$\dots$ is constant for all positions of
the line $OP_{1}P_{2}\dots$. These two theorems are generalizations of
those given by Newton on diameters and asymptotes. Either
is deducible from the other. In the second and third sections
these theorems are applied to conics and cubics; most of the
harmonic properties connected with a quadrilateral inscribed
in a conic are determined; and in particular the theorem on
an inscribed hexagon which is known by the name of \index{Pascal!ref.\ to}Pascal is
deduced. Pascal's essay was not published till 1779, and
the earliest printed enunciation of his theorem was that given
by Maclaurin. Amongst other propositions he shews that,
if a quadrilateral be inscribed in a cubic, and if the points
of intersection of the opposite sides also lie on the curve, then
the tangents to the cubic at any two opposite angles of the
quadrilateral will meet on the curve. In the fourth section
he considers some theorems on central force. The fifth section
contains some theorems on the description of curves through
given points. One of these (which includes \index{Pascal!ref.\ to}Pascal's as a particular
case) is that if a polygon be deformed so that while
each of its sides passes through a fixed point its angles (save
one) describe respectively curves of the $m$th, $n$th, $p$th,\,\ldots\
degrees, then shall a remaining angle describe a curve of the
degree $2mnp$\ldots; but if the given points be collinear, the
resulting curve will be only of the degree $mnp$\ldots. This essay
was reprinted with additions in the \emph{Philosophical Transactions}
for 1735.

\index{Fluxional calculus}\nblabel{png404a}The \emph{Treatise of Fluxions}, published in 1742, was the first
logical and systematic exposition of the method of fluxions.
The cause of its publication was an attack by \index{Berkeley, G.}Berkeley on the
principles of the \index{Calculus, infinitesimal}infinitesimal calculus. In it [art.~751, p.~610]
Maclaurin gave a proof of the theorem that
\[
f(x) = f(0) + xf'(0) + \frac{x^2}{|\underline{2}}f''(0) + \ldots.
\]
\index{Expansion,!offx@of $f(x)$}This was obtained in the manner given in many modern text-books
by assuming that $f(x)$ can be \index{Expansions@Expansion in series}expanded in a form like
\[
f(x) = A_0 + A_1x + A_2x^2 + \ldots,
\]
then, on differentiating and putting $x=0$ in the successive
results, the values of $A_0$, $A_1$,\,\ldots\ are obtained; but he did not
investigate the \index{Convergency}convergency of the series. The result had been
previously given in 1730 by \index{Stirling}James Stirling in his \emph{Methodus
Differentialis} [p.~102], and of course is at once deducible from
\index{Taylors@Taylor's theorem}Taylor's theorem. \index{Infinite series, difficulties in connection with}Maclaurin also here enunciated [art.~350,
p.~289] the important theorem that, if $\phi(x)$ be positive and
decrease as $x$ increases from $x=a$ to $x=\infty$, then the series
\[
\phi(a) + \phi(a+1) + \phi(a+2) + \ldots
\]
is \index{Convergency}convergent or divergent as the integral from $x=a$ to $x=\infty$ of
$\phi(x)$ is finite or infinite. The theorem had been given by
\index{Euler!ref.\ to}Euler\footnote
  {See Cantor, vol.~iii, p.~663.}
in 1732, but in so awkward a form that its value escaped
general attention. Maclaurin here also gave the correct theory of
\index{Maxima and minima, determination of}maxima and minima, and rules for finding and discriminating
multiple points.

\nblabel{png405a}This treatise is, however, especially valuable for the solutions
it contains of numerous problems in geometry, statics, the \index{Attraction, theories of}theory
of attractions, and astronomy. To solve these Maclaurin reverted
to classical methods, and so powerful did these processes
seem, when used by him, that \index{Clairaut!ref.\ to}Clairaut, after reading the work,
abandoned analysis, and attacked the problem of the figure of
the earth again by pure geometry. At a later time this part of
the book was described by \index{Lagrange!ref.\ to}Lagrange as the ``chef-d'\oe uvre de
géométrie qu'on peut comparer à tout ce qu'Archimède nous a
laissé de plus beau et de plus ingénieux.'' Maclaurin also
determined the attraction of a homogeneous ellipsoid at an
internal point, and gave some theorems on its attraction at an
external point; in attacking these questions he introduced
the conception of level surfaces, that is, surfaces at every point
of which the resultant attraction is perpendicular to the surface.
No further advance in the theory of attractions was made until
Lagrange in 1773 introduced the idea of the potential. \index{Hydrodynamics. Developed by!z3@Maclaurin}Maclaurin
also shewed that a spheroid was a possible form of
equilibrium of a mass of homogeneous liquid rotating about an
axis passing through its centre of mass. Finally he discussed
the \index{Tides, theory of}tides; this part had been previously published (in 1740) and
had received a prize from the French Academy.

Among Maclaurin's minor works is his \emph{Algebra}, published
in 1748, and founded on \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's \emph{Universal Arithmetic}. It
contains the results of some early papers of Maclaurin; notably
of two, written in 1726 and 1729, on the number of imaginary
roots of an equation, suggested by Newton's theorem; and of
one, written in 1729, containing the well-known rule for finding
equal roots by means of the derived equation. In this book
negative quantities are treated as being not less real than
positive quantities. To this work a treatise, entitled \emph{De
Linearum Geometricarum Proprietatibus Generalibus}, was added
as an appendix; besides the paper of 1720 above alluded to,
it contains some additional and elegant theorems. Maclaurin
also produced in 1728 an exposition of the Newtonian philosophy,
which is incorporated in the posthumous work printed in 1748.
Almost the last paper he wrote was one printed in the \emph{Philosophical
Transactions} for 1743 in which he discussed from a
mathematical point of view the form of a bee's cell.

Maclaurin was one of the most able mathematicians of the
eighteenth century, but his influence on the progress of British
mathematics was on the whole unfortunate. By himself
abandoning the use both of analysis and of the infinitesimal
calculus, he induced \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's countrymen to confine themselves
to Newton's methods, and it was not until about 1820, when
the differential calculus was introduced into the Cambridge
curriculum, that English mathematicians made any general use
of the more powerful methods of modern analysis.\index{Maclaurin|)}

\index{Stewart, Matthew}\nblabel{png406a}\nbparagraph{Stewart.} \index{Maclaurin!ref.\ to}Maclaurin was succeeded in his chair at Edinburgh
by his pupil \emph{Matthew Stewart}, born at Rothesay in 1717 and
died at Edinburgh on January~23, 1785, a mathematician of
considerable power, to whom I allude in passing, for his theorems
on the problem of three bodies, and for his discussion, treated by
transversals and involution, of the properties of the circle and
straight line.

\index{Simpson, Thomas|(}\nbparagraph{Simpson.\footnote
  {A sketch of Simpson's life, with a bibliography of his writings, by \index{Bevis and Hutton on Simpson}J.~Bevis
  and \index{Hutton, ref.\ to}C.~Hutton, was published in London in 1764. A short memoir is
  also prefixed to the later editions of his work on fluxions.}}
The last member of the English school whom
I need mention here is \emph{Thomas Simpson}, who was born in
Leicestershire on August~20, 1710, and died on May~14, 1761.
His father was a weaver, and he owed his education to his own
efforts. His mathematical interests were first aroused by the
solar eclipse which took place in 1724, and with the aid of a
fortune-telling pedlar he mastered \index{Cocker's arithmetic}Cocker's \emph{Arithmetic} and the
elements of algebra. He then gave up his weaving and became
an usher at a school, and by constant and laborious efforts
improved his mathematical education, so that by 1735 he was
able to solve several questions which had been recently proposed
and which involved the infinitesimal calculus. He next moved
to London, and in 1743 was appointed professor of mathematics
at Woolwich, a post which he continued to occupy till his death.

The works published by Simpson prove him to have been
a man of extraordinary natural genius and extreme industry.
The most important of them are his \emph{Fluxions}, 1737 and 1750,
with numerous applications to physics and astronomy; his \emph{Laws
of Chance} and his \emph{Essays}, 1740; his \index{Assurance, life}\index{Life assurance}theory of \emph{Annuities and
Reversions} (a branch of mathematics that is due to \index{Dodson on life assurance}James
Dodson, died in 1757, who was a master at Christ's Hospital,
London), with tables of the value of lives, 1742; his \emph{Dissertations},
1743, in which the figure of the earth, the force of
attraction at the surface of a nearly spherical body, the \index{Tides, theory of}theory
of the tides, and the law of astronomical refraction are discussed;
his \emph{Algebra}, 1745; his \emph{Geometry}, 1747; his \emph{Trigonometry},
1748, in which he introduced the current abbreviations for
the \index{Trigonometrical functions}trigonometrical functions; his \emph{Select Exercises}, 1752, containing
the solutions of numerous problems and a theory of
gunnery; and lastly, his \emph{Miscellaneous Tracts}, 1754.

The work last mentioned consists of eight memoirs, and these
contain his best known investigations. The first three papers
are on various problems in astronomy; the fourth is on the
theory of mean observations; the fifth and sixth on problems in
fluxions and algebra; the seventh contains a general solution of
the \index{Isoperimetrical problem}isoperimetrical problem; the eighth contains a discussion
of the third and ninth sections of the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia}, and their
application to the lunar orbit. In this last memoir Simpson
obtained a differential equation for the motion of the \index{Apse, motion of lunar}apse of the
lunar orbit similar to that arrived at by \index{Clairaut!ref.\ to}Clairaut, but instead of
solving it by successive approximations, he deduced a general
solution by indeterminate coefficients. The result agrees with
that given by Clairaut. Simpson solved this problem in 1747,
two years later than the publication of Clairaut's memoir,
but the solution was discovered independently of Clairaut's
researches, of which Simpson first heard in 1748.\index{Simpson, Thomas|)}

\nbChapter{XVIII}{Lagrange, Laplace, and their Contemporaries}{Lagrange, Laplace, etc.}{Circ.\ 1740--1830.}
  {A fourth volume of M.~Cantor's \emph{History}, covering the period from 1759
  to 1799, was brought out in 1907. It contains memoirs by S.~Günther
  on the mathematics of the period; by F.~Cajori on arithmetic, algebra, and
  numbers; by E.~Netto on series, imaginaries, \&c.; by V.~von~Braunmühl on
  trigonometry; by V.~Bobynin and G.~Loria on pure geometry; by V.~Kommerell
  on analytical geometry; by G.~Vivanti on the infinitesimal
  calculus; and by C.~R.~Wallner on differential equations.}
\index{Mechanics.!z4@Subsequently extended by (among others) D'Alembert, Maclaurin, Euler, Lagrange, Laplace, and Poisson, chapters \textsc{xvii, xviii}}\index{Geometry.!z2@Classical synthetic geometry, discussed or used by nearly all the mathematicians considered in the first period, chapters \textsc{ii-v}!also by Newton and his School, chapters \textsc{xvi, xviii}}\index{Wallner, C. R.}\index{Vivanti, G.}\index{Braunmühl, V. von}\index{Kommerell, V.}\index{Bobynin on Ahmes!ref.\ to}\index{Loria, ref.\ to}\index{Gunther@Günther, S.}\index{Cajori, F.}\index{Cantor, M., ref.\ to}\nblabel{png409a}

\nblabel{png409b}\First{The} \hyperref[chapter:XVII]{last chapter} contains the history of two separate schools---the
continental and the British. In the early years of the
eighteenth century the English school appeared vigorous and
fruitful, but decadence rapidly set in, and after the deaths of
\index{Maclaurin!ref.\ to}Maclaurin and \index{Simpson, Thomas!ref.\ to}Simpson no British mathematician appeared
who is at all comparable to the continental mathematicians of
the latter half of the eighteenth century. This fact is partly
explicable by the isolation of the school, partly by its tendency
to rely too exclusively on geometrical and fluxional methods.
Some attention was, however, given to practical science, but,
except for a few remarks at the end of this chapter, I do not
think it necessary to discuss English mathematics in detail,
until about 1820, when analytical methods again came into
vogue.

On the continent, under the influence of \index{Bernoulli, John!ref.\ to}John Bernoulli, the
calculus had become an instrument of great analytical power
expressed in an admirable notation---and for practical applications
it is impossible to over-estimate the value of a good
notation. The subject of mechanics remained, however, in much
the condition in which \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton had left it, until \index{D'Alembert!ref.\ to}D'Alembert, by
making use of the differential calculus, did something to extend
it. Universal gravitation as enunciated in the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia} was
accepted as an established fact, but the geometrical methods
adopted in proving it were difficult to follow or to use in
analogous problems; \index{Maclaurin!ref.\ to}Maclaurin, \index{Simpson, Thomas!ref.\ to}Simpson, and \index{Clairaut!ref.\ to}Clairaut may
be regarded as the last mathematicians of distinction who
employed them. Lastly, the Newtonian theory of light was
generally received as correct.

The leading mathematicians of the era on which we are now
entering are \index{Euler!ref.\ to}Euler, \index{Lagrange!ref.\ to}Lagrange, \index{Laplace!ref.\ to}Laplace, and \index{Legendre!ref.\ to}Legendre. Briefly
we may say that Euler extended, summed up, and completed
the work of his predecessors; while Lagrange with almost unrivalled
skill developed the infinitesimal calculus and theoretical
mechanics, and presented them in forms similar to those in
which we now know them. At the same time Laplace made
some additions to the infinitesimal calculus, and applied that
calculus to the theory of universal gravitation; he also created
a calculus of probabilities. Legendre invented spherical harmonic
analysis and elliptic integrals, and added to the theory
of numbers. The works of these writers are still standard
authorities. I shall content myself with a mere sketch of the
chief discoveries embodied in them, referring any one who wishes
to know more to the works themselves. Lagrange, Laplace,
and Legendre created a French school of mathematics of which
the younger members are divided into two groups; one (including
\index{Poisson!ref.\ to}Poisson and \index{Fourier!ref.\ to}Fourier) began to apply mathematical analysis to
physics, and the other (including \index{Monge!ref.\ to}Monge, \index{Carnot, Lazare!ref.\ to}Carnot, and \index{Poncelet!ref.\ to}Poncelet)
created modern geometry. Strictly speaking, some of the great
mathematicians of recent times, such as \index{Gauss!ref.\ to}Gauss and \index{Abel!ref.\ to}Abel, were
contemporaries of the mathematicians last named; but, except
for this remark, I think it convenient to defer any consideration
of them to the \hyperref[chapter:XIX]{next chapter}.

\section*{The development of analysis and mechanics.\nblabel{png411a}}

\index{Euler!ref.\ to}\nblabel{png411b}\nbparagraph{Euler.\footnote
  {The chief facts in Euler's life are given by \index{Fuss, ref.\ to}N.~Fuss, and a list of Euler's
  writings is prefixed to his \emph{Correspondence}, 2 vols., Petrograd, 1843; see also
  \emph{Index Operum Euleri} by \index{Hagen, J. G.}J.~G.~Hagen, Berlin, 1896. Euler's earlier
  works are discussed by \index{Cantor, M., ref.\ to}Cantor, chapters~cxi, cxiii, cxv, and cxvii. No
  complete edition of Euler's writings has been published, though the work
  has been begun twice.}}
\index{Euler|(}\emph{Leonhard Euler} was born at Bâle on April~15, 1707,
and died at Petrograd on September~7, 1783. He was
the son of a Lutheran minister who had settled at Bâle, and
was educated in his native town under the direction of \index{Bernoulli, John!ref.\ to}John
Bernoulli, with whose sons \index{Bernoulli, Daniel!ref.\ to}Daniel and \index{Bernoulli, Nicholas!ref.\ to}Nicholas he formed a
lifelong friendship. When, in 1725, the younger Bernoullis
went to Russia, on the invitation of the empress, they procured
a place there for Euler, which in 1733 he exchanged for the
chair of mathematics, then vacated by Daniel Bernoulli. The
severity of the climate affected his eyesight, and in 1735 he lost
the use of one eye completely. In 1741 he moved to Berlin at
the request, or rather command, of \index{Frederick the Great of Prussia, ref.\ to}Frederick the Great; here
he stayed till 1766, when he returned to Russia, and was
succeeded at Berlin by \index{Lagrange!ref.\ to}Lagrange. Within two or three years of
his going back to Petrograd he became blind; but in spite
of this, and although his house, together with many of his
papers, were burnt in 1771, he recast and improved most of his
earlier works. He died of apoplexy in 1783. He was married
twice.

I think we may sum up Euler's work by saying that he
created a good deal of analysis, and revised almost all the
branches of pure mathematics which were then known, filling
up the details, adding proofs, and arranging the whole in a
consistent form. Such work is very important, and it is
fortunate for science when it falls into hands as competent as
those of Euler.

Euler wrote an immense number of memoirs on all kinds of
mathematical subjects. His chief works, in which many of the
results of earlier memoirs are embodied, are as follows.

\nblabel{png412a}In the first place, he wrote in 1748 his \emph{Introductio in
Analysin Infinitorum}, which was intended to serve as an introduction
to pure analytical mathematics. This is divided into
two parts.

The first part of the \emph{Analysis Infinitorum} %[**TN: "Analysis" is in original text]
contains the bulk
of the matter which is to be found in modern text-books on
algebra, \index{Equations, differential!theory of}theory of equations, and \index{Trigonometry.!z9@Development of, by John Bernoulli!x2@Euler}trigonometry. In the algebra
he paid particular attention to the \index{Expansions@Expansion in series}expansion of various functions
in series, and to the summation of given series; and
pointed out explicitly that an \index{Infinite series, difficulties in connection with}infinite series cannot be safely
employed unless it is \index{Convergency}convergent. In the trigonometry, much
of which is founded on \index{Mayer, F. C.}F.~C.~Mayer's \emph{Arithmetic of Sines}, which
had been published in 1727, \index{Euler!ref.\ to}Euler developed the idea of \index{Bernoulli, John!ref.\ to}John
Bernoulli, that the subject was a branch of analysis and not a
mere appendage of astronomy or geometry. He also introduced
(contemporaneously with \index{Simpson, Thomas!ref.\ to}Simpson) the \index{Trigonometrical symbols, origin of}current abbreviations for
the \index{Secant}\index{Sine}\index{Trigonometrical functions}trigonometrical functions, and shewed that the trigonometrical
and exponential functions were connected by the
relation $\cos\theta + i\sin\theta = e^{i\theta}$.

Here, too [pp.~85, 90, 93], we meet the symbol \index{e@$e$, symbol for 2.71828\ldots}$e$ used to
denote the base of the Napierian logarithms, namely, the incommensurable
number $2.71828\ldots$, and the \index{P1@$\pi$,!z3@introduction of symbol|(}symbol $\pi$ used to denote
the incommensurable number $3.14159\ldots$. The use of a single
symbol to denote the number $2.71828\ldots$ seems to be due to
\index{Cotes!ref.\ to}Cotes, who denoted it by $M$; Euler in 1731 denoted it by $e$.
To the best of my knowiedge, \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton had been the first to
employ the literal exponential notation, and Euler, using the
form $a^z$, had taken $a$ as the base of any system of logarithms. It
is probable that the choice of $e$ for a particular base was determined
by its being the vowel consecutive to $a$. The use of a
single symbol to denote the number $3.14159\ldots$ appears to have
been introduced about the beginning of the eighteenth century.
\index{Jones, Wm.}W.~Jones in 1706 represented it by $\pi$, a symbol which had been
used by \index{Oughtred!ref.\ to}Oughtred in 1647, and by \index{Barrow!ref.\ to}Barrow a few years later, to
denote the periphery of a circle. John Bernoulli represented
the number by $c$; Euler in 1734 denoted it by $p$, and in
a letter of 1736 (in which he enunciated the theorem that the
sum of the squares of the reciprocals of the natural numbers
is $\pi^{2}/6$) he used the letter $c$; \index{Goldbach}Chr.~Goldbach in 1742 used $\pi$;
and after the publication of Euler's \emph{Analysis} the symbol $\pi$ was
generally employed.

The numbers $e$ and $\pi$ would enter into mathematical analysis
from whatever side the subject was approached. The latter
represents among other things the ratio of the circumference of
a circle to its diameter, but it is a mere accident that that is
taken for its definition. \index{Demorgana@De~Morgan, A.!ref.\ to}De~Morgan in the \emph{Budget of Paradoxes}
tells an anecdote which illustrates how little the usual definition
suggests its real origin. He was explaining to an actuary what
was the chance that at the end of a given time a certain proportion
of some group of people would be alive; and quoted the
actuarial formula involving $\pi$, which, in answer to a question, he
explained stood for the ratio of the circumference of a circle to
its diameter. His acquaintance, who had so far listened to the
explanation with interest, interrupted him and explained, ``My
dear friend, that must be a delusion; what can a circle have to
do with the number of people alive at the end of a given
time?''\index{P1@$\pi$,!z3@introduction of symbol|)}

\index{Calculus, infinitesimal|(}The second part of the \emph{Analysis Infinitorum} %[**TN: "Analysis" is in original text]
is on analytical
geometry. \index{Euler!ref.\ to}Euler commenced this part by dividing curves into
algebraical and transcendental, and established a variety of propositions
which are true for all algebraical curves. He then
applied these to the general equation of the second degree in
two dimensions, shewed that it represents the various \index{Conicoids}\index{Conics (Analytical).!q3@and Euler}conic
sections, and deduced most of their properties from the general
equation. \index{Quadrics}He also considered the \index{Curves, classification of}classification of cubic, quartic,
and other algebraical curves. He next discussed the question as
to what surfaces are represented by the general equation of the
second degree in three dimensions, and how they may be discriminated
one from the other: some of these surfaces had not
been previously investigated. In the course of this analysis he
laid down the rules for the transformation of co-ordinates in
space. Here also we find the earliest attempt to bring the
curvature of surfaces within the domain of mathematics, and the
first complete discussion of \index{Curves, tortuous}\index{Tortuous curves}tortuous curves.

\nblabel{png414a}The \emph{Analysis Infinitorum} %[**TN: "Analysis" is in original text]
was followed in 1755 by the
\emph{Institutiones Calculi Differentialis}, to which it was intended as
an introduction. This is the first text-book on the differential
calculus which has any claim to be regarded as complete, and it
may be said that until recently many modern treatises on the
subject are based on it; at the same time it should be added
that the exposition of the principles of the subject is often prolix
and obscure, and sometimes not altogether accurate.\index{Calculus, infinitesimal|)}

\nblabel{png414b}This series of works was completed by the publication in
three volumes in 1768 to 1770 of the \emph{Institutiones Calculi
Integralis}, in which the results of several of \index{Euler!ref.\ to}Euler's earlier
memoirs on the same subject and on \index{Differential equations}\index{Equations, differential}differential equations are
included. This, like the similar treatise on the differential
calculus, summed up what was then known on the subject, but
many of the theorems were recast and the proofs improved.
\index{Gamma function}\index{Beta function}The Beta and Gamma\footnote
  {The history of the Gamma function is given in a monograph by \index{Brunel on Gamma function}Brunel in
  the \emph{Mémoires de la société des sciences}, Bordeaux, 1886.}
functions were invented by Euler and
are discussed here, but only as illustrations of methods of
reduction and integration. His treatment of \index{Elliptic functions}elliptic integrals
is superficial; it was suggested by a theorem, given by \index{Landen}John
Landen in the \emph{Philosophical Transactions} for 1775, connecting
the arcs of a hyperbola and an ellipse. Euler's works that
form this trilogy have gone through numerous subsequent
editions.

The classic problems on \index{Isoperimetrical problem}isoperimetrical curves, the \index{Brachistochrone, the}brachistochrone
in a resisting medium, and the theory of \index{Geodesics}geodesics (all of
which had been suggested by his master, \index{Bernoulli, John!ref.\ to}John Bernoulli) had
engaged Euler's attention at an early date; and in solving them
he was led to the \index{Calculus of operations!of variations}\index{Variations, calculus of}calculus of variations. The idea of this was
given in his \emph{Curvarum Maximi Minimive Proprietate Gaudentium
Inventio}, published in 1741 and extended in 1744, but the
complete development of the new calculus was first effected by
\index{Lagrange!ref.\ to}Lagrange in 1759. The method used by Lagrange is described
in \index{Euler!ref.\ to}Euler's integral calculus, and is the same as that given in
most modern text-books on the subject.

\index{Algebra.!z8@Developed by (amongst others) Descartes!x4@and Euler|(}\nblabel{png415a}In 1770 Euler published his \emph{Vollständige Anleitung zur
Algebra}. A French translation, with numerous and valuable
additions by Lagrange, was brought out in 1774; and a
treatise on arithmetic by Euler was appended to it. The first
volume treats of determinate algebra. This contains one of
the earliest attempts to place the fundamental processes on a
scientific basis: the same subject had attracted \index{D'Alembert!ref.\ to}D'Alembert's
attention. This work also includes the proof of the \index{Expansion,!ofbin@of binomial}binomial
theorem for an unrestricted real index which is still known by
Euler's name; the proof is founded on the principle of the
permanence of equivalent forms, but Euler made no attempt to
investigate the convergency of the series: that he should have
omitted this essential step is the more curious as he had himself
recognized the necessity of considering the convergency of
infinite series: \index{Vandermonde}Vandermonde's proof given in 1764 suffers from
the same defect.

The second volume of the algebra treats of indeterminate
or Diophantine algebra. This contains the solutions of some
of the problems proposed by \index{Fermat!ref.\ to}Fermat, and which had hitherto
remained unsolved.

\index{Numbers, theory of.!Treatment of, by Pythagoras!q4@by Euler|(}As illustrating the simplicity and directness of Euler's
methods I give the substance of his demonstration,\footnote
  {\emph{Commentationes Arithmeticae Collectae}, Petrograd, 1849, vol.~ii, p.~514,
  art.~107. \index{Sylvester!ref.\ to}Sylvester published an analysis of the argument in \emph{Nature},
  December~15, 1887, vol.~xxxvii, p.~152.}
alluded to
above, that all even \index{Numbers,!perfect}\index{Perfect numbers}perfect numbers are included in Euclid's
formula, $2^{n-1}p,$ where $p$ stands for $2^{n}-1$ and is a prime.\footnote
  {\index{Eucn@Euc.!f36@\textsc{ix}, 36. ref.\ to}Euc.~ix,~36; see above, page~\pageref{xref:307}.}
Let $N$ be an even perfect number. $N$ is even, hence it can be
written in the form $2^{n-1}a$, where $a$ is not divisible by 2. $N$
is perfect, that is, is equal to the sum of all its integral subdivisors;
therefore (if the number itself be reckoned as one of
its divisors) it is equal to half the sum of all its integral divisors,
which we may denote by $\sum N$. Since $2N=\sum N$, we have
\begin{gather*}
2 × 2^{n-1}\alpha = \sum 2^{n-1}\alpha = \sum 2^{n-1} × \sum \alpha. \\
\therefore 2^{n}\alpha = (1 + 2 + \ldots + 2^{n-1})\sum \alpha = (2^{n}-1)\sum \alpha,
\end{gather*}
therefore $\alpha : \sum \alpha = 2^{n} - 1 : 2^{n} = p :p+1$. Hence $\alpha = \lambda p$, and
$\sum \alpha = \lambda(p + 1)$; and since the ratio $p:p + 1$ is in its lowest
terms, $\lambda$ must be a positive integer. Now, unless $\lambda=1$, we
have $1, \lambda, p$, and $\lambda p$ as factors of $\lambda p$; moreover, if $p$ be not
prime, there will be other factors also. Hence, unless $\lambda = 1$ and
$p$ be a prime, we have
\[
\sum \lambda p = 1 + \lambda + p + \lambda p + \ldots = (\lambda + 1)(p + 1)+\ldots
\]
But this is inconsistent with the result $\sum \lambda p = \sum \alpha = \lambda(p + 1)$.
Hence $\lambda$ must be equal to 1 and $p$ must be a prime. Therefore
$\alpha =p$, therefore $N= 2^{n-1} \alpha = 2^{n-1} (2^{n} - 1)$. I may add the
corollary that since $p$ is a prime, it follows that $n$ is a prime;
and the determination of what values of $n$ (less than 257)
make $p$ prime falls under Mersenne's rule.\index{Algebra.!z8@Developed by (amongst others) Descartes!x4@and Euler|)}\index{Numbers, theory of.!Treatment of, by Pythagoras!q4@by Euler|)}

\nblabel{png416a}The four works mentioned above comprise most of what
\index{Euler!ref.\ to}Euler produced in pure mathematics. He also wrote numerous
memoirs on nearly all the subjects of applied mathematics and
mathematical physics then studied: the chief novelties in them
are as follows.

In the mechanics of a rigid system he determined the
general equations of motion of a body about a fixed point,
which are ordinarily written in the form
\[
A \frac{d\omega_1}{dt}-(B-C)\omega_2 \omega_3 = L:
\]
and he gave the general equations of motion of a free body,
which are usually presented in the form
\[
\frac{d}{dt}(mu)-mv\theta_3 + mw\theta_2 = X, \text{ and } \frac{dh_1'}{dt}-h_2'\theta_3 + h_3'\theta_2 = L.
\]
He also defended and elaborated the theory of \index{Least action}\index{Action,!least}``least action''
which had been propounded by \index{Maupertuis, P. L. M.}Maupertuis in 1751 in his
\emph{Essai de cosmologie}[p.~70].

\index{Hydrodynamics. Developed by!z4@Euler}In hydrodynamics Euler established the general equations of
motion, which are commonly expressed in the form
\[
\frac{1}{\rho}\frac{dp}{dx} = X - \frac{du}{dt} - u\frac{du}{dx} - v\frac{du}{dy} - w\frac{du}{dz}.
\]
At the time of his death he was engaged in writing a treatise
on \index{Hydrostatics. Developed by!z6@and by Euler}hydromechanics in which the treatment of the subject would
have been completely recast.

His most important works on astronomy are his \emph{Theoria
Motuum Planetarum et Cometarum}, published in 1744; his
\emph{Theoria Motus Lunaris}, published in 1753; and his \emph{Theoria
Motuum Lunae}, published in 1772. In these he attacked the
\index{Three bodies, problem of}problem of three bodies: he supposed the body considered
(\emph{ex.~gr.}\ the moon) to carry three rectangular axes with it in
its motion, the axes moving parallel to themselves, and to
these axes all the motions were referred. This method is not
convenient, but it was from \index{Euler!ref.\ to}Euler's results that \index{Mayer, J. T.}Mayer\footnote
  {\emph{Johann Tobias Mayer}, born in Würtemberg in 1723, and died in 1762,
  was director of the English observatory at Göttingen. Most of his memoirs,
  other than his lunar tables, were published in 1775 under the title \emph{Opera
  Inedita.}}
constructed the lunar tables for which his widow in 1770 received
£5000 from the English parliament, and in recognition of
Euler's services a sum of £300 was also voted as an honorarium
to him.

\index{Wave theory (optics)}\index{Optics (geometrical).!z11@(physical)}Euler was much interested in optics. \index{Light,!physical theories of}In 1746 he discussed
the relative merits of the emission and \index{Undulatory theory (optics)}undulatory theories of
light; he on the whole preferred the latter. In 1770--71 he
published his optical researches in three volumes under the
title \emph{Dioptrica}.

He also wrote an elementary work on physics and the
fundamental principles of mathematical philosophy. This
originated from an invitation he received when he first went
to Berlin to give lessons on physics to the princess of Anhalt-Dessau.
These lectures were published in 1768--1772 in
three volumes under the title \emph{Lettres \dots sur quelques sujets de
physique \dots}, and for half a century remained a standard treatise
on the subject.

Of course Euler's magnificent works were not the only
text-books containing original matter produced at this time.
Amongst numerous writers I would specially single out \emph{Daniel
Bernoulli, Simpson, Lambert, Bézout, Trembley, }and \index{Arbogast!ref.\ to}\emph{Arbogast},
as having influenced the development of mathematics. To the
two first-mentioned I have already alluded in the \hyperref[chapter:XVII]{last chapter}.\index{Euler|)}

\index{Lambert|(}\nblabel{png418a}\nbparagraph{Lambert.\footnote
  {See \emph{Lambert nach seinem Leben und Wirken}, by \index{Huber on Lambert}D.~Huber, Bâle, 1829.
  Most of Lambert's memoirs are collected in his \emph{Beiträge zum Gebrauche der
  Mathematik}, published in four volumes, Berlin, 1765--1772.}}
 \emph{Johann Heinrich Lambert }was born at Mülhausen
on August~28, 1728, and died at Berlin on September~25,
1777. He was the son of a small tailor, and had to rely
on his own efforts for his education; from a clerk in some ironworks
he got a place in a newspaper office, and subsequently,
on the recommendation of the editor, he was appointed tutor in
a private family, which secured him the use of a good library
and sufficient leisure to use it. In 1759 he settled at Augsburg,
and in 1763 removed to Berlin where he was given a small
pension, and finally made editor of the Prussian astronomical
almanack.

Lambert's most important works were one on optics, issued
in 1759, which suggested to \index{Arago!ref.\ to}Arago the lines of investigation he
subsequently pursued; a treatise on perspective, published in
1759 (to which in 1768 an appendix giving practical applications
were added); and a treatise on comets, printed in 1761,
containing the well-known expression for the area of a focal
sector of a conic in terms of the chord and the bounding radii.
Besides these he communicated numerous papers to the Berlin
Academy. Of these the most important are his memoir in 1768
on transcendental magnitudes, in which he proved that \index{P1@$\pi$,!z2@incommensurability of}$\pi$ is
incommensurable (the proof is given in \index{Legendre!ref.\ to}Legendre's \emph{Géométrie},
and is there extended to $\pi^2$): his paper on \index{Trigonometry.!z9@Development of, by John Bernoulli!x3@and Lambert}trigonometry, read
in 1768, in which he developed \index{Moivre, de!ref.\ to}\index{Demoivre!ref.\ to}Demoivre's theorems on the
trigonometry of complex variables, and introduced the \index{Hyperbolic trigonometry}hyperbolic
sine and cosine\footnote
  {These functions are said to have been previously suggested by
  \index{Mayer, F. C.}F.~C.~Mayer, see \emph{Die Lehre von den Hyperbelfunktionen} by \index{Gunther@Günther, S.}S.~Günther, Halle,
  1881, and \emph{Beiträge zur Geschichte der neueren Mathematik}, Ansbach, 1881.}
denoted by the symbols $\sinh x, \cosh x$:
his essay entitled analytical observations, published in 1771,
which is the earliest attempt to form functional equations by
expressing the given properties in the language of the \index{Operations, calculus of}\index{Calculus of operations}differential
calculus, and then integrating his researches on non-Euclidean
geometry: lastly, his paper on vis viva, published in 1783, in
which for the first time he expressed \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's second law of
motion in the notation of the differential calculus.\index{Lambert|)}

\index{Arbogast}\index{Bezout@Bézout}\index{Trembley}\nblabel{png419a}\nbparagraph{Bézout.\nbnamespace Trembley.\nbnamespace Arbogast.} Of the other mathematicians
above mentioned I here add a few words. \emph{Étienne
Bézout}, born at Nemours on March~31, 1730, and died on
September~27, 1783, besides numerous minor works, wrote a
\emph{Théorie générale des équations algébriques}, published at Paris in
1779, which in particular contained much new and valuable
matter on the \index{Elimination, theory of}theory of elimination and \index{Symmetrical functions of roots of an equation}symmetrical functions
of the \index{Roots of equations,!symmetrical functions of}roots of an equation: he used \index{Determinants}determinants in a paper
in the \emph{Histoire de L'académie royale}, 1764, but did not treat
of the general theory. \emph{Jean Trembley}, born at Geneva in 1749,
and died on September~18, 1811, contributed to the development
of \index{Differential equations}\index{Equations, differential}differential equations, finite differences, and the calculus
of \index{Probabilities, theory of}probabilities. \emph{Louis François Antoine Arbogast}, born in
Alsace on October~4, 1759, and died at Strassburg, where he
was professor, on April~8, 1803, wrote on series and the derivatives
known by his name: he was the first writer to separate
the symbols of operation from those of quantity.

I do not wish to crowd my pages with an account of those
who have not distinctly advanced the subject, but I have
mentioned the above writers because their names are still well
known. We may, however, say that the discoveries of \index{Euler!ref.\ to}Euler
and \index{Lagrange!ref.\ to}Lagrange in the subjects which they treated were so complete
and far-reaching that what their less gifted contemporaries
added is not of sufficient importance to require mention in a
book of this nature.

\index{Lagrange|(}\nblabel{png419b}\nbparagraph{Lagrange.\footnote
  {Summaries of the life and works of Lagrange are given in the \emph{English
  Cyclopaedia} and the \emph{Encyclopaedia Britannica} (ninth edition), of which I
  have made considerable use: the former contains a bibliography of his
  writings. Lagrange's works, edited by \index{Serret}MM.~J.~A.~Serret and \index{Darboux}G.~Darboux,
  were published in 14~volumes, Paris, 1867--1892. \index{Delambre}Delambre's account of his
  life is printed in the first volume.}}
\emph{Joseph Louis Lagrange}, the greatest mathematician
of the eighteenth century, was born at Turin on
January~25, 1736, and died at Paris on April~10, 1813. His
father, who had charge of the Sardinian military chest, was
of good social position and wealthy, but before his son grew up
he had lost most of his property in speculations, and young
Lagrange had to rely for his position on his own abilities. He
was educated at the college of Turin, but it was not until he
was seventeen that he shewed any taste for mathematics---his
interest in the subject being first excited by a memoir by \index{Halley!ref.\ to}Halley,\footnote
  {On the excellence of the modern algebra in certain optical problems,
  \emph{Philosophical Transactions}, 1693, vol.~xviii, p.~960.}
across which he came by accident. Alone and unaided he threw
himself into mathematical studies; at the end of a year's
incessant toil he was already an accomplished mathematician,
and was made a lecturer in the artillery school.

The first fruit of Lagrange's labours here was his letter,
written when he was still only nineteen, to \index{Euler!ref.\ to}Euler, in which he
solved the \index{Isoperimetrical problem}isoperimetrical problem which for more than half a
century had been a subject of discussion. To effect the solution
(in which he sought to determine the form of a function so
that a formula in which it entered should satisfy a certain condition)
he enunciated the principles of the \index{Calculus of operations!of variations}\index{Variations, calculus of}calculus of variations.
Euler recognized the generality of the method adopted, and its
superiority to that used by himself; and with rare courtesy he
withheld a paper he had previously written, which covered some
of the same ground, in order that the young Italian might
have time to complete his work, and claim the undisputed
invention of the new calculus. The name of this branch of
analysis was suggested by Euler. This memoir at once placed
Lagrange in the front rank of mathematicians then living.

In 1758 Lagrange established with the aid of his pupils a
society, which was subsequently incorporated as the Turin
Academy, and in the five volumes of its transactions, usually
known as the \emph{Miscellanea Taurinensia}, most of his early
writings are to be found. \nblabel{png421a}Many of these are elaborate memoirs.
The first volume contains a memoir on the theory of the
\index{Sound, velocity of}propagation of sound; in this he indicates a mistake made by
\index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton, obtains the general differential equation for the motion,
and integrates it for motion in a straight line. This volume
also contains the complete solution of the problem of a \index{Vibrating string}\index{String, vibrating, theory of}string
vibrating transversely; in this paper he points out a lack
of generality in the solutions previously given by \index{Taylor (Brook)!ref.\ to}Taylor,
\index{D'Alembert!ref.\ to}D'Alembert, and \index{Euler!ref.\ to}Euler, and arrives at the conclusion that
the form of the curve at any time $t$ is given by the equation
$y = a\sin mx \sin nt$. The article concludes with a masterly
discussion of echoes, beats, and compound sounds. Other
articles in this volume are on \index{Recurring series}recurring series, \index{Probabilities, theory of}probabilities, and
the \index{Calculus of operations!of variations}\index{Variations, calculus of}calculus of variations.

The second volume contains a long paper embodying the
results of several memoirs in the first volume on the theory and
notation of the calculus of variations; and he illustrates its
use by deducing the \index{Action,!least}\index{Least action}principle of least action, and by solutions
of various problems in dynamics.

The third volume includes the solution of several dynamical
problems by means of the calculus of variations; some papers
on the integral calculus; a solution of \index{Fermat!ref.\ to}Fermat's problem
mentioned above, to find a number $x$ which will make $(x^{2}n+1)$
a square where $n$ is a given integer which is not a square; and
the general differential equations of motion for three bodies
moving under their mutual attractions.

In 1761 Lagrange stood without a rival as the foremost
mathematician living; but the unceasing labour of the preceding
nine years had seriously affected his health, and the
doctors refused to be responsible for his reason or life unless
he would take rest and exercise. Although his health was
temporarily restored his nervous system never quite recovered
its tone, and henceforth he constantly suffered from attacks of
profound melancholy.

The next work he produced was in 1764 on the \index{Libration of moon}libration of
the moon, and an explanation as to why the same face was
always turned to the earth, a problem which he treated by the
aid of \index{Virtual work}\index{Work, virtual}virtual work. His solution is especially interesting as
containing the germ of the idea of generalized equations
of motion, equations which he first formally proved in
1780.

He now started to go on a visit to London, but on the way
fell ill at Paris. There he was received with marked honour,
and it was with regret he left the brilliant society of that city
to return to his provincial life at Turin. His further stay in
Piedmont was, however, short. In 1766 \index{Euler!ref.\ to}Euler left Berlin, and
\index{Frederick the Great of Prussia, ref.\ to}Frederick the Great immediately wrote expressing the wish of
``the greatest king in Europe'' to have ``the greatest mathematician
in Europe'' resident at his court. Lagrange accepted
the offer and spent the next twenty years in Prussia, where he
produced not only the long series of memoirs published in the
Berlin and Turin transactions, but his monumental work, the
\emph{Mécanique analytique}. His residence at Berlin commenced
with an unfortunate mistake. Finding most of his colleagues
married, and assured by their wives that it was the only way
to be happy, he married; his wife soon died, but the union was
not a happy one.

Lagrange was a favourite of the king, who used frequently
to discourse to him on the advantages of perfect regularity of
life. The lesson went home, and thenceforth Lagrange studied
his mind and body as though they were machines, and found
by experiment the exact amount of work which he was able to
do without breaking down. Every night he set himself a
definite task for the next day, and on completing any branch
of a subject he wrote a short analysis to see what points in the
demonstrations or in the subject-matter were capable of improvement.
He always thought out the subject of his papers
before he began to compose them, and usually wrote them
straight off without a single erasure or correction.

His mental activity during these twenty years was amazing.
Not only did he produce his splendid \emph{Mécanique analytique},
but he contributed between one and two hundred papers to
the Academies of Berlin, Turin, and Paris. Some of these are
really treatises, and all without exception are of a high order
of excellence. Except for a short time when he was ill he
produced on an average about one memoir a month. Of these
I note the following as among the most important.

First, his contributions to the fourth and fifth volumes,
1766--1773, of the \emph{Miscellanea Taurinensia}; of which the most
important was the one in 1771, in which he discussed how
numerous \index{Astronomy.!z20@Developed by (amongst others) Clairaut!q1@Lagrange}astronomical observations should be combined so as to
give the most probable result. And later, his contributions to
the first two volumes, 1784--1785, of the transactions of the
Turin Academy; to the first of which he contributed a paper
on the pressure exerted by fluids in motion, and to the second
an article on integration by infinite series, and the kind of
problems for which it is suitable.

Most of the memoirs sent to Paris were on astronomical
questions, and among these I ought particularly to mention
his memoir on the Jovian system in 1766, his essay on the
\index{Three bodies, problem of}problem of three bodies in 1772, his work on the secular
equation of the moon in 1773, and his treatise on \index{Disturbing forces}cometary
perturbations in 1778. These were all written on subjects
proposed by the French Academy, and in each case the prize
was awarded to him.

The greater number of his papers during this time were,
however, contributed to the Berlin Academy. Several of them
deal with questions on \emph{algebra}. In particular I may mention
the following. (i)~His discussion of the solution in integers of
indeterminate quadratics, 1769, and generally of \index{Equations, differential!indeterminate}indeterminate
equations, 1770. (ii)~His tract on the \index{Elimination, theory of}theory of elimination,
1770. (iii)~His memoirs on a general process for solving an
algebraical equation of any degree, 1770 and 1771; this method
fails for equations of an order above the fourth, because it then
involves the solution of an equation of higher dimensions than
the one proposed, but it gives all the solutions of his predecessors
as modifications of a single principle. (iv)~The complete solution
of a \index{Binomial equations}binomial equation of any degree; this is contained in the
memoirs last mentioned. (v)~Lastly, in 1773, his treatment of
\index{Determinants}determinants of the second and third order, and of invariants.

Several of his early papers also deal with questions connected
with the neglected but singularly fascinating subject
of the \index{Numbers, theory of.!Treatment of, by Pythagoras!q5@by Lagrange}\emph{theory of numbers}. Among these are the following.
(i)~His proof of the theorem that every integer which is not
a square can be expressed as the sum of two, three, or four
integral squares, 1770. (ii)~His proof of \index{Wilson's theorem}Wilson's theorem that
if $n$ be a prime, then $\inlinetabular{@{}|@{}c@{}}{\,$n-1$\\\cline{1-1}} + 1$ is always a multiple of $n$,
1771. (iii)~His memoirs of 1773, 1775, and 1777, which
give the demonstrations of several results enunciated by \index{Fermat!ref.\ to}Fermat,
and not previously proved. (iv)~And, lastly, his method for
determining the factors of numbers of the form $x^{2} + ay^{2}$.

There are also numerous articles on various points of \emph{analytical
geometry}. In two of them, written rather later, in 1792 and
1793, he reduced the equations of the \index{Quadrics}quadrics (or \index{Conicoids}conicoids) to
their canonical forms.

During the years from 1772 to 1785 he contributed a long
series of memoirs which created the science of \index{Differential equations}\index{Equations, differential}\emph{differential
equations}, at any rate as far as partial differential equations
are concerned. I do not think that any previous writer had
done anything beyond considering equations of some particular
form. A large part of these results were collected in the
second edition of \index{Euler!ref.\ to}Euler's integral calculus which was published
in 1794.

\nblabel{png424a}Lagrange's papers on \emph{mechanics} require no separate mention
here as the results arrived at are embodied in the \emph{Mécanique
analytique} which is described below.

\index{Astronomy.!z20@Developed by (amongst others) Clairaut!q1@Lagrange}Lastly, there are numerous memoirs on problems in \emph{astronomy}.
Of these the most important are the following. (i)~On the
\index{Attraction, theories of}attraction of ellipsoids, 1773: this is founded on \index{Maclaurin!ref.\ to}Maclaurin's
work. (ii)~On the secular equation of the moon, 1773; also
noticeable for the earliest introduction of the idea of the
potential. The potential of a body at any point is the sum
of the mass of every element of the body when divided by its
distance from the point. Lagrange shewed that if the potential
of a body at an external point were known, the attraction in
any direction could be at once found. The theory of the
potential was elaborated in a paper sent to Berlin in 1777.
(iii)~On the motion of the nodes of a \index{Planetary motions}planet's orbit, 1774.
(iv)~On the \index{Planetary motions!stability}stability of the planetary orbits, 1776. (v)~Two
memoirs in which the method of determining the orbit of a
comet from three observations is completely worked out, 1778
and 1783: this has not indeed proved practically available,
but his system of calculating the perturbations by means of
mechanical quadratures has formed the basis of most subsequent
researches on the subject. (vi)~His determination of the secular
and periodic variations of the elements of the planets, 1781--1784:
the upper limits assigned for these agree closely with those
obtained later by Leverrier, and Lagrange proceeded as far as
the knowledge then possessed of the masses of the planets
permitted. (vii)~Three memoirs on the method of \index{Interpolation, method of}interpolation,
1783, 1792, and 1793: the part of \index{Differences, finite}\index{Finite differences}finite differences dealing
therewith is now in the same stage as that in which Lagrange
left it.

Over and above these various papers he composed his great
treatise, the \emph{Mécanique analytique}. In this he lays down the
law of \index{Work, virtual}\index{Virtual work}virtual work, and from that one fundamental principle,
by the aid of the calculus of variations, deduces the whole
of mechanics, both of solids and fluids. The object of the
book is to shew that the subject is implicitly included in a
single principle, and to give general formulae from which any
particular result can be obtained. The method of \index{Coord@Co-ordinates!generalized}\index{Generalized co-ordinates}generalized
co-ordinates by which he obtained this result is perhaps the
most brilliant result of his analysis. Instead of following the
motion of each individual part of a material system, as
\index{D'Alembert!ref.\ to}D'Alembert and \index{Euler!ref.\ to}Euler had done, he shewed that, if we determine
its configuration by a sufficient number of variables
whose number is the same as that of the degrees of freedom
possessed by the system, then the kinetic and potential energies
of the system can be expressed in terms of these variables, and
the differential equations of motion thence deduced by simple
differentiation. For example, in dynamics of a rigid system
he replaces the consideration of the particular problem by
the general equation which is now usually written in the form
\[
\frac{d}{dt}\frac{\partial T}{\partial \theta} - \frac{\partial T}{\partial \theta} + \frac{\partial V}{\partial \theta}=0.
\]
Amongst other theorems here given are the proposition that the
kinetic energy imparted by given impulses to a material system
under given constraints is a maximum, and a more general statement
of the principle of \index{Least action}least action than had been given by
\index{Maupertuis, P. L. M.}Maupertuis or \index{Euler!ref.\ to}Euler. All the analysis is so elegant that
\index{Hamilton, Sir Wm.!ref.\ to}Sir William Rowan Hamilton said the work could be only
described as a scientific poem. Lagrange held that mechanics
was really a branch of pure mathematics analogous to a geometry
of four dimensions, namely, the time and the three co-ordinates
of the point in space;\footnote
  {On the development of this idea, see \index{Minkowski, H.}H.~Minkowski, \emph{Raum und Zeit}, Leipzig, 1909.}
 and it is said that he prided himself
that from the beginning to the end of the work there was not a
single diagram. At first no printer could be found who would
publish the book; but \index{Legendre!ref.\ to}Legendre at last persuaded a Paris firm
to undertake it, and it was issued in 1788.

In 1787 \index{Frederick the Great of Prussia, ref.\ to}Frederick died, and Lagrange, who had found
the climate of Berlin trying, gladly accepted the offer of
\index{Louis XVI. of France, ref.\ to}Louis XVI. to migrate to Paris. He received similar invitations
from Spain and Naples. In France he was received with
every mark of distinction, and special apartments in the Louvre
were prepared for his reception. At the beginning of his
residence here he was seized with an attack of melancholy,
and even the printed copy of his \emph{Mécanique} on which he had
worked for a quarter of a century lay for more than two years
unopened on his desk. Curiosity as to the results of the
French revolution first stirred him out of his lethargy, a
curiosity which soon turned to alarm as the revolution
developed. It was about the same time, 1792, that the unaccountable
sadness of his life and his timidity moved the
compassion of a young girl who insisted on marrying him, and
proved a devoted wife to whom he became warmly attached.
Although the decree of October 1793, which ordered all
foreigners to leave France, specially exempted him by name,
he was preparing to escape when he was offered the presidency
of the commission for the reform of weights and measures.
The choice of the units finally selected was largely due to him,
and it was mainly owing to his influence that the \index{Decimal measures}decimal
subdivision was accepted by the commission of 1799.

Though Lagrange had determined to escape from France
while there was yet time, he was never in any danger; and
the different revolutionary governments (and, at a later time,
\index{Napoleon I.}Napoleon) loaded him with honours and distinctions. A
striking testimony to the respect in which he was held was
shown in 1796 when the French commissary in Italy was
ordered to attend in full state on Lagrange's father, and tender
the congratulations of the republic on the achievements of his
son, who ``had done honour to all mankind by his genius, and
whom it was the special glory of Piedmont to have produced.''
It may be added that Napoleon, when he attained power,
warmly encouraged scientific studies in France, and was a
liberal benefactor of them.

In 1795 Lagrange was appointed to a mathematical chair at
the newly-established École normale, which enjoyed only a
brief existence of four months. His lectures here were quite
elementary, and contain nothing of any special importance, but
they were published because the professors had to ``pledge
themselves to the representatives of the people and to each
other neither to read nor to repeat from memory,'' and the
discourses were ordered to be taken down in shorthand in order
to enable the deputies to see how the professors acquitted
themselves.

On the establishment of the École polytechnique in 1797
Lagrange was made a professor; and his lectures there are
described by mathematicians who had the good fortune to be
able to attend them, as almost perfect both in form and matter.
Beginning with the merest elements, he led his hearers on until,
almost unknown to themselves, they were themselves extending
the bounds of the subject: above all he impressed on his pupils
the advantage of always using general methods expressed in a
symmetrical notation.

\nblabel{png428a}His lectures on the \index{Differentials}\index{Equations, differential!theory of}\index{Calculus, infinitesimal}differential calculus form the basis of his
\emph{Théorie des fonctions analytiques} which was published in 1797.
This work is the extension of an idea contained in a paper he
had sent to the Berlin Memoirs in 1772, and its object is to
substitute for the differential calculus a group of theorems based
on the development of algebraic functions in series. A somewhat
similar method had been previously used by \index{Landen}John Landen
in his \emph{Residual Analysis}, published in London in 1758.
Lagrange believed that he could thus get rid of those difficulties,
connected with the use of infinitely large and infinitely
small quantities, to which some philosophers objected in the
usual treatment of the differential calculus. The book is divided
into three parts: of these, the first treats of the general theory
of functions, and gives an algebraic proof of \index{Taylors@Taylor's theorem}Taylor's theorem,
the validity of which is, however, open to question; the second
deals with applications to geometry; and the third with applications
to mechanics. Another treatise on the same lines was
his \emph{Leçons sur le calcul des fonctions}, issued in 1804. These
works may be considered as the starting-point for the researches
of \index{Cauchy!ref.\ to}Cauchy, \index{Jacobi!ref.\ to}Jacobi, and \index{Weierstrass, K.!ref.\ to}Weierstrass, and are interesting from the
historical point of view.

Lagrange, however, did not himself object to the \index{Infinitesimals, use of}use of
infinitesimals in the differential calculus; and in the preface
to the second edition of the \emph{Mécanique}, which was issued in
1811, he justifies their employment, and concludes by saying
that ``when we have grasped the spirit of the infinitesimal
method, and have verified the exactness of its results either by
the \index{Prime and ultimate ratios}geometrical method of prime and ultimate ratios, or by the
analytical method of derived functions, we may employ infinitely
small quantities as a sure and valuable means of shortening and
simplifying our proofs.''

\nblabel{png428b}His \emph{Résolution des équations numériques}, published in 1798,
was also the fruit of his lectures at the Polytechnic. In this he
gives the method of approximating to the real roots of an
equation by means of \index{Continued fractions}\index{Fractions, continued}continued fractions, and enunciates several
other theorems. In a note at the end he shows how \index{Fermat!ref.\ to}Fermat's
theorem that $a^{p-1} - 1 \equiv 0 (\bmod p)$, where $p$ is a prime and $a$ is
prime to $p$, may be applied to give the complete algebraical
solution of any \index{Binomial equations}binomial equation. \index{Roots of equations,!position of}\index{Equations, differential!position of roots}He also here explains how
the equation whose roots are the squares of the differences of
the roots of the original equation may be used so as to give
considerable information as to the position and nature of those
roots.

The theory of the planetary motions had formed the subject
of some of the most remarkable of Lagrange's Berlin papers.
In 1806 the subject was reopened by \index{Poisson!ref.\ to}Poisson, who, in a paper
read before the French Academy, showed that Lagrange's
formulae led to certain limits for the stability of the orbits.
Lagrange, who was present, now discussed the whole subject
afresh, and in a memoir communicated to the Academy in
1808 explained how, by the variation of arbitrary constants, the
periodical and secular inequalities of any system of mutually
interacting bodies could be determined.

In 1810 Lagrange commenced a thorough revision of the
\emph{Mécanique analytique,} but he was able to complete only about
two-thirds of it before his death.

In appearance he was of medium height, and slightly formed,
with pale blue eyes and a colourless complexion. In character
he was nervous and timid, he detested controversy, and to avoid
it willingly allowed others to take the credit for what he had
himself done.

\nblabel{png429a}Lagrange's interests were essentially those of a student of
pure mathematics: he sought and obtained far-reaching abstract
results, and was content to leave the applications to others.
Indeed, no inconsiderable part of the discoveries of his great
contemporary, \index{Laplace!ref.\ to}Laplace, consists of the application of the
Lagrangian formulae to the facts of nature; for example,
Laplace's conclusions on the \index{Sound, velocity of}velocity of sound and the \index{Moon, secular acceleration of}secular
acceleration of the moon are implicitly involved in Lagrange's
results. The only difficulty in understanding Lagrange is that
of the subject-matter and the extreme generality of his processes;
but his analysis is ``as lucid and luminous as it is
symmetrical and ingenious.''

A recent writer speaking of Lagrange says truly that he
took a prominent part in the advancement of almost every
branch of pure mathematics. Like \index{Diophantus!ref.\ to}Diophantus and \index{Fermat!ref.\ to}Fermat, he
possessed a special genius for the theory of numbers, and in this
subject he gave solutions of many of the problems which had
been proposed by Fermat, and added some theorems of his own.
He developed the calculus of variations. To him, too, the theory
of differential equations is indebted for its position as a science
rather than a collection of ingenious artifices for the solution of
particular problems. To the calculus of \index{Differences, finite}\index{Finite differences}finite differences he
contributed the formula of \index{Interpolation, method of}interpolation which bears his name.
But above all he impressed on mechanics (which it will be
remembered he considered a branch of pure mathematics) that
generality and completeness towards which his labours invariably tended.\index{Lagrange|)}

\index{Laplace|(}\nblabel{png430a}\nbparagraph{Laplace.\footnote
  {The following account of Laplace's life and writings is mainly
  founded on the articles in the \emph{English Cyclopaedia} and the \emph{Encyclopaedia
  Britannica.} Laplace's works were published in seven volumes by the
  French government in 1843--7; and a new edition with considerable
  additional matter was issued at Paris in six volumes, 1878--84.}}
 \emph{Pierre Simon Laplace} was born at Beaumont-en-Auge
in Normandy on March~23, 1749, and died at Paris on
March~5, 1827. He was the son of a small cottager or perhaps
a farm-labourer, and owed his education to the interest excited
in some wealthy neighbours by his abilities and engaging
presence. Very little is known of his early years, for when he
became distinguished he had the pettiness to hold himself aloof
both from his relatives and from those who had assisted him.
It would seem that from a pupil he became an usher in the
school at Beaumont; but, having procured a letter of introduction
to \index{D'Alembert!ref.\ to}D'Alembert, he went to Paris to push his fortune. A
paper on the principles of mechanics excited D'Alembert's
interest, and on his recommendation a place in the military
school was offered to Laplace.

\nblabel{png431a}Secure of a competency, Laplace now threw himself into
original research, and in the next seventeen years, 1771--1787,
he produced much of his original work in astronomy. This
commenced with a memoir, read before the French Academy
in 1773, in which he shewed that the planetary motions were
stable, and carried the proof as far as the cubes of the eccentricities
and inclinations. This was followed by several papers
on points in the integral calculus, \index{Differences, finite}\index{Finite differences}finite differences, differential
equations, and astronomy.

During the years 1784--1787 he produced some memoirs of
exceptional power, Prominent among these is one read in 1784,
and reprinted in the third volume of the \emph{Mécanique céleste,} in
which he completely determined the \index{Attraction, theories of}attraction of a spheroid on
a particle outside it. \nblabel{png431b}This is memorable for the introduction
into \index{Harmonic analysis}analysis of \index{Spherical harmonics}spherical harmonics or \index{Laplace's coefficients}Laplace's coefficients, as
also for the development of the use of the \index{Potential, the|(}potential---a name
first given by Green in 1828.

If the co-ordinates of two points be $(r, \mu, \omega)$ and $(r', \mu', \omega')$,
and if $r' \nbnlt\, r$, then the reciprocal of the distance between them
can be expanded in powers of $r/r'$, and the respective coefficients
are Laplace's coefficients. Their utility arises from the fact that
every function of the co-ordinates of a point on a sphere can be
expanded in a series of them. It should be stated that the
similar coefficients for space of two dimensions, together with
some of their properties, had been previously given by \index{Legendre!ref.\ to}Legendre
in a paper sent to the French Academy in 1783. Legendre had
good reason to complain of the way in which he was treated in
this matter.

This paper is also remarkable for the development of the
idea of the potential, which was appropriated from \index{Lagrange!ref.\ to}Lagrange,\footnote
  {See the \emph{Bulletin} of the New York Mathematical Society, 1892, vol.~i,
  pp.~66--74.}
who had used it in his memoirs of 1773, 1777, and 1780. Laplace
shewed that the potential always satisfies the differential equation
\[
\nabla^{2}V =
  \frac{\partial^{2}V}{\partial x^{2}} +
  \frac{\partial^{2}V}{\partial y^{2}} +
  \frac{\partial^{2}V}{\partial z^{2}} = 0,
\]
and on this result his subsequent work on attractions was based.
The quantity $\nabla^{2}V$ has been termed the concentration of $V$, and
its value at any point indicates the excess of the value of $V$
there over its mean value in the neighbourhood of the point.
Laplace's equation, or the more general form $\nabla^{2}V = -4\pi\rho$,
appears in all branches of mathematical physics. According to
some writers this follows at once from the fact that $\nabla^{2}V$ is a
scalar operator; or the equation may represent analytically
some general law of nature which has not been yet reduced to
words; or possibly it might be regarded by a \index{Kant, ref.\ to}Kantian as the
outward sign of one of the necessary forms through which all
phenomena are perceived.\index{Potential, the|)}

\index{Astronomy.!z20@Developed by (amongst others) Clairaut!q2@Laplace|(}\index{Planetary motions|(}\nblabel{png432a}This memoir was followed by another on planetary inequalities,
which was presented in three sections in 1784, 1785, and
1786. This deals mainly with the explanation of the ``great
inequality'' of Jupiter and Saturn. Laplace shewed by general
considerations that the mutual action of two planets could never
largely affect the eccentricities and inclinations of their orbits;
and that the peculiarities of the Jovian system were due to the
near approach to commensurability of the mean motions of
Jupiter and Saturn: further developments of these theorems
on planetary motion were given in his two memoirs of 1788
and 1789. It was on these data that \index{Delambre}Delambre computed his
astronomical tables.

{\nbstretchyspace The year 1787 was rendered memorable by Laplace's explanation
and analysis of the relation between the lunar acceleration
and the secular changes in the eccentricity of the earth's orbit:}
this investigation completed the proof of the \index{Planetary motions!stability}stability of the
whole solar system on the assumption that it consists of a
collection of rigid bodies moving in a vacuum. All the memoirs
above alluded to were presented to the French Academy,
and they are printed in the \emph{Mémoires présentés par divers
savans}.

Laplace now set himself the task to write a work which
should ``offer a complete solution of the great mechanical
problem presented by the solar system, and bring theory to
coincide so closely with observation that empirical equations
should no longer find a place in astronomical tables.'' The
result is embodied in the \emph{Exposition du système du monde} and
\nblabel{png433a}the \emph{Mécanique céleste}.

The former was published in 1796, and gives a general
explanation of the phenomena, but omits all details. It contains
a summary of the history of astronomy: this summary
procured for its author the honour of admission to the forty
of the French Academy; it is commonly esteemed one of the
masterpieces of French literature, though it is not altogether
reliable for the later periods of which it treats.

\index{Nebular hypothesis|(}\nblabel{png433b}The nebular hypothesis was here enunciated.\footnote
  {On hypotheses as to the origin of the solar system, see \index{Poincaré, H.}H.~Poincaré,
  \emph{Hypothèses cosmogoniques}, Paris, 1911.}
According to
this hypothesis the solar system has been evolved from a quantity
of incandescent gas rotating round an axis through its centre
of mass. As it cooled the gas contracted and successive
rings broke off from its outer edge. These rings in their turn
cooled, and finally condensed into the planets, while the sun
represents the central core which is still left. On this view we
should expect that the more distant planets would be older than
those nearer the sun. The subject is one of great difficulty, and
though it seems certain that the solar system has a common
origin, there are various features which appear almost inexplicable
on the nebular hypothesis as enunciated by Laplace.

\index{Meteoric hypothesis}\nblabel{png433c}Another theory which avoids many of the difficulties raised
by Laplace's hypothesis has recently found favour. According
to this, the origin of the solar system is to be found in the
gradual aggregation of meteorites which swarm through our
system, and perhaps through space. These meteorites which
are normally cold may, by repeated collisions, be heated, melted,
or even vaporized, and the resulting mass would, by the effect
of gravity, be condensed into planet-like bodies---the larger
aggregations so formed becoming the chief bodies of the solar
system. To account for these collisions and condensations it
is supposed that a vast number of meteorites were at some
distant epoch situated in a spiral nebula, and that condensations
and collisions took place at certain knots or intersections of
orbits. As the resulting planetary masses cooled, moons or rings
would be formed either by collisions of outlying parts or in the
manner suggested in Laplace's hypothesis. This theory seems
to be primarily due to \index{Lockyer, Sir Norman}Sir Norman Lockyer. It does not
conflict with any of the known facts of cosmical science, but
as yet our knowledge of the facts is so limited that it would be
madness to dogmatize on the subject. Recent investigations
have shown that our moon broke off from the earth while the
latter was in a plastic condition owing to \index{Tidal friction}tidal friction. Hence
its origin is neither nebular nor meteoric.

Probably the best modern opinion inclines to the view that
nebular condensation, meteoric condensation, tidal friction, and
possibly other causes as yet unsuggested, have all played their
part in the evolution of the system.

The idea of the nebular hypothesis had been outlined by
\index{Kant, ref.\ to}Kant\footnote
  {See \emph{Kant's Cosmogony}, edited by \index{Hastie on Kant}W.~Hastie, Glasgow, 1900.}
in 1755, and he had also suggested meteoric aggregations
and tidal friction as causes affecting the formation of the
solar system: it is probable that Laplace was not aware of
this.\index{Nebular hypothesis|)}

According to the rule published by \index{Titius of Wittemberg}Titius of Wittemberg
in 1766---but generally known as \index{Bode's law}Bode's law, from the fact
that attention was called to it by Johann Elert Bode in
1778---the distances of the planets from the sun are nearly in
the ratio of the numbers $0 + 4$, $3 + 4$, $6 + 4$, $12 + 4$, \&c., the
$(n + 2)$th term being $(2^{n} × 3) + 4$. It would be an interesting
fact if this could be deduced from the nebular, meteoric, or any
other hypotheses, but so far as I am aware only one writer has
made any serious attempt to do so, and his conclusion seems
to be that the law is not sufficiently exact to be more than a
convenient means of remembering the general result.

Laplace's analytical discussion of the solar system is given
in his \emph{Mécanique céleste} published in five volumes. An analysis
of the contents is given in the \emph{English Cyclopaedia}. The first
two volumes, published in 1799, contain methods for calculating
the motions of the planets, determining their figures, and resolving
tidal problems. The third and fourth volumes, published
in 1802 and 1805, contain applications of these methods, and
several astronomical tables. The fifth volume, published in
1825, is mainly historical, but it gives as appendices the results
of Laplace's latest researches. Laplace's own investigations
embodied in it are so numerous and valuable that it is regrettable
to have to add that many results are appropriated from
writers with scanty or no acknowledgment, and the conclusions---which
have been described as the organized result of a century
of patient toil---are frequently mentioned as if they were due to
Laplace.

The matter of the \emph{Mécanique céleste} is excellent, but it is
by no means easy reading. \index{Biot!ref.\ to}Biot, who assisted Laplace in
revising it for the press, says that Laplace himself was frequently
unable to recover the details in the chain of reasoning,
and, if satisfied that the conclusions were correct, he was
content to insert the constantly recurring formula, ``II est aisé
à voir.'' The \emph{Mécanique céleste} is not only the translation of
the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia} into the language of the differential calculus,
but it completes parts of which \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton had been unable to
fill in the details. \index{Tisserand}F.~F.~Tisserand's recent work may be taken
as the modern presentation of dynamical astronomy on classical
lines, but Laplace's treatise will always remain a standard
authority.\index{Planetary motions|)}\index{Astronomy.!z20@Developed by (amongst others) Clairaut!q2@Laplace|)}

Laplace went in state to beg \index{Napoleon I.}Napoleon to accept a copy of
his work, and the following account of the interview is well
authenticated, and so characteristic of all the parties concerned
that I quote it in full. Someone had told Napoleon that the
book contained no mention of the name of God; Napoleon,
who was fond of putting embarrassing questions, received it
with the remark, ``M.~Laplace, they tell me you have written
this large book on the system of the universe, and have never
even mentioned its Creator.'' Laplace, who, though the most
supple of politicians, was as stiff as a martyr on every point of
his philosophy, drew himself up and answered bluntly, ``Je
n'avais pas besoin de cette hypothèse-là.'' \index{Napoleon I.}Napoleon, greatly
amused, told this reply to \index{Lagrange!ref.\ to}Lagrange, who exclaimed, ``Ah!
c'est une belle hypothèse; ça explique beaucoup de choses.''

\nblabel{png436a}In 1812 Laplace issued his \index{Probabilities, theory of|(}\emph{Théorie analytique des probabilités}.\footnote
  {A summary of Laplace's reasoning is given in the article on Probability
  in the \emph{Encyclopaedia Metropolitana}.}
The theory is stated to be only common sense expressed
in mathematical language. The method of estimating
the ratio of the number of favourable cases to the whole
number of possible cases had been indicated by Laplace
in a paper written in 1779. It consists in treating the successive
values of any function as the coefficients in the expansion
of another function with reference to a different variable.
The latter is therefore called the generating function of the
former. Laplace then shews how, by means of interpolation,
these coefficients may be determined from the generating function.
Next he attacks the converse problem, and from the
coefficients he finds the generating function; this is effected by
the solution of an equation in \index{Differences, finite}\index{Finite differences}finite differences. The method
is cumbersome, and in consequence of the increased power of
analysis is now rarely used.

\nblabel{png436b}This treatise includes an exposition of the method of \index{Least squares}least
squares, a remarkable testimony to Laplace's command over the
processes of analysis. The method of least squares for the combination
of numerous observations had been given empirically
by \index{Gauss!ref.\ to}Gauss and \index{Legendre!ref.\ to}Legendre, but the fourth chapter of this work
contains a formal proof of it, on which the whole of the \index{Errors, theory of}theory
of errors has been since based. This was effected only by a
most intricate analysis specially invented for the purpose, but
the form in which it is presented is so meagre and unsatisfactory
that in spite of the uniform accuracy of the results it was
at one time questioned whether Laplace had actually gone through
the difficult work he so briefly and often incorrectly indicates.

In 1819 Laplace published a popular account of his work
on probability. This book bears the same relation to the
\emph{Théorie des probabilités} that the \emph{Système du monde} does to
the \emph{Mécanique céleste}.\index{Probabilities, theory of|)}

\nblabel{png437a}Amongst the minor discoveries of Laplace in pure mathematics
I may mention his discussion (simultaneously with \index{Vandermonde}Vandermonde)
of the general theory of \index{Determinants}determinants in 1772; his
proof that every equation of an even degree must have at least
one real quadratic factor; his reduction of the solution of linear
differential equations to definite integrals; and his solution of
the linear partial differential equation of the second order. He
was also the first to consider the difficult problems involved in
equations of \index{Differences, finite!mixed}mixed differences, and to prove that the solution of
an equation in \index{Differences, finite}\index{Finite differences}finite differences of the first degree and the
second order might be always obtained in the form of a
\index{Fractions, continued}\index{Continued fractions}continued fraction. Besides these original discoveries he
determined, in his theory of probabilities, the values of a
number of the more common definite integrals; and in the
same book gave the general proof of the theorem enunciated
by \index{Lagrange!ref.\ to}Lagrange for the development of any implicit function in
a series by means of differential coefficients.

\index{Hydrodynamics. Developed by!z5@and Laplace}In theoretical physics the theory of \index{Capillarity}capillary attraction
is due to Laplace, who accepted the idea propounded by
\index{Hauksbee on capillarity}Hauksbee in the \emph{Philosophical Transactions} for 1709, that
the phenomenon was due to a force of attraction which was
insensible at sensible distances. The part which deals with
the action of a solid on a liquid and the mutual action of two
liquids was not worked out thoroughly, but ultimately was
completed by \index{Gauss!ref.\ to}Gauss: \index{Neumann, C.}Neumann later filled in a few details.
In 1862 \index{Kelvin, Lord}Lord Kelvin (Sir William Thomson) shewed that, if
we assume the molecular constitution of matter, the laws of
capillary attraction can be deduced from the Newtonian law of
gravitation.

Laplace in 1816 was the first to point out explicitly why
\index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's theory of vibratory motion gave an incorrect value for
the \index{Sound, velocity of}velocity of sound. The actual velocity is greater than that
calculated by Newton in consequence of the heat developed by
the sudden compression of the air which increases the elasticity
and therefore the velocity of the sound transmitted. Laplace's
investigations in practical physics were confined to those carried
on by him jointly with \index{Lavoisier}Lavoisier in the years 1782 to 1784 on
the specific heat of various bodies.

\nblabel{png438a}Laplace seems to have regarded analysis merely as a means
of attacking physical problems, though the ability with which
he invented the necessary analysis is almost phenomenal. As
long as his results were true he took but little trouble to explain
the steps by which he arrived at them; he never studied
elegance or symmetry in his processes, and it was sufficient
for him if he could by any means solve the particular question
he was discussing.

It would have been well for Laplace's reputation if he had
been content with his scientific work, but above all things he
coveted social fame. The skill and rapidity with which he
managed to change his politics as occasion required would be
amusing had they not been so servile. As \index{Napoleon I.}Napoleon's power
increased Laplace abandoned his republican principles (which,
since they had faithfully reflected the opinions of the party in
power, had themselves gone through numerous changes) and
begged the first consul to give him the post of minister of the
interior. Napoleon, who desired the support of men of science,
agreed to the proposal; but a little less than six weeks saw
the close of Laplace's political career. Napoleon's memorandum
on his dismissal is as follows: ``Géomètre de premier rang,
Laplace ne tarda pas à se montrer administrateur plus que
médiocre; dès son premier travail nous reconnûmes que nous
nous étions trompé. Laplace ne saisissait aucune question sous
son véritable point de vue: il cherchait des subtilités partout,
n'avait que des idées problématiques, et portait enfin l'esprit des
`infiniment petits' jusque dans l'administration.''

Although Laplace was removed from office it was desirable
to retain his allegiance. He was accordingly raised to the
senate, and to the third volume of the \emph{Mécanique céleste} he
prefixed a note that of all the truths therein contained the most
precious to the author was the declaration he thus made of his
devotion towards the peacemaker of Europe. In copies sold
after the restoration this was struck out. In 1814 it was
evident that the empire was falling; Laplace hastened to
tender his services to the Bourbons, and on the restoration
was rewarded with the title of marquis: the contempt that his
more honest colleagues felt for his conduct in the matter may
be read in the pages of \index{Courier on Laplace}Paul Louis Courier. His knowledge
was useful on the numerous scientific commissions on which
he served, and probably accounts for the manner in which his
political insincerity was overlooked; but the pettiness of his
character must not make us forget how great were his services
to science.

\nblabel{png439a}That Laplace was vain and selfish is not denied by his
warmest admirers; his conduct to the benefactors of his youth
and his political friends was ungrateful and contemptible; while
his appropriation of the results of those who were comparatively
unknown seems to be well established and is absolutely indefensible---of
those whom he thus treated three subsequently
rose to distinction (\index{Legendre!ref.\ to}Legendre and \index{Fourier!ref.\ to}Fourier in France and \index{Young, Thos.!ref.\ to}Young
in England) and never forgot the injustice of which they had
been the victims. On the other side it may be said that on
some questions he shewed independence of character, and he
never concealed his views on religion, philosophy, or science,
however distasteful they might be to the authorities in power:
it should be also added that towards the close of his life, and
especially to the work of his pupils, Laplace was both generous and
appreciative, and in one case suppressed a paper of his own in
order that a pupil might have the sole credit of the investigation.\index{Laplace|)}

\index{Legendre|(}\nblabel{png439b}\nbparagraph{Legendre.} \emph{Adrian Marie Legendre} was born at Toulouse
on September~18, 1752, and died at Paris on January~10, 1833.
The leading events of his life are very simple and may be
summed up briefly. He was educated at the Mazarin College
in Paris, appointed professor at the military school in Paris
in 1777, was a member of the Anglo-French commission of
1787 to connect Greenwich and Paris \index{Geodesics}geodetically; served on
several of the public commissions from 1792 to 1810; was made
a professor at the Normal school in 1795; and subsequently
held a few minor government appointments. The influence
of Laplace was steadily exerted against his obtaining office or
public recognition, and Legendre, who was a timid student,
accepted the obscurity to which the hostility of his colleague
condemned him.

Legendre's analysis is of a high order of excellence, and is
second only to that produced by \index{Lagrange!ref.\ to}Lagrange and Laplace, though
it is not so original. His chief works are his \emph{Géométrie}, his
\emph{Théorie des nombres}, his \emph{Exercices de calcul intégral,} and his
\emph{Fonctions elliptiques}. These include the results of his various
papers on these subjects. Besides these he wrote a treatise
which gave the rule for the method of \index{Least squares}least squares, and two
groups of memoirs, one on the \index{Attraction, theories of}theory of attractions, and the
other on geodetical operations.

\nblabel{png440a}The memoirs on attractions are analyzed and discussed in
\index{Todhunter, ref.\ to}Todhunter's \emph{History of the Theories of Attraction.} The earliest
of these memoirs, presented in 1783, was on the attraction
of spheroids. \index{Circular harmonics}\index{Spherical harmonics}\index{Harmonic analysis}This contains the introduction of \index{Legendre's coefficients}Legendre's
coefficients, which are sometimes called \index{Harmonic analysis}circular \index{Zonal harmonics}(or zonal)
harmonics, and which are particular cases of \index{Laplace's coefficients}Laplace's coefficients;
it also includes the solution of a problem in which
the \index{Potential, the}potential is used. The second memoir was communicated
in 1784, and is on the form of equilibrium of a mass of
rotating liquid which is approximately spherical. The third,
written in 1786, is on the attraction of confocal ellipsoids.
The fourth is on the figure which a fluid planet would assume,
and its law of density.

His papers on geodesy are three in number, and were
presented to the Academy in 1787 and 1788. The most
important result is that by which a spherical triangle may
be treated as plane, provided certain corrections are applied
to the angles. In connection with this subject he paid considerable
attention to geodesics.

{\nbstretchyspace The method of \index{Least squares}least squares was enunciated in his \emph{Nouvelles
méthodes} published in 1806, to which supplements were added
in 1810 and 1820.} \index{Gauss!ref.\ to}Gauss independently had arrived at the
same result, had used it in 1795, and published it and the
\index{Facility, law of}law of facility in 1809. Laplace was the earliest writer to
give a proof of it; this was in 1812.

Of the other books produced by Legendre, the one most
widely known is his \emph{Éléments de géométrie} which was published
in 1794, and was at one time widely adopted on the continent
as a substitute for Euclid. The later editions contain the
elements of trigonometry, and proofs of the irrationality of
\index{P1@$\pi$,!z2@incommensurability of}$\pi$ and $\pi^{2}$. An appendix on the difficult question of the \index{Parallel lines}theory
of parallel lines was issued in 1803, and is bound up with most
of the subsequent editions.

\index{Numbers, theory of.!Treatment of, by Pythagoras!q6@by Legendre|(}\nblabel{png441a}His \emph{Théorie des nombres} was published in 1798, and appendices
were added in 1816 and 1825; the third edition,
issued in two volumes in 1830, includes the results of his
various later papers, and still remains a standard work on the
subject. It may be said that he here carried the subject as
far as was possible by the application of ordinary algebra; but
he did not realize that it might be regarded as a higher
arithmetic, and so form a distinct subject in mathematics.

{\nbstretchyspace \index{Quadratic reciprocity}\index{Primes!distribution of|(}\nblabel{png441b}The law of quadratic reciprocity, which connects any two
odd primes, was first proved in this book, but the result had
been enunciated in a memoir of 1785.} Gauss called the proposition
``the gem of arithmetic,'' and no less than six separate
proofs are to be found in his works. The theorem is as follows.
If $p$ be a prime and $n$ be prime to $p$, then we know that the
remainder when $n^{(p-1)/2}$ is divided by $p$ is either $+ 1$ or $- 1$.
Legendre denoted this remainder by $(n/p)$. \index{Residues, theory of}When the remainder
is $+1$ it is possible to find a square number which
when divided by $p$ leaves a remainder $n$, that is, $n$ is a \index{Quadratic residues}quadratic
residue of $p$; when the remainder is $- 1$ there exists no such
square number, and $n$ is a non-residue of $p$. The law of
quadratic reciprocity is expressed by the theorem that, if $a$ and
$b$ be any odd primes, then
\[
(a/b)(b/a) = (-1)^{(a-1)(b-1)/4};
\]
thus, if $b$ be a residue of $a$, then $a$ is also a residue of $b$, unless
both of the primes $a$ and $b$ are of the form $4m + 3$. In other
words, if $a$ and $b$ be odd primes, we know that
\[
a^{(b-1)/2}\equiv\pm 1 (\bmod\ b), \text{ and } b^{(a-1)/2}\equiv ± 1 (\bmod\ a);
\]
and, by Legendre's law, the two ambiguities will be either both
positive or both negative, unless $a$ and $b$ are both of the form
$4m + 3$. Thus, if one odd prime be a non-residue of another,
then the latter will be a non-residue of the former. \index{Gauss!ref.\ to}Gauss
and \index{Kummer!ref.\ to}Kummer have subsequently proved similar laws of \index{Cubic reciprocity}cubic
and \index{Biquadratic reciprocity}biquadratic reciprocity; and an important branch of the
theory of numbers has been based on these researches.\index{Primes!distribution of|)}

This work also contains the useful theorem by which, when
it is possible, an indeterminate equation of the second degree
can be reduced to the form $ax^{2} + by^{2}+ cz^{2} = 0$. Legendre here
discussed the forms of numbers which can be expressed as the
sum of three squares; and he proved [art.~404] that the number
of primes less than $n$ is approximately $n/(\log_{e} n - 1.08366)$.\index{Numbers, theory of.!Treatment of, by Pythagoras!q6@by Legendre|)}

\nblabel{png442a}The \emph{Exercices de calcul intégral} was published in three
volumes, 1811, 1817, 1826. Of these the third and most of
the first are devoted to \index{Elliptic functions}elliptic functions; the bulk of this
being ultimately included in the \emph{Fonctions elliptiques}. The
contents of the remainder of the treatise are of a miscellaneous
character; they include integration by series, definite integrals,
and in particular an elaborate discussion of the \index{Beta function}Beta and the
\index{Gamma function}Gamma functions.

The \emph{Traité des fonctions elliptiques} was issued in two volumes
in 1825 and 1826, and is the most important of Legendre's
works. A third volume was added a few weeks before his
death, and contains three memoirs on the researches of \index{Abel!ref.\ to}Abel and
\index{Jacobi!ref.\ to}Jacobi. Legendre's investigations had commenced with a paper
written in 1786 on elliptic arcs, but here and in his other papers
he treated the subject merely as a problem in the integral
calculus, and did not see that it might be considered as a
higher trigonometry, and so constitute a distinct branch of
analysis. Tables of the elliptic integrals were constructed by
him. The modern treatment of the subject is founded on that
of \index{Abel!ref.\ to}Abel and \index{Jacobi!ref.\ to}Jacobi. The superiority of their methods was at
once recognized by Legendre, and almost the last act of his
life was to recommend those discoveries which he knew would
consign his own labours to comparative oblivion.

This may serve to remind us of a fact which I wish to
specially emphasize, namely, that \index{Gauss!ref.\ to}Gauss, \index{Abel!ref.\ to}Abel, Jacobi, and some
others of the mathematicians alluded to in the \hyperref[chapter:XIX]{next chapter}, were
contemporaries of the members of the French school.\index{Legendre|)}

\index{Pfaff}\nblabel{png443a}\nbparagraph{Pfaff.} I may here mention another writer who also made
a special study of the integral calculus. This was \emph{Johann
Friederich Pfaff}, born at Stuttgart on Dec.~22, 1765, and died
at Halle on April~21, 1825, who was described by \index{Laplace!ref.\ to}Laplace as
the most eminent mathematician in Germany at the beginning
of this century, a description which, had it not been for Gauss's
existence, would have been true enough.

Pfaff was the precursor of the German school, which under
Gauss and his followers largely determined the lines on which
mathematics developed during the nineteenth century. He was
an intimate friend of Gauss, and in fact the two mathematicians
lived together at Helmstadt during the year 1798, after Gauss
had finished his university course. Pfaff's chief work was his
(unfinished) \emph{Disquisitiones Analyticae} on the integral calculus,
published in 1797; and his most important memoirs were either
on the calculus or on \index{Differential equations}\index{Equations, differential}differential equations: on the latter subject
his paper read before the Berlin Academy in 1814 is noticeable.


\section*{The creation of modern geometry.\nblabel{png443b}}

While \index{Euler!ref.\ to}Euler, \index{Lagrange!ref.\ to}Lagrange, \index{Laplace!ref.\ to}Laplace, and \index{Legendre!ref.\ to}Legendre were perfecting
analysis, the members of another group of French
mathematicians were extending the range of geometry by
methods similar to those previously used by \index{Desargues!ref.\ to}Desargues and
\index{Pascal!ref.\ to}Pascal. \index{Geometry.!z8@Modern synthetic geometry, originated with Desargues!x5@Monge, Carnot, and Poncelet|(}\index{Conic Sections (Geometrical).!Treatment of, by modern synthetic geometry|(}The revival of the study of synthetic geometry is
largely due to \index{Poncelet!ref.\ to}Poncelet, but the subject is also associated with
the names of \index{Monge!ref.\ to}Monge and \index{Carnot, Lazare!ref.\ to}L.~Carnot; its great development in
more recent times is mainly due to \index{Steiner!ref.\ to}Steiner, \index{Von Staudt!ref.\ to}\index{Staudt, von!ref.\ to}von Staudt, and
\index{Cremona, L.}Cremona.

\index{Monge|(}\nblabel{png444a}\nbparagraph{Monge.\footnote
  {On the authorities for Monge's life and works, see the note by \index{Brocard, H., on Monge}H.~Brocard
  in \emph{L'Intermédiaire des mathématiciens}, 1906, vol.~xiii, pp.~118,~119.}}
\emph{Gaspard Monge} was born at Beaune on May~10,
1746, and died at Paris on July~28, 1818. He was the son of
a small pedlar, and was educated in the schools of the Oratorians,
in one of which he subsequently became an usher. A plan of
Beaune which he had made fell into the hands of an officer who
recommended the military authorities to admit him to their
training-school at Mézières. His birth, however, precluded his
receiving a commission in the army, but his attendance at an
annexe of the school where surveying and drawing were taught
was tolerated, though he was told that he was not sufficiently
well born to be allowed to attempt problems which required
calculation. At last his opportunity came. A plan of a fortress
having to be drawn from the data supplied by certain observations,
he did it by a geometrical construction. At first the
officer in charge refused to receive it, because etiquette required
that not less than a certain time should be used in making such
drawings, but the superiority of the method over that then
taught was so obvious that it was accepted; and in 1768
Monge was made professor, on the understanding that the
results of his descriptive geometry were to be a military secret
confined to officers above a certain rank.

In 1780 he was appointed to a chair of mathematics in Paris,
and this with some provincial appointments which he held gave
him a comfortable income. The earliest paper of any special
importance which he communicated to the French Academy was
one in 1781, in which he discussed the lines of \index{Curvature, lines of}\index{Lines of curvature}curvature drawn
on a surface. These had been first considered by \index{Euler!ref.\ to}Euler in 1760,
and defined as those normal sections whose curvature was a
maximum or a minimum. Monge treated them as the locus of
those points on the surface at which successive normals intersect,
and thus obtained the general \index{Differential equations}\index{Equations, differential}differential equation. He applied
his results to the central quadrics in 1795. In 1786 he published
his well-known work on statics.

Monge eagerly embraced the doctrines of the revolution.
In 1792 he became minister of the marine, and assisted the
committee of public safety in utilizing science for the defence
of the republic. When the Terrorists obtained power he was
denounced, and escaped the guillotine only by a hasty flight.
On his return in 1794 he was made a professor at the short-lived
Normal school, where he gave lectures on descriptive
geometry; the notes of these were published under the regulation
above alluded to. In 1796 he went to Italy on the roving
commission which was sent with orders to compel the various
Italian towns to offer pictures, sculpture, or other works of art
that they might possess, as a present or in lieu of contributions
to the French republic for removal to Paris. In 1798 he
accepted a mission to Rome, and after executing it joined
\index{Napoleon I.}Napoleon in Egypt. Thence after the naval and military
victories of England he escaped to France.

Monge then settled down at Paris, and was made professor
at the Polytechnic school, where he gave lectures on descriptive
geometry; these were published in 1800 in the form of a text-book
entitled \emph{Géo\-métrie descriptive.} This work contains propositions
on the form and relative position of geometrical figures
deduced by the use of transversals. The theory of perspective
is considered; this includes the art of representing in two
dimensions geometrical objects which are of three dimensions,
a problem which Monge usually solved by the aid of two
diagrams, one being the plan and the other the elevation.
Monge also discussed the question as to whether, if in solving
a problem certain subsidiary quantities introduced to facilitate
the solution become imaginary, the validity of the solution is
thereby impaired, and he shewed that the result would not be
affected. On the restoration he was deprived of his offices and
honours, a degradation which preyed on his mind and which he
did not long survive.

Most of his miscellaneous papers are embodied in his works,
\emph{Application de l'algèbre à la géométrie}, published in 1805, and
\emph{Application de l'analyse à la géométrie}, the fourth edition of
which, published in 1819, was revised by him just before his
death. It contains among other results his solution of a partial
\index{Differential equations}\index{Equations, differential}differential equation of the second order.\index{Monge|)}

\index{Carnot, Lazare}\nblabel{png446a}\nbparagraph{Carnot.\footnote
  {See the \emph{éloge} by \index{Arago!ref.\ to}Arago, which, like most obituary notices, is a panegyric
  rather than an impartial biography.}}
\emph{Lazare Nicholas Marguerite Carnot}, born at
Nolay on May~13, 1753, and died at Magdeburg on Aug.~22,
1823, was educated at Burgundy, and obtained a commission
in the engineer corps of Condé. Although in the army, he
continued his mathematical studies in which he felt great
interest. His first work, published in 1784, was on machines;
it contains a statement which foreshadows the \index{Conservation of energy}\index{Energy, conservation of}principle of
energy as applied to a falling weight, and the earliest proof of
the fact that kinetic energy is lost in the collision of imperfectly
elastic bodies. On the outbreak of the revolution in 1789 he
threw himself into politics. In 1793 he was elected on the
committee of public safety, and the victories of the French army
were largely due to his powers of organization and enforcing
discipline. He continued to occupy a prominent place in every
successive form of government till 1796 when, having opposed
\index{Napoleon I.}Napoleon's \emph{coup d'état}, he had to fly from France. He took
refuge in Geneva, and there in 1797 issued his \emph{Réflexions sur la
métaphysique du calcul infinitésimal}: in this he amplifies views
previously expounded by \index{Berkeley, G.}Berkeley and \index{Lagrange!ref.\ to}Lagrange. In 1802 he
assisted Napoleon, but his sincere republican convictions were
inconsistent with the retention of office. In 1803 he produced
his \emph{Géométrie de position}. This work deals with projective rather
than descriptive geometry, it also contains an elaborate discussion
of the geometrical meaning of negative roots of an algebraical
equation. In 1814 he offered his services to fight for France,
though not for the empire; and on the restoration he was exiled.

\index{Poncelet}\nbparagraph{Poncelet.\footnote
  {See \emph{La Vie et les ouvrages de Poncelet}, by \index{Didion and Dupin on Poncelet}I.~Didion and \index{Dupin, ref.\ to}C.~Dupin, Paris,
  1869.}}
 \emph{Jean Victor Poncelet}, born at Metz on July~1,
1788, and died at Paris on Dec.~22, 1867, held a commission
in the French engineers. Having been made a prisoner in the
French retreat from Moscow in 1812 he occupied his enforced
leisure by writing the \emph{Traité des propriétés projectives des
figures}, published in 1822, which was long one of the best
known text-books on modern geometry. By means of projection,
reciprocation, and homologous figures, he established
all the chief properties of conics and quadrics. He also treated
the theory of polygons. His treatise on practical mechanics in
1826, his memoir on water-mills in 1826, and his report on
the English machinery and tools exhibited at the International
Exhibition held in London in 1851 deserve mention. He
contributed numerous articles to Crelle's journal; the most
valuable of these deal with the explanation, by the aid of the
\index{Continuity, principle of}doctrine of continuity, of imaginary solutions in geometrical
problems.\index{Conic Sections (Geometrical).!Treatment of, by modern synthetic geometry|)}\index{Geometry.!z8@Modern synthetic geometry, originated with Desargues!x5@Monge, Carnot, and Poncelet|)}


\section*{The development of mathematical physics.\nblabel{png447a}}

It will be noticed that \index{Lagrange!ref.\ to}Lagrange, \index{Laplace!ref.\ to}Laplace, and \index{Legendre!ref.\ to}Legendre
mostly occupied themselves with analysis, geometry, and astronomy.
I am inclined to regard \index{Cauchy!ref.\ to}Cauchy and the French mathematicians
of the present day as belonging to a different school
of thought to that considered in this chapter, and I place them
amongst modern mathematicians, but I think that \index{Fourier!ref.\ to}Fourier,
\index{Poisson!ref.\ to}Poisson, and the majority of their contemporaries, are the lineal
successors of Lagrange and Laplace. If this view be correct, we
may say that the successors of Lagrange and Laplace devoted
much of their attention to the application of mathematical
analysis to physics. Before considering these mathematicians
I may mention the distinguished English experimental physicists
who were their contemporaries, and whose merits have only
recently received an adequate recognition. Chief among these
are Cavendish and \index{Young, Thos.!ref.\ to}Young.

\index{Cavendish, H.}\nblabel{png447b}\nbparagraph{Cavendish.\footnote
  {An account of his life by \index{Wilson on Cavendish}G.~Wilson will be found in the first volume
  of the publications of the Cavendish Society, London, 1851. His \emph{Electrical}
  \emph{Researches} were edited by \index{Maxwell, J. C.}J.~C.~Maxwell, and published at Cambridge in
  1879.}}
The Honourable \emph{Henry Cavendish} was born at
Nice on October~10, 1731, and died in London on February~4,
1810. His tastes for scientific research and mathematics were
formed at Cambridge, where he resided from 1749 to 1753. He
created experimental electricity, and was one of the earliest
writers to treat chemistry as an exact science. I mention him
here on account of his experiment in 1798 to determine the
\index{Earth,!density of}density of the earth, by estimating its attraction as compared
with that of two given lead balls: the result is that the mean
density of the earth is about five and a half times that of water.
This experiment was carried out in accordance with a suggestion
which had been first made by \index{Mitchell, J.}John Mitchell (1724--1793), a
fellow of Queens' College, Cambridge, who had died before he
was able to carry it into effect.

\index{Rumford, Count}\nblabel{png448a}\nbparagraph{Rumford.\footnote
  {An edition of Rumford's works, edited by \index{Ellis, G., on Rumford}George Ellis, accompanied by
  a biography, was published by the American Academy of Sciences at Boston
  in 1872.}}
\index{Thomson, Sir Benjamin}\emph{Sir Benjamin Thomson, Count Rumford,} born
at Concord on March~26, 1753, and died at Auteuil on August~21,
1815, was of English descent, and fought on the side of the
loyalists in the American War of Secession: on the conclusion
of peace he settled in England, but subsequently entered the
service of Bavaria, where his powers of organization proved of
great value in civil as well as military affairs. At a later period
he again resided in England, and when there founded the \index{Royal Institution of London}Royal
Institution. The majority of his papers were communicated to
the Royal Society of London; of these the most important is
his memoir in which he showed that heat and work are mutually
convertible.

\index{Young, Thos.}\nbparagraph{Young.\footnote
  {Young's collected works and a memoir on his life were published by \index{Peacock!ref.\ to}G.~Peacock,
  four volumes, London, 1855.}}
Among the most eminent physicists of his time
was \emph{Thomas Young,} who was born at Milverton on June~13,
1773, and died in London on May~10, 1829. He seems as a
boy to have been somewhat of a prodigy, being well read in
modern languages and literature, as well as in science; he always
kept up his literary tastes, and it was he who in 1819 first
suggested the key to decipher the \index{Hieroglyphics, Egyptian}Egyptian hieroglyphics, which
\index{Champollion, ref.\ to}J.~F.~Champollion used so successfully. Young was destined
to be a doctor, and after attending lectures at Edinburgh and
Göttingen entered at Emmanuel College, Cambridge, from which
he took his degree in 1799; and to his stay at the University
he attributed much of his future distinction. His medical
career was not particularly successful, and his favourite maxim
that a medical diagnosis is only a balance of probabilities was
not appreciated by his patients, who looked for certainty in
return for their fee. Fortunately his private means were ample.
\index{Optics (geometrical).!z11@(physical)}Several papers contributed to various learned societies from
1798 onwards prove him to have been a mathematician of
considerable power; but the researches which have immortalised
his name are those by which he \index{Diffraction}laid down the \index{Light,!physical theories of}laws of \index{Interference, principle of}interference
of \index{Wave theory (optics)}waves and of light, and was thus able to suggest the
means by which the chief difficulties then felt in the way of the
acceptance of the \index{Undulatory theory (optics)}undulatory theory of light could be overcome.

\index{Dalton, J.}\nblabel{png449a}\nbparagraph{Dalton.\footnote
  {See the \emph{Memoir of Dalton}, by \index{Smith, R. A., on Dalton}R.~A.~Smith, London, 1856; and \index{Henry, W. C., on Dalton}W.~C.~Henry's
  memoir in the \emph{Cavendish Society Transactions}, London, 1854.}}
Another distinguished writer of the same period
was \emph{John Dalton}, who was born in Cumberland on September~5,
1766, and died at Manchester on July~27, 1844. Dalton
investigated the tension of vapours, and the law of the expansion
of a gas under changes of temperature. He also founded the
\index{Atomic theory in chemistry}atomic theory in chemistry.

\index{Experiments, necessity of}It will be gathered from these notes that the English school
of physicists at the beginning of this century were mostly
concerned with the experimental side of the subject. But in
fact no satisfactory theory could be formed without some similar
careful determination of the facts. The most eminent French
physicists of the same time were \index{Fourier!ref.\ to}Fourier, \index{Poisson!ref.\ to}Poisson, Ampère,
and \index{Fresnel!ref.\ to}Fresnel. Their method of treating the subject is more
mathematical than that of their English contemporaries, and
the two first named were distinguished for general mathematical
ability.

\index{Fourier|(}\nblabel{png450a}\nbparagraph{Fourier.\footnote
  {An edition of his works, edited by \index{Darboux}G.~Darboux, was published in two
  volumes, Paris, 1888, 1890.}}
The first of these French physicists was \emph{Jean
Baptiste Joseph Fourier}, who was born at Auxerre on March~21,
1768, and died at Paris on May~16, 1830. He was the son of
a tailor, and was educated by the Benedictines. The commissions
in the scientific corps of the army were, as is still the case
in Russia, reserved for those of good birth, and being thus
ineligible he accepted a military lectureship on mathematics.
He took a prominent part in his own district in promoting the
revolution, and was rewarded by an appointment in 1795 in the
Normal school, and subsequently by a chair in the Polytechnic
school.

Fourier went with \index{Napoleon I.}Napoleon on his Eastern expedition in
1798, and was made governor of Lower Egypt. Cut off from
France by the English fleet, he organised the workshops on
which the French army had to rely for their munitions of war.
He also contributed several mathematical papers to the Egyptian
Institute which Napoleon founded at Cairo, with a view of
weakening English influence in the East. After the British
victories and the capitulation of the French under \index{Menou, General, ref.\ to}General
Menou in 1801, Fourier returned to France, and was made
prefect of Grenoble, and it was while there that he made his
\index{Heat, theory of}experiments on the propagation of heat. He moved to Paris
in 1816. In 1822 he published his \emph{Théorie analytique de la
chaleur}, in which he bases his reasoning on \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's law of
cooling, namely, that the flow of heat between two adjacent
molecules is proportional to the infinitely small difference of
their temperatures. In this work he shows that any function
of a variable, whether continuous or discontinuous, can be
expanded in a \index{Harmonic series}series of sines of multiples of the variable---a
result which is constantly used in modern analysis. \index{Lagrange!ref.\ to}Lagrange
had given particular cases of the \index{Fourier's theorem}theorem, and had implied that
the method was general, but he had not pursued the subject.
\index{Dirichlet, Lejeune!ref.\ to}Dirichlet was the first to give a satisfactory demonstration
of it.

Fourier left an unfinished work on determinate equations
which was edited by \index{Navier on Fourier}Navier, and published in 1831; this
contains much original matter, in particular there is a demonstration
of \index{Equations, differential!position of roots}\index{Roots of equations,!position of}Fourier's theorem on the position of the roots of an
algebraical equation. \index{Lagrange!ref.\ to}Lagrange had shewn how the roots of an
algebraical equation might be separated by means of another
equation whose roots were the squares of the differences of the
roots of the original equation. \index{Budan}Budan, in 1807 and 1811, had
enunciated the theorem generally known by the name of
Fourier, but the demonstration was not altogether satisfactory.
Fourier's proof is the same as that usually given in text-books
on the theory of equations. The final solution of the
problem was given in 1829 by \index{Sturm, ref.\ to}Jacques Charles Fran\c cois Sturm
(1803--1855).\index{Fourier|)}

\index{Carnot, Sadi}\nblabel{png451a}\nbparagraph{Sadi Carnot.\footnote
  {A sketch of S.~Carnot's life and an English translation of his \emph{Réflexions}
  was published by \index{Thurston on Carnot}R.~H.~Thurston, London and New York, 1890.}}
Among \index{Fourier!ref.\ to}Fourier's contemporaries who were
interested in the \index{Heat, theory of}theory of heat the most eminent was \emph{Sadi
Carnot}, a son of the eminent geometrician mentioned above.
Sadi Carnot was born at Paris in 1796, and died there of
cholera in August 1832; he was an officer in the French
army. In 1824 he issued a short work entitled \emph{Réflexions
sur la puissance motrice du feu}, in which he attempted to
determine in what way heat produced its mechanical effect.
He made the mistake of assuming that heat was material, but
his essay may be taken as initiating the modern theory of
\index{Thermodynamics}thermodynamics.

\index{Poisson|(}\nblabel{png451b}\nbparagraph{Poisson.\footnote
  {Memoirs of Poisson will be found in the \emph{Encyclopaedia Britannica}, the
  \emph{Transactions of the Royal Astronomical Society}, vol.~v, and \index{Arago!ref.\ to}Arago's \emph{Éloges},
  vol.~ii; the latter contains a bibliography of Poisson's papers and works.}}
\emph{Siméon Denis Poisson}, born at Pithiviers on
June~21, 1781, and died at Paris on April~25, 1840, is almost
equally distinguished for his applications of mathematics to
mechanics and to physics. His father had been a private
soldier, and on his retirement was given some small administrative
post in his native village; when the revolution broke
out he appears to have assumed the government of the place,
and, being left undisturbed, became a person of some local
importance. The boy was put out to nurse, and he used to tell
how one day his father, coming to see him, found that the nurse
had gone out, on pleasure bent, having left him suspended by a
small cord attached to a nail fixed in the wall. This, she
explained, was a necessary precaution to prevent him from
perishing under the teeth of the various animals and animalculae
that roamed on the floor. Poisson used to add that his gymnastic
efforts carried him incessantly from one side to the other, and it
was thus in his tenderest infancy that he commenced those
studies on the \index{Pendulum, motion of}pendulum that were to occupy so large a part of
his mature age.

He was educated by his father, and destined much against
his will to be a doctor. His uncle offered to teach him the art,
and began by making him prick the veins of cabbage-leaves
with a lancet. When perfect in this, he was allowed to put on
blisters; but in almost the first case he did this by himself, the
patient died in a few hours, and though all the medical practitioners
of the place assured him that ``the event was a very
common one,'' he vowed he would have nothing more to do with
the profession.

{\nbstretchyspace Poisson, on his return home after this adventure, discovered
amongst the official papers sent to his father a copy of the
questions set at the Polytechnic school, and at once found his
career.} At the age of seventeen he entered the Polytechnic, and
his abilities excited the interest of \index{Lagrange!ref.\ to}Lagrange and \index{Laplace!ref.\ to}Laplace, whose
friendship he retained to the end of their lives. A memoir on
\index{Finite differences}\index{Differences, finite}finite differences which he wrote when only eighteen was
reported on so favourably by \index{Legendre!ref.\ to}Legendre that it was ordered to be
published in the \emph{Recueil des savants étrangers}. As soon as he
had finished his course he was made a lecturer at the school,
and he continued through his life to hold various government
scientific posts and professorships. He was somewhat of a
socialist, and remained a rigid republican till 1815, when, with
a view to making another empire impossible, he joined the
legitimists. He took, however, no active part in politics, and
made the study of mathematics his amusement as well as his
business.

His works and memoirs are between three and four hundred
in number. The chief treatises which he wrote were his \emph{Traité
de mécanique},\footnote
  {Among Poisson's contemporaries who studied mechanics and of whose
  works he made use I may mention \index{Poinsot}\emph{Louis Poinsot}, who was born in Paris on
  Jan.~3, 1777, and died there on Dec.~5, 1859. In his \emph{Statique}, published in
  1803, he treated the subject without any explicit reference to dynamics. The
  theory of couples is largely due to him (1806), as also the motion of a body
  in space under the action of no forces.}
published in two volumes, 1811 and 1833, which
was long a standard work; his \emph{Théorie nouvelle de l'action
capillaire, }1831; his \emph{Théorie mathématique de la chaleur}, 1835,
to which a supplement was added in 1837; and his \emph{Recherches
sur la probabilité des jugements}, 1837. He had intended, if he
had lived, to write a work which should cover all mathematical
physics and in which the results of the three books last named
would have been incorporated.

Of his memoirs in pure mathematics the most important are
those on definite integrals, and \index{Fourier!ref.\ to}Fourier's series, their application
to physical problems constituting one of his chief claims to distinction;
his essays on the \index{Variations, calculus of}calculus of variations; and his
papers on the probability of the mean results of observations.\footnote
  {See the \emph{Journal de l'école polytechnique} from 1813 to 1823, and the
  \emph{Mémoires de l'académie} for 1823; the \emph{Mémoires de l'académie}, 1833; and
  the \emph{Connaissance des temps}, 1827 and following years. Most of his memoirs
  were published in the three periodicals here mentioned.}

Perhaps the most remarkable of his memoirs in applied
mathematics are those on the theory of electrostatics and
\index{Magnetism}magnetism, which originated a new branch of mathematical
physics; he supposed that the results were due to the attractions
and repulsions of imponderable particles. The most
important of those on physical astronomy are the two read in
1806 (printed in 1809) on the secular inequalities of the mean
\index{Planetary motions!stability}motions of the planets, and on the variation of arbitrary
constants introduced into the solutions of questions on
mechanics; in these Poisson discusses the question of the
stability of the planetary orbits (which \index{Lagrange!ref.\ to}Lagrange had already
proved to the first degree of approximation for the \index{Disturbing forces}disturbing
forces), and shews that the result can be extended to the third
order of small quantities: these were the memoirs which led
to Lagrange's famous memoir of 1808. Poisson also published
a paper in 1821 on the \index{Libration of moon}libration of the moon; and another in
1827 on the motion of the earth about its centre of gravity.
His most important memoirs on the \index{Attraction, theories of}theory of attraction are
one in 1829 on the attraction of spheroids, and another in 1835
on the attraction of a homogeneous ellipsoid: the substitution
of the correct equation involving the \index{Potential, the}potential, namely,
$\nabla^2 V = - 4\pi\rho$, for \index{Laplace!ref.\ to}Laplace's form of it, $\nabla^2 V = 0$, was first published\footnote
  {In the \emph{Bulletin des sciences} of the \emph{Société philomatique.}}
in 1813. Lastly, I may mention his memoir in 1825
on the theory of \index{Wave theory (optics)}waves.\index{Poisson|)}

\index{Ampère}\nblabel{png454a}\nbparagraph{Ampère.\footnote
  {See \index{Valson, ref.\ to}C.~A.~Valson's \emph{Étude sur la vie et les ouvrages d'Ampère}, Lyons, 1885.}}
\emph{André Marie Ampère} was born at Lyons on
January~22, 1775, and died at Marseilles on June~10, 1836.
He was widely read in all branches of learning, and lectured
and wrote on many of them, but after the year 1809, when he
was made professor of analysis at the Polytechnic school in
Paris, he confined himself almost entirely to mathematics and
science. His papers on the connection between \index{Electricity}electricity and
magnetism were written in 1820. According to his theory,
propounded in 1826, a molecule of matter which can be
magnetized is traversed by a closed electric current, and
magnetization is produced by any cause which makes the
direction of these currents in the different molecules of the
body approach parallelism.

\index{Fresnel}\nbparagraph{Fresnel. \nbnamespace Biot.} \emph{Augustin Jean Fresnel}, born at Broglie on
May~10, 1788, and died at Ville-d'Avray on July~14, 1827, was
a civil engineer by profession, but he devoted his leisure to the
\index{Optics (geometrical).!z11@(physical)}study of physical optics. \index{Light,!physical theories of}\index{Undulatory theory (optics)}The undulatory theory of light, which
\index{Hooke!ref.\ to}Hooke, \index{Huygens!ref.\ to}Huygens, and \index{Euler!ref.\ to}Euler had supported on \emph{a priori} grounds,
had been based on experiment by the researches of \index{Young, Thos.!ref.\ to}Young.
Fresnel deduced the mathematical consequences of these experiments,
and explained the phenomena of \index{Interference, principle of}\index{Diffraction}interference both of
ordinary and \index{Polarization of light}polarized light. Fresnel's friend and contemporary,
\index{Biot}\emph{Jean Baptiste Biot}, who was born at Paris on April~21, 1774,
and died there in 1862, requires a word or two in passing.
Most of his mathematical work was in connection with the
subject of optics, and especially the polarization of light. His
systematic works were produced within the years 1805 and
1817; a selection of his more valuable memoirs was published
in Paris in 1858.

\index{Arago|(}\nblabel{png455a}\nbparagraph{Arago.\footnote
  {Arago's works, which include \emph{éloges} on many of the leading mathematicians
  of the last five or six centuries, have been edited by \index{Barral on Arago}M.~J.~A.~Barral,
  and published in fourteen volumes, Paris, 1856--57. An autobiography is
  prefixed to the first volume.}}
\emph{Fran\c{c}ois Jean Dominique Arago} was born at
Estagel in the Pyrenees on February~26, 1786, and died in
Paris on October~2, 1853. He was educated at the Polytechnic
school, Paris, and we gather from his autobiography that
however distinguished were the professors of that institution
they were remarkably incapable of imparting their knowledge
or maintaining discipline.

\index{Earth,!dimensions of}\index{Degree, length of}In 1804 Arago was made secretary to the observatory at
Paris, and from 1806 to 1809 he was engaged in measuring a
meridian arc in order to determine the exact length of a metre.
He was then appointed to a leading post in the observatory,
given a residence there, and made a professor at the Polytechnic
school, where he enjoyed a marked success as a lecturer. He
subsequently gave popular lectures on astronomy, which were
both lucid and accurate---a combination of qualities which was
rarer then than now. He reorganized the national observatory,
the management of which had long been inefficient, but in doing
this his want of tact and courtesy raised many unnecessary
difficulties. He remained to the end a consistent republican,
and after the \emph{coup d'état} of 1852, though half blind and dying,
he resigned his post as astronomer rather than take the oath of
allegiance. It is to the credit of \index{Napoleon III.}Napoleon III. that he gave
directions that the old man should be in no way disturbed, and
should be left free to say and do what he liked.

Arago's earliest physical researches were on the pressure of
steam at different temperatures, and the velocity of sound, 1818
to 1822. His \index{Magnetism}magnetic observations mostly took place from
1823 to 1826. He discovered what has been called rotatory
magnetism, and the fact that most bodies could be magnetized;
these discoveries were completed and explained by \index{Faraday, ref.\ to}Faraday.
He warmly supported Fresnel's optical theories, and the two
philosophers conducted together those experiments on the
\index{Polarization of light}polarization of light which led to the inference that the vibrations
of the luminiferous ether were transverse to the direction
of motion, and that polarization consisted in a resolution of
rectilinear motion into components at right angles to each other.
The subsequent invention of the polariscope and discovery of
rotatory polarization are due to Arago. The general idea of the
experimental determination of the \index{Light,!velocity of}velocity of light in the
manner subsequently effected by \index{Fizeau, ref.\ to}Fizeau and \index{Foucault, ref.\ to}Foucault was
suggested by him in 1838, but his failing eyesight prevented
his arranging the details or making the experiments.\index{Arago|)}

It will be noticed that some of the last members of the
French school were alive at a comparatively recent date, but
nearly all their mathematical work was done before the year
1830. They are the direct successors of the French writers who
flourished at the commencement of the nineteenth century, and
seem to have been out of touch with the great German mathematicians
of the early part of it, on whose researches much of
the best work of that century is based; they are thus placed
here, though their writings are in some cases of a later date
than those of \index{Gauss!ref.\ to}Gauss, \index{Abel!ref.\ to}Abel, and \index{Jacobi!ref.\ to}Jacobi.


\section*{The introduction of analysis into England.\nblabel{png456a}}

The complete isolation of the English school and its devotion
to geometrical methods are the most marked features in its
history during the latter half of the eighteenth century; and
the absence of any considerable contribution to the advancement
of mathematical science was a natural consequence. One result
of this was that the energy of English men of science was
largely devoted to practical physics and practical astronomy,
which were in consequence studied in Britain perhaps more
than elsewhere.

\index{Ivory}\nblabel{png457a}\nbparagraph{Ivory.} Almost the only English mathematician at the
beginning of this century who used analytical methods, and
whose work requires mention here, is Ivory, to whom the celebrated
\index{Attraction, theories of}theorem in attractions is due. \emph{Sir James Ivory} was
born in Dundee in 1765, and died on September~21, 1842.
After graduating at St.~Andrews he became the managing
partner in a flax-spinning company in Forfarshire, but continued
to devote most of his leisure to mathematics. In 1804 he was
made professor at the Royal Military College at Marlow, which
was subsequently moved to Sandhurst; he was knighted in
1831. He contributed numerous papers to the \emph{Philosophical
Transactions}, the most remarkable being those on attractions.
In one of these, in 1809, he shewed how the attraction of a
homogeneous ellipsoid on an external point is a multiple of that
of another ellipsoid on an internal point: the latter can be
easily obtained. He criticized \index{Laplace!ref.\ to}Laplace's solution of the method
of \index{Least squares}least squares with unnecessary bitterness, and in terms which
shewed that he had failed to understand it.

\index{Cambridge, university of|(}\index{Analysis,!Cambridge School|(}\nblabel{png457b}\nbparagraph{The Cambridge Analytical School.} Towards the beginning
of the last century the more thoughtful members of the Cambridge
school of mathematics began to recognize that their isolation
from their continental contemporaries was a serious evil. The
earliest attempt in England to explain the notation and methods
of the calculus as used on the continent was due to \index{Woodhouse!ref.\ to}Woodhouse,
who stands out as the apostle of the new movement. It is
doubtful if he could have brought the analytical methods into
vogue by himself; but his views were enthusiastically adopted
by three students, \index{Peacock!ref.\ to}Peacock, \index{Babbage!ref.\ to}Babbage, and \index{Herschel, Sir John!ref.\ to}Herschel, who succeeded
in carrying out the reforms he had suggested. In a
book which will fall into the hands of few but English readers
I may be pardoned for making space for a few remarks on these
four mathematicians, though otherwise a notice of them would
not be required in a work of this kind.\footnote
  {The following account is condensed from my \index{Ball, W. W. R., ref.\ to}\emph{History of the Study of
  Mathematics at Cambridge}, Cambridge, 1889.}
The original stimulus
came from French sources, and I therefore place these remarks
at the close of my account of the French school; but I should
add that the English mathematicians of this century at once
struck out a line independent of their French contemporaries.

\index{Woodhouse|(}\nblabel{png458a}\nbparagraph{Woodhouse.} \emph{Robert Woodhouse} was born at Norwich on
April~28, 1773; was educated at Caius College, Cambridge, of
which society he was subsequently a fellow; was Plumian professor
in the university; and continued to live at Cambridge till
his death on December~23, 1827.

Woodhouse's earliest work, entitled the \emph{Principles of Analytical
Calculation}, was published at Cambridge in 1803. In
this he explained the differential notation and strongly pressed
the employment of it; but he severely criticized the methods
used by continental writers, and their constant assumption of
non-evident principles. This was followed in 1809 by a trigonometry
(plane and spherical), and in 1810 by a historical treatise
on the calculus of variations and \index{Isoperimetrical problem}isoperimetrical problems. He
next produced an astronomy; of which the first book (usually
bound in two volumes), on practical and descriptive astronomy,
was issued in 1812, and the second book, containing an account
of the treatment of physical astronomy by \index{Laplace!ref.\ to}Laplace and other
continental writers, was issued in 1818. All these works deal
critically with the scientific foundation of the subjects considered---a
point which is not unfrequently neglected in modern text-books.

A man like Woodhouse, of scrupulous honour, universally
respected, a trained logician, and with a caustic wit, was well
fitted to introduce a new system; and the fact that when he
first called attention to the continental analysis he exposed the
unsoundness of some of the usual methods of establishing it,
more like an opponent than a partisan, was as politic as it
was honest. Woodhouse did not exercise much influence on
the majority of his contemporaries, and the movement might
have died away for the time being if it had not been for the
advocacy of \index{Peacock!ref.\ to}Peacock, \index{Babbage!ref.\ to}Babbage, and Herschel, who formed an
Analytical Society, with the object of advocating the general use
in the university of analytical methods and of the differential
notation.\index{Woodhouse|)}

\index{Peacock}\nblabel{png459a}\nbparagraph{Peacock.} \emph{George Peacock}, who was the most influential of
the early members of the new school, was born at Denton on
April~9, 1791. He was educated at Trinity College, Cambridge,
of which society he was subsequently a fellow and tutor. The
establishment of the university observatory was mainly due to
his efforts, and in 1836 he was appointed to the Lowndean
professorship of astronomy and geometry. In 1839 he was
made dean of Ely, and resided there till his death on Nov.~8,
1858. Although Peacock's influence on English mathematicians
was considerable, he has left but few memorials of his work;
but I may note that his report on progress in analysis, 1833,
commenced those valuable summaries of current scientific progress
which enrich many of the annual volumes of the \emph{Transactions} of
the \index{British Association}British Association.

\index{Babbage}\nbparagraph{Babbage.} Another important member of the Analytical
Society was \emph{Charles Babbage}, who was born at Totnes on Dec.
26, 1792; he entered at Trinity College, Cambridge, in 1810;
subsequently became Lucasian professor in the university; and
died in London on Oct.~18, 1871. It was he who gave the
name to the Analytical Society, which, he stated, was formed
to advocate ``the principles of pure \emph{d}-ism as opposed to the \emph{dot}-age
of the university''. In 1820 the \index{Astronomical Society, London}Astronomical Society was
founded mainly through his efforts, and at a later time, 1830 to
1832, he took a prominent part in the foundation of the British
Association. He will be remembered for his mathematical
memoirs on the calculus of functions, and his invention of an
\index{Arithmetical!machine}\index{Calculating machine}analytical machine which could not only perform the ordinary
processes of arithmetic, but could tabulate the values of any function
and print the results.

\index{Herschel, Sir John}\nbparagraph{Herschel.} The third of those who helped to bring analytical
methods into general use in England was the son of \index{Herschel, Sir William}Sir William
Herschel (1738--1822), the most illustrious astronomer of the
latter half of the eighteenth century and the creator of modern
stellar astronomy. \emph{Sir John Frederick William Herschel} was born
on March~7, 1792, educated at St.~John's College, Cambridge,
and died on May~11, 1871. His earliest original work was a
paper on \index{Cotes!ref.\ to}Cotes's theorem, and it was followed by others on
mathematical analysis, but his desire to complete his father's
work led ultimately to his taking up astronomy. His papers
on light and astronomy contain a clear exposition of the
principles which underlie the mathematical treatment of those
subjects.

In 1813 the Analytical Society published a volume of
memoirs, of which the preface and the first paper (on continued
products) are due to \index{Babbage!ref.\ to}Babbage; and three years later they
issued a translation of \index{Lacroix}Lacroix's \emph{Traité élémentaire du calcul
différentiel et du calcul intégral}. In 1817, and again in 1819,
the differential notation was used in the university examinations,
and after 1820 its use was well established. The Analytical
Society followed up this rapid victory by the issue in 1820 of
two volumes of examples illustrative of the new method; one
by \index{Peacock!ref.\ to}Peacock on the differential and integral calculus, and the
other by Herschel on the calculus of \index{Finite differences}\index{Differences, finite}finite differences. Since
then English works on the infinitesimal calculus have abandoned
the exclusive use of the fluxional notation. It should be noticed
in passing that \index{Lagrange!ref.\ to}Lagrange and \index{Laplace!ref.\ to}Laplace, like the majority of other
modern writers, employ both the fluxional and the differential
notation; it was the exclusive adoption of the former that was
so hampering.

Amongst those who materially assisted in extending the
use of the new analysis were \index{Whewell, W.}William Whewell (1794--1866)
and \index{Airy, G. B.}George Biddell Airy (1801--1892), both Fellows of Trinity
College, Cambridge. The former issued in 1819 a work on
mechanics, and the latter, who was a pupil of Peacock, published
in 1826 his \emph{Tracts}, in which the new method was applied with
great success to various physical problems. The efforts of the
society were supplemented by the rapid publication of good
text-books in which analysis was freely used. The employment
of analytical methods spread from Cambridge over the rest of
Britain, and by 1830 these methods had come into general use
there.\index{Analysis,!Cambridge School|)}\index{Cambridge, university of|)}

\nbChapter{XIX}{Mathematics of the Nineteenth Century}{Nineteenth Century Mathematics}{}{}
\index{Recent mathematics, chapter \textsc{xix}}\index{Geometry.!z6@Analytical geometry!x3@discussed or used by nearly all the mathematicians considered in the third period, chapters \textsc{xiv-xix}|)}\index{Astronomy.!z20@Developed by (amongst others) Clairaut!q3@and in recent times by Gauss and others, chapter \textsc{xix}}\nblabel{png462a}

\nblabel{png462b}\First{The} nineteenth century saw the creation of numerous new
departments of pure mathematics---notably of a theory of
numbers, or higher arithmetic; of theories of forms and
groups, or a higher algebra; of theories of functions of
multiple periodicity, or a higher trigonometry; and of a
general theory of functions, embracing extensive regions of
higher analysis. Further, the developments of synthetic and
analytical geometry created what practically were new subjects.
The foundations of the subject and underlying assumptions
(notably in arithmetic, geometry, and the calculus) were also
subjected to a rigorous scrutiny. Lastly, the application of
mathematics to physical problems revolutionized the foundations
and treatment of that subject. Numerous Schools, Journals,
and Teaching Posts were established, and the facilities for the
study of mathematics were greatly extended.

Developments, such as these, may be taken as opening a
new period in the history of the subject, and I recognize that in
the future a writer who divides the history of mathematics as I
have done would probably treat the mathematics of the seventeenth
and eighteenth centuries as forming one period, and
would treat the mathematics of the nineteenth century as
commencing a new period. This, however, would imply a
tolerably complete and systematic account of the development
of the subject in the nineteenth century. \nblabel{png463a}But evidently it is
impossible for me to discuss adequately the mathematics of a
time so near to us, and the works of mathematicians some of
whom are living and some of whom I have met and known.
\nblabel{png463b}Hence I make no attempt to give a complete account of the
mathematics of the nineteenth century, but as a sort of appendix
to the preceding chapters I mention the more striking features
in the history of recent pure mathematics, in which I include
theoretical dynamics and astronomy; I do not, however, propose
to discuss in general the recent application of mathematics to
physics.

\nblabel{png463c}In only a few cases do I give an account of the life and
works of the mathematicians mentioned; but I have added brief
notes about some of those to whom the development of any
branch of the subject is chiefly due, and an indication of that
part of it to which they have directed most attention. Even
with these limitations it has been very difficult to put together a
connected account of the mathematics of recent times; and I
wish to repeat explicitly that I do not suggest, nor do I wish
my readers to suppose, that my notes on a subject give the
names of all the chief writers who have studied it. In fact the
quantity of matter produced has been so enormous that no one
can expect to do more than make himself acquainted with the
works produced in some special branch or branches. As an
illustration of this remark I may add that the committee
appointed by the Royal Society to report on a catalogue of
periodical literature estimated, in 1900, that more than 1500
memoirs on pure mathematics were then issued annually, and
more than 40,000 a year on scientific subjects.

{\nbstretchyspace Most histories of mathematics do not treat of the work
produced during this century. The chief exceptions with which
I am acquainted are \index{Adhémar, R. d'}R.~d'Adhémar's \emph{L'{\OE}uvre mathématique
du xix\textsuperscript{e} siècle}; \index{Fink, K.}K.~Fink's \emph{Geschichte der Mathematik}, Tübingen,
1890; \index{Gerhardt, ref.\ to}E.~J.~Gerhardt's \emph{Geschichte der Mathematik in Deutschland},
Munich, 1877; \index{Gunther@Günther, S.}S.~Günther's \emph{Verm.\ Unt.\ zur Geschichte
der mathematischen Wissenschaften}, Leipzig, 1876, and \emph{Ziele und
Resultate der neueren mathematisch-historischen Forschung},
Erlangen, 1876; \index{Hagen, J. G.}J.~G.~Hagen, \emph{Synopsis der höheren Mathematik},
3 volumes, Berlin, 1891, 1893, 1906; a short dissertation by \index{Hankel, ref.\ to}H.~Hankel,
entitled \emph{Die Entwickelung der Mathematik in den letzten
Jahrhunderten}, Tübingen, 1885; a \emph{Discours} on the professors
at the Sorbonne by \index{Hermite!ref.\ to}C.~Hermite in the \emph{Bulletin des sciences
mathématiques}, 1890; \index{Klein, F. C.}F.~C.\allowbreak~Klein's \emph{Lectures on Mathematics},
Evanston Colloquium, New York and London, 1894; \index{Lampe, ref.\ to}E.~Lampe's
\emph{Die reine Mathematik in den Jahren 1884--1899}, Berlin, 1899;
the eleventh and twelfth volumes of \index{Marie, ref.\ to}Marie's \emph{Histoire des
sciences}, in which are some notes on mathematicians who were
born in the last century; \index{Painlevé, P.}P.~Painlevé's \emph{Les Sciences mathématiques
au xix$^{e}$ siècle}; a chapter by \index{Smith, D. E.}D.~E.~Smith in \emph{Higher
Mathematics}, by \index{Merriman, M.}M.~Merriman and \index{Woodward, R. S.}R.~S.~Woodward, New York,
1900; and \index{Volterra, V., ref.\ to}V.~Volterra's lecture at the \index{Rome Congress}Rome Congress, 1908,
``On the history of mathematics in Italy during the latter half of
the nineteenth century.''}

A few histories of the development of particular subjects
have been written---such as those by \index{Todhunter, ref.\ to}Isaac Todhunter on the
\index{Attraction, theories of}theories of attraction and on the calculus of probabilities; those
by \index{Muir, T.}T.~Muir on determinants, that by \index{Braunmühl, A. von}A.~von Braunmühl on
trigonometry, that by \index{Reiff, R.}R.~Reiff on infinite series, that by \index{Loria, ref.\ to}G.~Loria,
\emph{Il passato ed il presente delle principali teorie geometriche},
and that by \index{Engel, F., on Grassmann!ref.\ to}F.~Engel and \index{Stäckel, P.}P.~Stäckel on the theory of parallels.
The transactions of some of the scientific societies and academies
also contain reports on the progress in different branches of the
subject, while information on the memoirs by particular mathematicians
is given in the invaluable volumes of \index{Poggendorff, J. C.}J.~C.~Poggendorff's
\emph{Biographisch-literarisches Handwörterbuch zur Geschichte der
exacten Wissenschaften}, Leipzig. The \emph{Encyklopädie der mathematischen
Wissenschaften}, which is now in course of issue, aims
at representing the present state of knowledge in pure and
applied mathematics, and doubtless in some branches of mathematics
it will supersede these reports. The French translation
of this encyclopaedia contains numerous and valuable additions.
I have found these authorities and these reports useful, and I
have derived further assistance in writing this chapter from the
obituary notices in the proceedings of various learned Societies.
I am also indebted to information kindly furnished me by
various friends, and if I do not further dwell on this, it is only
that I would not seem to make them responsible for my errors
and omissions.

A period of exceptional intellectual activity in any subject
is usually followed by one of comparative stagnation; and
after the deaths of \index{Lagrange!ref.\ to}Lagrange, \index{Laplace!ref.\ to}Laplace, \index{Legendre!ref.\ to}Legendre, and \index{Poisson!ref.\ to}Poisson,
the French school, which had occupied so prominent a position
at the beginning of this century, ceased for some years to
produce much new work. Some of the mathematicians whom
I intend to mention first, \index{Gauss!ref.\ to}Gauss, \index{Abel!ref.\ to}Abel, and \index{Jacobi!ref.\ to}Jacobi, were
contemporaries of the later years of the French mathematicians
just named, but their writings appear to me to belong to a
different school, and thus are properly placed at the beginning
of a fresh chapter.

There is no mathematician of this century whose writings
have had a greater effect than those of Gauss; nor is it on only
one branch of the science that his influence has left a permanent
mark. I cannot, therefore, commence my account of the
mathematics of recent times better than by describing very
briefly his more important researches.

\index{Gauss|(}\nblabel{png465a}\nbparagraph{Gauss.\footnote
  {Biographies of Gauss have been published by \index{Hanselmann@Hänselmann, L., on Gauss}L.~Hänselmann, Leipzig,
  1878, and by \index{Walterhausen, S. von}S.~von Walterhausen, Leipzig, 1856. The Royal Society of
  Göttingen undertook the issue of a collection of Gauss's works, and nine
  volumes are already published. Further additions are expected, and some
  hints of what may be expected have been given by \index{Klein, F. C.}F~C.~Klein.}}
\emph{Karl Friedrich Gauss} was born at Brunswick on
April~23, 1777, and died at Göttingen on February~23, 1855.
His father was a bricklayer, and Gauss was indebted for a
liberal education (much against the will of his parents, who
wished to profit by his wages as a labourer) to the notice which
his talents procured from the reigning duke. In 1792 he was
sent to the Caroline College, and by 1795 professors and pupils
alike admitted that he knew all that the former could teach
him: it was while there that he investigated the method of
\index{Least squares}least squares, and proved by induction the \index{Quadratic reciprocity}law of quadratic
reciprocity. Thence he went to Göttingen, where he studied
under \index{Kästner}Kästner: many of his discoveries in the \index{Numbers, theory of.!Treatment of, by Pythagoras!q7@by Gauss and other mathematicians of recent times}theory of
numbers were made while a student here. In 1798 he
returned to Brunswick, where he earned a somewhat precarious
livelihood by private tuition.

In 1799 Gauss published a demonstration that every integral
algebraical function of one variable can be expressed as a product
of real linear or quadratic factors. Hence every algebraical
\index{Equations, differential!number of roots}equation has a \index{Roots of equations,!number of}root of the form $a + bi$, a theorem of which he
gave later two other distinct proofs. His \emph{Disquisitiones
Arithmeticae} appeared in 1801. A large part of this had
been submitted as a memoir to the French Academy in the
preceding year, and had been rejected in a most regrettable
manner; Gauss was deeply hurt, and his reluctance to publish
his investigations may be partly attributable to this unfortunate
incident.

\index{Planetary motions}\nblabel{png466a}The next discovery of Gauss was in a totally different
department of mathematics. The absence of any planet in the
space between Mars and Jupiter, where \index{Bode's law}Bode's law would have
led observers to expect one, had been long remarked, but it
was not till 1801 that any one of the numerous group of minor
planets which occupy that space was observed. The discovery
was made by \index{Piazzi of Palermo}G.~Piazzi of Palermo; and was the more interesting
as its announcement occurred simultaneously with a publication
by \index{Hegel, ref.\ to}Hegel in which he severely criticised astronomers for not
paying more attention to philosophy,---a science, said he, which
would at once have shewn them that there could not possibly
be more than seven planets, and a study of which would therefore
have prevented an absurd waste of time in looking for
what in the nature of things could never be found. The new
planet was named \index{Ceres, the planet}Ceres, but it was seen under conditions
which appeared to render it impracticable to forecast its orbit.
The observations were fortunately communicated to Gauss; he
calculated its elements, and his analysis put him in the first
rank of theoretical astronomers.

The attention excited by these investigations procured for
him in 1807 the offer of a chair at Petrograd, which he
declined. In the same year he was appointed director of the
Göttingen Observatory and professor of Astronomy there.
These offices he retained to his death; and after his appointment
he never slept away from his Observatory except on one
occasion when he attended a scientific congress at Berlin. His
lectures were singularly lucid and perfect in form, and it is
said that he used here to give the analysis by which he had
arrived at his various results, and which is so conspicuously
absent from his published demonstrations; but for fear his
auditors should lose the thread of his discourse, he never
willingly permitted them to take notes.

I have already mentioned Gauss's publications in 1799,
1801, and 1802. For some years after 1807 his time was
mainly occupied by work connected with his Observatory. In
1809 he published at Hamburg his \emph{Theoria Motus Corporum
Coelestium}, a treatise which contributed largely to the improvement
of practical astronomy, and introduced the principle
of curvilinear triangulation; and on the same subject, but
connected with observations in general, we have his memoir
\emph{Theoria Combinationis Observationum Erroribus Minimis
Obnoxia}, with a second part and a supplement.

Somewhat later he took up the subject of \index{Geodesy}geodesy, acting
from 1821 to 1848 as scientific adviser to the Danish and
Hanoverian Governments for the survey then in progress;
his papers of 1843 and 1866, \emph{Ueber Gegenstände der höhern
Geodäsie}, contain his researches on the subject.

\nblabel{png467a}Gauss's researches on \index{Electricity}electricity and \index{Magnetism}magnetism date from
about the year 1830. His first paper on the theory of
magnetism, entitled \emph{Intensitas Vis Magneticae Terrestris ad
Mensuram Absolutam Revocata}, was published in 1833. A few
months afterwards he, together with \index{Weber, W. E.}W.~E.~Weber, invented
the declination instrument and the bifilar magnetometer;
and in the same year they erected at Göttingen a magnetic
observatory free from iron (as \index{Humboldt}\index{Von Humboldt}Humboldt and \index{Arago!ref.\ to}Arago had
previously done on a smaller scale) where they made magnetic
observations, and in particular showed that it was practicable
to send telegraphic signals. In connection with this Observatory
Gauss founded an association with the object of securing
continuous observations at fixed times. The volumes of their
publications, \emph{Resultate aus der Beobachtungen des magnetischen
Vereins} for 1838 and 1839, contain two important memoirs by
Gauss: one on the general theory of earth-magnetism, and the
other on the theory of forces attracting according to the inverse
square of the distance.

Gauss, like \index{Poisson!ref.\ to}Poisson, treated the phenomena in electrostatics
as due to attractions and repulsions between imponderable
particles. \index{Kelvin, Lord}Lord Kelvin, then William Thomson (1824--1907),
of Glasgow, shewed in 1846 that the effects might also be
supposed analogous to a flow of heat from various sources of
electricity properly distributed.

In electrodynamics Gauss arrived (in 1835) at a result
equivalent to that given by \index{Weber, W. E.}W.~E.~Weber of Göttingen in
1846, namely, that the attraction between two electrified
particles $e$ and $e'$, whose distance apart is $r$, depends on their
relative motion and position according to the formula
\[
e e' r^{-2} \{ 1 + (r\ddot{r} - \tfrac{1}{2}\dot{r}^2)^2 c^{-2} \}.
\]

Gauss, however, held that no hypothesis was satisfactory which
rested on a formula and was not a consequence of a physical
conjecture, and as he could not frame a plausible physical conjecture
he abandoned the subject.

Such conjectures were proposed by \index{Riemann!ref.\ to}Riemann in 1858, and by
\index{Neumann, C.}C.~Neumann, now of Leipzig, and \index{Betti, E.}E.~Betti (1823--1892) of Pisa
in 1868, but \index{Von Helmholtz}\index{Helmholtz, von, ref.\ to}Helmholtz in 1870, 1873, and 1874 showed that
they were untenable. A simpler view which regards all electric
and magnetic phenomena as stresses and motions of a material
elastic medium had been outlined by \index{Faraday, ref.\ to}Michael Faraday (1791--1867),
and was elaborated by \index{Maxwell, J. C.}James Clerk Maxwell (1831--1879)
of Cambridge in 1873; the latter, by the use of \index{Generalized co-ordinates}\index{Coord@Co-ordinates!generalized}generalised
co-ordinates, was able to deduce the consequences, and the agreement
with experiment is close. Maxwell concluded by showing
that if the medium were the same as the so-called \index{Ether, luminiferous}luminiferous
ether, the \index{Light,!velocity of}velocity of light would be equal to the ratio of the
electromagnetic and electrostatic units, and subsequent experiments
have tended to confirm this conclusion. The theories
previously current had assumed the existence of a simple elastic
solid or an action between matter and ether.

The above and other electric theories were classified by
\index{Thomson, Sir J. J.}J.~J.~Thomson of Cambridge, in a report to the British
Association in 1885, into those not founded on the principle
of the \index{Energy, conservation of}\index{Conservation of energy}conservation of energy (such as those of \index{Ampère!ref.\ to}Ampère, \index{Grassmann!ref.\ to}Grassmann,
\index{Stefan}Stefan, and \index{Korteweg}Korteweg); those which rest on assumptions
concerning the velocities and positions of electrified particles
(such as those of Gauss, \index{Weber, W. E.}W.~E.~Weber, \index{Riemann!ref.\ to}Riemann, and \index{Clausius, R. J. E.}R.~J.~E.~Clausius);
those which require the existence of a kind of energy
of which we have no other knowledge (such as the theory of \index{Neumann, C.}C.~Neumann);
those which rest on dynamical considerations, but
in which no account is taken of the action of the dielectric (such
as the theory of \index{Neumann, F. E.}F.~E.~Neumann); and, finally, those which rest
on dynamical considerations and in which the action of the
dielectric is considered (such as Maxwell's theory). In the
report these theories are described, criticised, and compared with
the results of experiments.

\index{Optics (geometrical).!Discussed by (among others) Euclid!z8@Gauss}Gauss's researches on optics, and especially on systems
of lenses, were published in 1840 in his \emph{Dioptrische Untersuchungen}.

From this sketch it will be seen that the ground covered
by Gauss's researches was extraordinarily wide, and it may be
added that in many cases his investigations served to initiate
new lines of work. He was, however, the last of the great
mathematicians whose interests were nearly universal: since his
time the literature of most branches of mathematics has grown
so fast that mathematicians have been forced to specialise in
some particular department or departments. I will now mention
very briefly some of the most important of his discoveries in
pure mathematics.

\nblabel{png470a}His most celebrated work in pure mathematics is the \emph{Disquisitiones
Arithmeticae}, which has proved a starting-point for
several valuable investigations on the \index{Numbers, theory of.!Treatment of, by Pythagoras!q7@by Gauss and other mathematicians of recent times}theory of numbers. This
treatise and \index{Legendre!ref.\ to}Legendre's \emph{Théorie des nombres} remain standard
works on the theory of numbers; but, just as in his discussion
of \index{Elliptic functions}elliptic functions Legendre failed to rise to the conception
of a new subject, and confined himself to regarding their theory
as a chapter in the integral calculus, so he treated the theory of
numbers as a chapter in algebra. Gauss, however, realised that
the theory of discrete magnitudes or higher arithmetic was of
a different kind from that of continuous magnitudes or algebra,
and he introduced a new notation and new methods of analysis,
of which subsequent writers have generally availed themselves.
The theory of numbers may be divided into two main divisions,
namely, the \index{Congruences}theory of congruences and the \index{Forms in algebra!in theory of numbers}theory of forms.
Both divisions were discussed by Gauss. In particular the
\emph{Disquisitiones Arithmeticae} introduced the modern theory of
congruences of the first and second orders, and to this Gauss
reduced indeterminate analysis. In it also he discussed the
solution of \index{Binomial equations}binomial equations of the form $x^n = 1$: this involves
the celebrated theorem that it is possible to construct, by
elementary geometry, \index{Polygons, regular}regular polygons of $2^m (2^n + 1)$ sides,
where $m$ and $n$ are integers and $2^n + 1$ is a prime---a discovery
he had made in 1796. He developed the theory of ternary quadratic
forms involving two indeterminates. He also investigated
the theory of \index{Determinants}determinants, and it was on Gauss's results that
\index{Jacobi!ref.\ to}Jacobi based his researches on that subject.

The theory of functions of double periodicity had its origin
in the discoveries of \index{Abel!ref.\ to}Abel and Jacobi, which I describe later.
Both these mathematicians arrived at the \index{Theta functions}theta functions, which
play so large a part in the theory of the subject. Gauss, however,
had independently, and indeed at a far earlier date,
discovered these functions and some of their properties, having
been led to them by certain integrals which occurred in the
\index{Attraction, theories of}\emph{Determinatio Attractionis}, to evaluate which he invented the
transformation now associated with the name of \index{Jacobi!ref.\ to}Jacobi. Though
Gauss at a later time communicated the fact to Jacobi, he did
not publish his researches; they occur in a series of note-books
of a date not later than 1808, and are included in his collected
works.

\nblabel{png471a}Of the remaining memoirs in pure mathematics the most
remarkable are those on the theory of \index{Biquadratic residues}biquadratic \index{Residues, theory of}residues
(wherein the notion of \index{Complex numbers}complex numbers of the form $a + bi$ was
first introduced into the theory of numbers), in which are included
several tables, and notably one of the number of the
classes of binary quadratic forms; that relating to the proof of
the theorem that every algebraical equation has a real or
imaginary root; that on the summation of \index{Expansions@Expansion in series}series; and, lastly,
one on interpolation. His introduction of rigorous tests for the
\index{Infinite series, difficulties in connection with}\index{Convergency}convergency of infinite series is worthy of attention. Specially
noticeable also are his investigations on \index{Hyperg2@Hyper-geometric series}hypergeometric
series; these contain a discussion of the \index{Gamma function}gamma function.
This subject has since become one of considerable importance,
and has been written on by (among others) \index{Kummer!ref.\ to}Kummer
and \index{Riemann!ref.\ to}Riemann; later the original conceptions were greatly
extended, and numerous memoirs on it and its extensions
have appeared. I should also mention Gauss's theorems on the
\index{Curvature of surfaces}curvature of surfaces, wherein he devised a new and general
method of treatment which has led to many new results.
Finally, we have his important memoir on the conformal
representation of one surface upon another, in which
the results given by \index{Lagrange!ref.\ to}Lagrange for surfaces of revolution are
generalised for all surfaces. It would seem also that Gauss
had discovered some of the properties of \index{Quaternions}\index{Planetary motions}quaternions,
though these investigations were not published until a few
years ago.

In the theory of attractions we have a paper on the attraction
of homogeneous ellipsoids: the already-mentioned memoir of
1839, on the theory of forces attracting according to the
inverse square of the distance; and the memoir, \emph{Determinatio}
\emph{Attractionis}, in which it is shown that the secular variations,
which the elements of the \index{Planetary motions}orbit of a planet experience from
the attraction of another planet which disturbs it, are the same
as if the mass of the disturbing planet were distributed over
its orbit into an elliptic ring in such a manner that equal masses
of the ring would correspond to arcs of the orbit described
in equal times.\index{Gauss|)}

\nblabel{png472a}The great masters of modern analysis are \index{Lagrange!ref.\ to}Lagrange, \index{Laplace!ref.\ to}Laplace,
and Gauss, who were contemporaries. It is interesting to note
the marked contrast in their styles. Lagrange is perfect both
in form and matter, he is careful to explain his procedure,
and though his arguments are general they are easy to follow.
Laplace, on the other hand, explains nothing, is indifferent to
style, and, if satisfied that his results are correct, is content
to leave them either with no proof or with a faulty one. Gauss
is as exact and elegant as Lagrange, but even more difficult
to follow than Laplace, for he removes every trace of the
analysis by which he reached his results, and studies to give
a proof which, while rigorous, shall be as concise and synthetical
as possible.

\index{Dirichlet, Lejeune|(}\nblabel{png472b}\nbparagraph{Dirichlet.\footnote
  {Dirichlet's works, edited by \index{Kronecker, L.}L.~Kronecker, were issued in two volumes,
  Berlin, 1889, 1897. His lectures on the theory of numbers were edited by
  \index{Dedekind, J. W. R., ref.\ to}J.~W.~R.~Dedekind, third edition, Brunswick, 1879--81. His investigations
  on the theory of the \index{Potential, the}potential were edited by \index{Grube on Dirichlet}F.~Grube, second edition, Leipzig,
  1887. His researches on definite integrate have been edited by \index{Arendt, G., on Dirichlet}G.~Arendt,
  Brunswick, 1904. There is a note on some of his researches by \index{Borchardt!ref.\ to}C.~W.~Borchardt
  in \emph{Crelle's Journal}, vol.~lvii, 1859, pp.~91--92.}}
One of Gauss's pupils to whom I may here
allude is Lejeune Dirichlet, whose masterly exposition of the
discoveries of \index{Jacobi!ref.\ to}Jacobi (who was his father-in-law) and of \index{Gauss!ref.\ to}Gauss
has unduly overshadowed his own original investigations on
similar subjects. \emph{Peter Gustav Lejeune Dirichlet} was born at
Düren on February~13, 1805, and died at Göttingen on May~5,
1859. He held successively professorships at Breslau and
Berlin, and on Gauss's death in 1855 was appointed to succeed
him as professor of the higher mathematics at Göttingen. He
intended to finish Gauss's incomplete works, for which he was
admirably fitted, but his early death prevented this. He produced,
however, several memoirs which have considerably facilitated
the comprehension of some of \index{Gauss!ref.\ to}Gauss's more abstruse methods.
Of Dirichlet's original researches the most celebrated are those
dealing with the establishment of \index{Fourier's theorem}Fourier's theorem, those in
the \index{Numbers, theory of.!Treatment of, by Pythagoras!q7@by Gauss and other mathematicians of recent times|(}theory of numbers on asymptotic laws (that is, laws which
approximate more closely to accuracy as the numbers concerned
become larger), and those on \index{Primes}primes.\index{Dirichlet, Lejeune|)}

It is convenient to take Gauss's researches as the starting-point
for the discussion of various subjects. Hence the length
with which I have alluded to them.

\tbskip
\nblabel{png473a}\nbtopictitle{The Theory of Numbers, \emph{or} Higher Arithmetic.} The researches
of Gauss on the theory of numbers were continued or supplemented
by \index{Jacobi!ref.\ to}\emph{Jacobi}, who first proved the law of \index{Cubic reciprocity}cubic reciprocity;
discussed the \index{Residues, theory of}theory of residues; and, in his \emph{Canon Arithmeticus},
gave a table of residues of prime roots. Dirichlet also paid
some attention to this subject.

\index{Eisenstein}\nblabel{png473b}\nbparagraph{Eisenstein.\footnote
  {For a sketch of Eisenstein's life and researches see \emph{Abhandlungen zur
  Geschichte der Mathematik}, 1895, p.~143 \emph{et seq}.}}
The subject was next taken up by \emph{Ferdinand
Gotthold Eisenstein}, a professor at the University of Berlin, who
was born at Berlin on April~16, 1823, and died there on
October~11, 1852. \index{Forms in algebra!in theory of numbers|(}The solution of the problem of the representation
of numbers by binary quadratic forms is one of
the great achievements of Gauss, and the fundamental principles
upon which the treatment of such questions rest were given by
him in the \emph{Disquisitiones Arithmeticae}. Gauss there added
some results relating to ternary quadratic forms, but the general
extension from two to three indeterminates was the work of
Eisenstein, who, in his memoir \emph{Neue Theoreme der höheren
Arithmetik}, defined the ordinal and generic characters of ternary
quadratic forms of an uneven \index{Determinants}determinant; and, in the case of
definite forms, assigned the weight of any order or genus; but
he did not consider forms of an even determinant, nor give any
demonstrations of his work.

Eisenstein also considered the theorems relating to the
possibility of representing a number as a sum of squares, and
showed that the general theorem was limited to eight squares.
The solutions in the cases of two, four, and six squares may
be obtained by means of \index{Elliptic functions}elliptic functions, but the cases in
which the number of squares is uneven involve special processes
peculiar to the theory of numbers. Eisenstein gave the
solution in the case of three squares. He also left a statement
of the solution he had obtained in the case of five squares;\footnote
  {\emph{Crelle's Journal}, vol.~xxxv, 1847, p.~368.}
but his results were published without proofs, and apply only
to numbers which are not divisible by a square.

\index{Smith, Henry|(}\nblabel{png474a}\nbparagraph{Henry Smith.\footnote
  {Smith's collected mathematical works, edited by \index{Glaisher}J.~W.~L.~Glaisher, and
  prefaced by a biographical sketch and other papers, were published in two
  volumes, Oxford, 1894. The following account is extracted from the obituary
  notice in the monthly notices of the Astronomical Society, 1884, pp.~138--149.}}
One of the most original mathematicians
of the school founded by \index{Gauss!ref.\ to}Gauss was Henry Smith. \emph{Henry
John Stephen Smith} was born in London on November~2,
1826, and died at Oxford on February~9, 1883. He was
educated at Rugby, and at Balliol College, Oxford, of which
latter society he was a fellow; and in 1861 he was elected
Savilian professor of Geometry at Oxford, where he resided till
his death.

The subject in connection with which Smith's name is specially
associated is the theory of numbers, and to this he devoted the
years from 1854 to 1864. The results of his historical researches
were given in his report published in parts in the \emph{Transactions}
of the British Association from 1859 to 1865. This report
contains an account of what had been done on the subject to
that time together with some additional matter. The chief
outcome of his own original work on the subject is included
in two memoirs printed in the \emph{Philosophical Transactions} for
1861 and 1867; the first being on linear indeterminate equations
and \index{Congruences}congruences, and the second on the orders and genera of
ternary quadratic forms. In the latter memoir demonstrations
of \index{Eisenstein!ref.\ to}Eisenstein's results and their extension to ternary quadratic
forms of an even determinant were supplied, and a complete
classification of ternary quadratic forms was given.

Smith, however, did not confine himself to the case of three
indeterminates, but succeeded in establishing the principles on
which the extension to the general case of $n$ indeterminates
depends, and obtained the general formulae---thus effecting
the greatest advance made in the subject since the publication
of \index{Gauss!ref.\ to}Gauss's work. In the account of his methods and results
which appeared in the \emph{Proceedings} of the Royal Society,\footnote
  {See vol.~xiii, 1864, pp.~199--203, and vol.~xvi, 1868, pp.~197--208.}
Smith remarked that the theorems relating to the representation of
numbers by four squares and other simple quadratic forms,
are deducible by a uniform method from the principles there
indicated, as also are the theorems relating to the representation
of numbers by six and eight squares. He then proceeded to
say that as the series of theorems relating to the representation
of numbers by sums of squares ceases, for the reason assigned
by \index{Eisenstein!ref.\ to}Eisenstein, when the number of squares surpasses eight, it
was desirable to complete it. The results for even squares were
known. The principal theorems relating to the case of five
squares had been given by Eisenstein, but he had considered only
those numbers which are not divisible by a square, and he had
not considered the case of seven squares. Smith here completed
the enunciation of the theorems for the case of five squares, and
added the corresponding theorems for the case of seven squares.

This paper was the occasion of a dramatic incident in the
history of mathematics. Fourteen years later, in ignorance of
Smith's work, the demonstration and completion of Eisenstein's
theorems for five squares were set by the \index{Academy,!the French}\index{French Academy}French Academy as
the subject of their ``Grand prix des sciences mathématiques.''
Smith wrote out the demonstration of his general theorems so
far as was required to prove the results in the special case of
five squares, and only a month after his death, in March 1883,
the prize was awarded to him, another prize being also awarded
to \index{Minkowski, H.}H.~Minkowski of Bonn. No episode could bring out in a
more striking light the extent of Smith's researches than that
a question, of which he had given the solution in 1867, as a
corollary from general formulae which governed the whole class
of investigations to which it belonged, should have been regarded
by the French Academy as one whose solution was of such
difficulty and importance as to be worthy of their great prize.
It has been also a matter of comment that they should have
known so little of contemporary English and German researches
on the subject as to be unaware that the result of the problem
they were proposing was then lying in their own library.\index{Smith, Henry|)}

\index{Glaisher}J.~W.~L.~Glaisher of Cambridge has recently extended\footnote
  {For a summary of his results see his paper in the \emph{Proceedings of the
  \index{London Mathematical Society}London Mathematical Society}, 1907, vol.~v, second series, pp.~479--490.}
these results, and investigated, by the aid of \index{Elliptic functions}elliptic functions, the
number of representations of a number as the sum of $2n$ squares
where $n$ is not greater than 9.

Among Smith's other investigations I may specially mention
his geometrical memoir, \emph{Sur quelques problèmes cubiques et
biquadratiques}, for which in 1868 he was awarded the \index{Steiner!ref.\ to}Steiner
prize of the Berlin Academy. In a paper which he contributed
to the \emph{Atti} of the Accademia dei Lincei for 1877 he established
a very remarkable analytical relation connecting the modular
equation of order $n$, and the theory of binary quadratic forms
belonging to the positive determinant~$n$. In this paper the
modular curve is represented analytically by a curve in such a
manner as to present an actual geometrical image of the complete
systems of the reduced quadratic forms belonging to the determinant,
and a geometrical interpretation is given to the ideas of
``class,'' ``equivalence,'' and ``reduced form.'' He was also the
author of important papers in which he succeeded in extending
to complex quadratic forms many of \index{Gauss!ref.\ to}Gauss's investigations
relating to real quadratic forms. He was led by his researches
on the theory of numbers to the theory of \index{Elliptic functions}elliptic functions, and
the results he arrived at, especially on the theories of the \index{Theta functions}theta
and \index{Omega function}omega functions, are of importance.

\index{Kummer}\nblabel{png476a}\nbparagraph{Kummer.} The theory of \index{Primes!distribution of}primes received a somewhat unexpected
development by E.~E.~Kummer of Berlin, who was
born in 1810 and died in 1893. In particular he treated
higher complex members of the form $a + \sum bj$, where $j$ is a complex
root of $j^p - 1 = 0$, $p$ being a prime. His theory brought out
the unexpected result that the proposition that a number can be
resolved into the product of powers of primes in one and only
one way is not necessarily true of every \index{Complex numbers}complex number. This
led to the theory of ideal primes, a theory which was developed
later by \index{Dedekind, J. W. R., ref.\ to}J.~W.~R.~Dedekind. Kummer also extended \index{Gauss!ref.\ to}Gauss's
theorems on \index{Quadratic residues}quadratic residues to residues of a higher order,
and wrote on the transformations of \index{Hyperg1@Hyper-geometric functions}hypergeometric functions.

\nblabel{png477a}The theory of numbers, as treated to-day, may be said to
originate with Gauss. I have already mentioned very briefly
the investigations of \index{Dirichlet, Lejeune!ref.\ to}\index{Jacobi!ref.\ to}\emph{Jacobi, Dirichlet, \index{Eisenstein!ref.\ to}Eisenstein, \index{Smith, Henry!ref.\ to}Henry Smith},
and \index{Kummer!ref.\ to}\emph{Kummer}. I content myself with adding some notes on the
subsequent development of certain branches of the theory.\footnote
  {See \index{Smith, H. J. S.}H.~J.~S.~Smith, \emph{Report on the Theory of Numbers} in vol.~i of his
  works, and \index{Stolz, O.}O.~Stolz, \emph{Groessen und Zahlen}, Leipzig, 1891.}

{\nbstretchyspace \index{Tchebycheff}The distribution of primes has been discussed in particular by
\emph{P.~L.\allowbreak~Tchebycheff}\footnote
  {Tchebycheff's collected works, edited by \index{Markoff on Tchebycheff}H.~Markoff and \index{Sonin on Tchebycheff}N.~Sonin, have
  been published in two volumes. A French translation was issued 1900, 1907.}
(1821--1894) of Petrograd, \index{Riemann!ref.\ to}\emph{G.~F.~B.~Riemann},
and \index{Sylvester!ref.\ to}\emph{J.~J.~Sylvester}.} Riemann's short tract on the number of
primes which lie between two given numbers affords a striking
instance of his analytical powers. \index{Legendre!ref.\ to}Legendre had previously
shown that the number of primes less than $n$ is approximately
$n/(\log_e n - 1.08366)$; but Riemann went farther, and this tract
and a memoir by Tchebycheff contain nearly all that has been
done yet in connection with a problem of so obvious a character,
that it has suggested itself to all who have considered the theory
of numbers, and yet which overtaxed the powers even of \index{Lagrange!ref.\ to}Lagrange
and Gauss. In this paper also Riemann stated that all the roots
of $\Gamma(\frac{1}{2}s + 1)(s - 1)\pi^{-\delta/2}\zeta(s)$ are of the form $\frac{1}{2} + it$ where $t$ is
real. It is believed that the theorem is true, but as yet it has
defied all attempts to prove it. Riemann's work in this connection
has proved the starting-point for researches by \index{Hadamard, J. S.}J.~S.~Hadamard,
\index{Mangoldt, H. C. F. von}H.~C.~F.~von Mangoldt, and other recent writers.

The partition of numbers, a problem to which \index{Euler!ref.\ to}Euler had
paid considerable attention, has been treated by \index{Cayley!ref.\ to}\emph{A.~Cayley,
\index{Sylvester!ref.\ to}J.~J.~Sylvester}, and \index{MacMahon, P. A.}\emph{P.~A.~MacMahon}. The representation of
numbers in special forms, the possible divisors of numbers
of specified forms, and general theorems concerned with the
divisors of numbers, have been discussed by \index{Liouville, J.}\emph{J.~Liouville}
(1809--1882), the editor from 1836 to 1874 of the well-known
mathematical journal, and by \index{Glaisher}\emph{J.~W.~L.~Glaisher} of Cambridge.
The subject of quadratic binomials has been studied by \index{Cauchy!ref.\ to}\emph{A.~L.~Cauchy};
of ternary and quadratic forms by \index{Kronecker, L.}\emph{L.~Kronecker}\footnote
  {See the \emph{Bulletin} of the New York (American) Mathematical Society, vol.~i,
  1891--2, pp.~173--184.}
(1823--1891) of Berlin; and of ternary forms by \index{Hermite!ref.\ to}\emph{C.~Hermite}
of Paris.

The most common text-books are, perhaps, that by \index{Stolz, O.}O.~Stolz
of Innspruck, %[**TN: old alternative spelling of Innsbruck]
Leipzig, 1885--6; that by \index{Mathews, G. B., on numbers}G.~B.~Mathews, Cambridge,
1892; that by E.~Lucas, Paris, 1891; and those by
\index{Bachmann, P.}P.~Bachmann, Leipzig, 1892--1905. Possibly it may be found
hereafter that the subject is approached better on other lines
than those now usual.

The conception of \emph{Number} has also been discussed at
considerable length during the last quarter of the nineteenth
century. \index{Numbers,!transcendent}Transcendent numbers had formed the subject of two
memoirs by Liouville, but were subsequently treated as a
distinct branch of mathematics, notably by L.~Kronecker and
\index{Cantor, G.}G.~Cantor. \index{Numbers,!irrational}Irrational numbers and the nature of numbers
have also been treated from first principles, in particular by
\index{Weierstrass, K.!ref.\ to}K.~Weierstrass, \index{Dedekind, J. W. R., ref.\ to}J.~W.~R.~Dedekind,\footnote
  {Dedekind's Essays may serve as an introduction to the subject. They
  have been translated into English, Chicago, 1901.}
\index{Meray@Méray, H. C. R.}H.~C.~R.~Méray, G.~Cantor,
\index{Peano, G.}G.~Peano, and \index{Russell, B. A. W.}B.~A.~W.~Russell. This subject has attracted much
attention of late years, and is now one of the most flourishing
branches of modern mathematics. Transfinite, cardinal, and
ordinal arithmetic, and the theory of sets of points, may be
mentioned as prominent divisions. The theory of aggregates is
related to this subject, and has been treated by G.~Cantor, \index{Bois-Raymond, P. du}P.~du
Bois-Raymond, \index{Schonflies@Schönflies A.}A.~Schönflies, \index{Zermelo, E.}E.~Zermelo, and B.~A.~W.~Russell.\index{Numbers, theory of.!Treatment of, by Pythagoras!q7@by Gauss and other mathematicians of recent times|)}\index{Forms in algebra!in theory of numbers|)}

\tbskip
\index{Elliptic functions}\index{Abelian functions}\nbtopictitle{Elliptic and Abelian Functions, \emph{or} Higher Trigonometry.\footnote
  {See the introduction to \emph{Elliptische Functionen}, by \index{Enneper, A., ref.\ to}A.~Enneper, second
  edition (ed.\ by \index{Muller2@Müller, F.}F.~Müller), Halle, 1890; and \emph{Geschichte der Theorie der elliptischen
  Transcendenten}, by \index{Konigsberger@Königsberger, L.}L.~Königsberger, Leipzig, 1879. On the history
  of Abelian functions see the \emph{Transactions of the British Association}, vol.~lxvii,
  London, 1897, pp.~246--286.}}
\nblabel{png479a}The theory of functions of double and multiple periodicity
is another subject to which much attention has been paid during
this century. I have already mentioned that as early as 1808
\index{Gauss!ref.\ to}Gauss had discovered the \index{Theta functions}theta functions and some of their
properties, but his investigations remained for many years concealed
in his note-books; and it was to the researches made %[**TN: hyphen added to 'notebooks' for consistency]
between 1820 and 1830 by \index{Abel!ref.\ to}Abel and \index{Jacobi!ref.\ to}Jacobi that the modern
development of the subject is due. Their treatment of it has
completely superseded that used by \index{Legendre!ref.\ to}Legendre, and they are
justly reckoned as the creators of this branch of mathematics.

\index{Abel|(}\nblabel{png479b}\nbparagraph{Abel.\footnote
  {The life of Abel by \index{Bjerknes on Abel}C.~A.~Bjerknes was published at Stockholm in 1880,
  and another by \index{Pesloüan, L. de}L.~de~Pesloüan at Paris in 1906. Two editions of Abel's
  works have been published, of which the last, edited by \index{Sylow and Lie on Abel}Sylow and \index{Lie!ref.\ to}Lie, and
  issued at Christiania in two volumes in 1881, is the more complete. See also
  the Abel centenary volume, Christiania, 1902; and a memoir by \index{Mittag-Leffler}G.~Mittag-Leffler.}}
\emph{Niels Henrick Abel} was born at Findoe, in Norway,
on August~5, 1802, and died at Arendal on April~6, 1829, at
the age of twenty-six. His memoirs on elliptic functions,
originally published in \emph{Crelle's Journal} (of which he was one of
the founders), treat the subject from the point of view of the
theory of equations and algebraic forms, a treatment to which
his researches naturally led him.

The important and very general result known as Abel's
theorem, which was subsequently applied by \index{Riemann!ref.\ to}Riemann to the
theory of transcendental functions, was sent to the French
Academy in 1826, but was not printed until 1841: its publication
then was due to inquiries made by Jacobi, in consequence
of a statement on the subject by \index{Holmboe on Abel}B.~Holmboe in his edition of
Abel's works issued in 1839. It is far from easy to state Abel's
theorem intelligently and yet concisely, but, broadly speaking,
it may be described as a theorem for evaluating the sum of a
number of integrals which have the same integrand, but different
limits---these limits being the roots of an algebraic equation.
The theorem gives the sum of the integrals in terms of the constants
occurring in this equation and in the integrand. We
may regard the inverse of the integral of this integrand as a
new transcendental function, and if so the theorem furnishes a
property of this function. For instance, if \index{Abel's theorem}Abel's theorem be
applied to the integrand $(1 - x^2)^{-1/2}$ it gives the \index{Trigonometry, addition formulae}addition theorem
for the circular (or \index{Trigonometrical functions}trigonometrical) functions.

The name of \index{Abelian functions}Abelian function has been given to the higher
transcendents of multiple periodicity which were first discussed
by Abel. The Abelian functions connected with a curve $f(x, y)$
are of the form $\int u dx$ where $u$ is a rational function of $x$ and $y$.
The theory of Abelian functions has been studied by a very large
number of modern writers.

Abel criticised the use of \index{Infinite series, difficulties in connection with}infinite series, and discovered the
well-known theorem which furnishes a test for the validity of
the result obtained by multiplying one infinite series by
another. He also proved\footnote
  {See Abel, \emph{\OE{}uvres}, 1881, vol.~i, pp.~219--250; and \index{Barnes, E. W.}E.~W.~Barnes,
  \emph{Quarterly Journal of Mathematics}, vol.~xxxviii, 1907, pp.~108--116.}
the binomial theorem for the
\index{Expansions@Expansion in series}expansion of $(1 + x)^n$ when $x$ and $n$ are complex. As
illustrating his fertility of ideas I may, in passing, notice his
celebrated demonstration that it is impossible to express a root
of the general \index{Quintic equation}quintic equation in terms of its coefficients by
means of a finite number of radicals and rational functions; this
theorem was the more important since it definitely limited a field
of mathematics which had previously attracted numerous writers.
I should add that this theorem had been enunciated as early as
1798 by \index{Ruffini}Paolo Ruffini, an Italian physician practising at
Modena; but I believe that the proof he gave was deficient in
generality.\index{Abel|)}

\index{Jacobi|(}\nblabel{png480a}\nbparagraph{Jacobi.\footnote
  {See \index{Gerhardt, ref.\ to}C.~J.~Gerhardt's \emph{Geschichte der Mathematik in Deutschland}, Munich,
  1877. Jacobi's collected works were edited by \index{Dirichlet, Lejeune!ref.\ to}Dirichlet, three volumes, Berlin,
  1846--71, and accompanied by a biography, 1852; a new edition, under the
  supervision of \index{Borchardt!ref.\ to}C.~W.~Borchardt and \index{Weierstrass, K.!ref.\ to}K.~Weierstrass, was issued at Berlin in
  seven volumes, 1881--91. See also \index{Konigsberger@Königsberger, L.}L.~Königsberger's \emph{C.~G.~J.~Jacobi}, Leipzig,
  1904.}}
\emph{Carl Gustav Jacob Jacobi}, born of Jewish parents
at Potsdam on Dec.~10, 1804, and died at Berlin on Feb.~18,
1851, was educated at the University of Berlin, where he obtained
the degree of Doctor of Philosophy in 1825. In 1827 he
became extraordinary professor of Mathematics at Königsberg,
and in 1829 was promoted to be an ordinary professor. This
chair he occupied till 1842, when the Prussian Government gave
him a pension, and he moved to Berlin, where he continued to
live till his death in 1851. He was the greatest mathematical
teacher of his generation, and his lectures, though somewhat
unsystematic in arrangement, stimulated and influenced the
more able of his pupils to an extent almost unprecedented at
the time.

Jacobi's most celebrated investigations are those on \index{Elliptic functions}elliptic
functions, the modern notation in which is substantially due to
him, and the theory of which he established simultaneously with \index{Abel!ref.\ to}Abel, but independently of him. Jacobi's results are given in
his treatise on elliptic functions, published in 1829, and in some
later papers in \emph{Crelle's Journal}; they are earlier than
\index{Weierstrass, K.!ref.\ to}Weierstrass's researches which are mentioned below. The correspondence
between \index{Legendre!ref.\ to}Legendre and Jacobi on elliptic functions has been
reprinted in the first volume of Jacobi's collected works. Jacobi,
like \index{Abel!ref.\ to}Abel, recognised that elliptic functions were not merely a
group of theorems on integration, but that they were types
of a new kind of function, namely, one of double periodicity;
hence he paid particular attention to the theory of the \index{Theta functions}theta
function. The following passage,\footnote
  {See Jacobi's collected works, vol.~i, 1881, p.~87.}
in which he explains
this view, is sufficiently interesting to deserve textual reproduction:---

\begin{nbQuote}
E quo, cum universam, quae fingi potest, amplectatur periodicitatem
analyticam elucet, functiones ellipticas non aliis adnumerari debere
transcendentibus, quae quibusdam gaudent elegantiis, fortasse pluribus
illas aut maioribus, sed speciem quandam iis inesse perfecti et absoluti.
\end{nbQuote}

Among Jacobi's other investigations I may specially single
out his papers on \index{Determinants}Determinants, which did a great deal to bring
them into general use; and particularly his introduction of the
\index{Jacobians}Jacobian, that is, of the functional determinant formed by the
$n^{2}$ partial differential coefficients of the first order of $n$ given
functions of $n$ independent variables. I ought also to mention
his papers on Abelian transcendents; his investigations on the
theory of numbers, to which I have already alluded; his important
memoirs on the theory of \index{Differential equations}\index{Equations, differential}differential equations, both
ordinary and partial; his development of the \index{Calculus of operations!of variations}\index{Variations, calculus of}calculus of variations;
and his contributions to the \index{Three bodies, problem of}problem of three bodies,
and other particular dynamical problems. Most of the results of
the researches last named are included in his \emph{Vorlesungen über
Dynamik}.\index{Jacobi|)}

\index{Riemann|(}\nblabel{png482a}\nbparagraph{Riemann.\footnote
  {Riemann's collected works, edited by \index{Weber, H.}H.~Weber and prefaced by an
  account of his life by \index{Dedekind, J. W. R., ref.\ to}Dedekind, were published at Leipzig, second edition,
  1892; an important supplement, edited by \index{Nother@Nöther, M.}M.~Nöther and \index{Wirtinger, W.}W.~Wirtinger, was
  issued in 1902. His lectures on \index{Elliptic functions}elliptic functions, edited by \index{Stahl, H. B. L.}H.~B.~L.~Stahl,
  were published separately, Leipzig, 1899. Another short biography of
  Riemann has been written by \index{Schering, ref.\ to}E.~J.~Schering, Göttingen, 1867.}}
\emph{Georg Friedrich Bernhard Riemann} was born
at Breselenz on Sept.~17, 1826, and died at Selasca on July~20,
1866. He studied at Göttingen under \index{Gauss!ref.\ to}Gauss, and subsequently
at Berlin under \index{Jacobi!ref.\ to}Jacobi, \index{Dirichlet, Lejeune!ref.\ to}Dirichlet, \index{Steiner!ref.\ to}Steiner, and \index{Eisenstein!ref.\ to}Eisenstein, all
of whom were professors there at the same time. In spite of
poverty and sickness he struggled to pursue his researches. In
1857 he was made professor at Göttingen, general recognition
of his powers soon followed, but in 1862 his health began to
give way, and four years later he died, working, to the end,
cheerfully and courageously.

Riemann must be esteemed one of the most profound and
brilliant mathematicians of his time; he was a creative genius.
The amount of matter he produced is small, but its originality
and power are manifest\allowbreak---his investigations on functions and on
geometry, in particular, initiating developments of great importance.

His earliest paper, written in 1850, was on the general theory
of functions of a complex variable. This gave rise to a new method
of treating the theory of functions. The development of this
method is specially due to the Göttingen school. \index{Functions, notation for!theory of}In 1854
Riemann wrote his celebrated memoir on the hypotheses on
which geometry is founded: to this subject I allude below.
This was succeeded by memoirs on \index{Elliptic functions}elliptic functions and on
the \index{Primes!distribution of}distribution of primes: these have been already mentioned.
He also investigated the conformal representation of areas, one
on the other: a problem subsequently treated by \index{Schwarz, H. A.}H.~A.~Schwarz
and \index{Schottky, F. H.}F.~H.~Schottky, both of Berlin. Lastly, in multiple periodic
functions, it is hardly too much to say that in his memoir in
\emph{Borchardt's Journal} for 1857, he did for the \index{Abelian functions}Abelian functions
what \index{Abel!ref.\ to}Abel had done for the elliptic functions. A posthumous
fragment on linear \index{Differential equations}\index{Equations, differential}differential equations with algebraic coefficients
has served as the foundation of important work by \index{Schlesinger, L.}L.~Schlesinger.\index{Riemann|)}

\nblabel{png483a}I have already alluded to the researches of \index{Legendre!ref.\ to}\emph{Legendre}, \index{Gauss!ref.\ to}\emph{Gauss},
\index{Abel!ref.\ to}\emph{Abel}, \index{Jacobi!ref.\ to}\emph{Jacobi}, and \index{Riemann!ref.\ to}\emph{Riemann} on elliptic and Abelian functions.
The subject has been also discussed by (among other writers)
\index{Rosenhain, J. G.}\emph{J.~G.~Rosenhain} (1816--1887) of Königsberg, who wrote (in
1844) on the hyperelliptic, and double \index{Theta functions}theta functions; \index{Gopel, A.}\emph{A.~Göpel}
(1812--1847) of Berlin, who discussed\footnote
  {See \emph{Crelle's Journal}, vol.~xxxv, 1847, pp.~277--312; an obituary notice,
  by Jacobi, is given on pp.~313--317.}
hyperelliptic functions;
\emph{L.~Kronecker}\footnote
  {Kronecker's collected works in four volumes, edited by \index{Hensel, K.}K.~Hensel, are
  now in course of publication at Leipzig, 1895, \&c.}
of Berlin, who wrote on elliptic functions;
\index{Konigsberger@Königsberger, L.}\emph{L.~Königsberger}\footnote%[**TN: 'Königberger' changed to 'Königsberger' for consistency]
  {See Königsberger's lectures, published at Leipzig in 1874.} %[**TN: 'Königberger' changed to 'Königsberger' for consistency]
of Heidelberg and \index{Brioschi, F.}\emph{F.~Brioschi}\footnote
  {His collected works were published in two volumes, Milan, 1901, 1902.}
(1824--1897) of Milan,
both of whom wrote on elliptic and hyperelliptic functions; \index{Smith, Henry!ref.\ to}\emph{Henry
Smith} of Oxford, who discussed the transformation theory, the
theta and \index{Omega function}omega functions, and certain functions of the modulus;
\index{Cayley!ref.\ to}\emph{A.~Cayley} of Cambridge, who was the first to work out (in 1845)
the theory of doubly infinite products and determine their periodicity,
and who has written at length on the connection between
the researches of \index{Legendre!ref.\ to}Legendre and \index{Jacobi!ref.\ to}Jacobi; and \index{Hermite!ref.\ to}\emph{C.~Hermite} of Paris,
whose researches are mostly concerned with the transformation
theory and the higher development of the theta functions.

\index{Weierstrass, K.}\nblabel{png484a}\nbparagraph{Weierstrass.\footnote
  {Weierstrass's collected works are now in course of issue, Berlin, 1894, \&c.
  Sketches of his career by \index{Mittag-Leffler}G.~Mittag-Leffler and \index{Poincaré, H.}H.~Poincaré are given in \emph{Acta
  Mathematica}, 1897, vol.~xxi, pp.~79--82, and 1899, vol.~xxii, pp.~1--18.}}
The subject of higher trigonometry was put
on a somewhat different footing by the researches of Weierstrass.
\emph{Karl Weierstrass}, born in Westphalia on October~31, 1815, and
died at Berlin on February~19, 1897, was one of the greatest
mathematicians of the nineteenth century. He took no part in
public affairs; his life was uneventful; and he spent the last
forty years of it at Berlin, where he was professor.

With two branches of pure mathematics---\index{Elliptic functions}elliptic and \index{Abelian functions}Abelian
functions, and the theory of functions---his name is inseparably
connected. His earlier researches on elliptic functions related
to the theta functions, which he treated under a modified form
in which they are expressible in powers of the modulus. At a
later period he developed a method for treating all elliptic functions
in a symmetrical manner. \index{Jacobi!ref.\ to}Jacobi had shown that a
function of $n$ variables might have $2n$ periods. Accordingly
Weierstrass sought the most general expressions for such functions,
and showed that they enjoyed properties analogous to
those of the hyperelliptic functions. Hence the properties of
the latter functions could be reduced as particular cases of
general results.

{\nbstretchyspace He was naturally led to this method of treating hyperelliptic
functions by his researches on the general theory of functions;
these co-ordinated and comprised various lines of investigation
previously treated independently.} In particular he constructed
a theory of uniform analytic functions. The representation of
functions by infinite products and series also claimed his especial
attention. Besides functions he also wrote or lectured on the
nature of the assumptions made in analysis, on the \index{Calculus of operations!of variations}\index{Variations, calculus of}calculus of
variations, and on the theory of minima surfaces. His methods
are noticeable for their wide-reaching and general character.
Recent investigations on elliptic functions have been largely
based on Weierstrass's method.

\nblabel{png485a}Among other prominent mathematicians who have recently
written on \index{Elliptic functions}elliptic and hyperelliptic functions, I may mention
the names of \index{Halphen, G. H.}\emph{G.~H.~Halphen}\footnote
  {See Halphen's collected works, 3 vols., Paris, 1916, 1918, 1921. A
  sketch of his life and work is given in \index{Liouville, J.}\emph{Liouville's Journal} for 1889, pp.~345--359,
  and in the \emph{Comptes Rendus}, 1890, vol.~cx, pp.~489--497.}
(1844--1889), an officer in the
French army, whose investigations were largely founded on
\index{Weierstrass, K.!ref.\ to}Weierstrass's work; \index{Klein, F. C.}\emph{F.~C.~Klein} of Göttingen, who has written
on Abelian functions, elliptic modular functions, and hyperelliptic
functions; \index{Schwarz, H. A.}\emph{H.~A.~Schwarz} of Berlin; \index{Weber, H.}\emph{H.~Weber} of Strassburg;
\index{Nother@Nöther, M.}\emph{M.~Nöther} of Erlangen; \index{Stahl, H. B. L.}\emph{H.~B.~L.~Stahl} of Tübingen; \index{Frobenius}\emph{F.~G.~Frobenius}
of Berlin; \index{Glaisher}\emph{J.~W.~L. Glaisher} of Cambridge, who has
in particular developed the theory of the \index{Zeta function}zeta function; and
\index{Baker, H. F.!ref.\ to}\emph{H.~F.~Baker} of Cambridge.

\index{Functions, notation for!theory of}The usual text-books of to-day on elliptic functions are
those by \index{Tannery, J., on elliptic functions}\index{Tannery, S. P., ref.\ to}J.~Tannery and \index{Molk on elliptic functions}J.~Molk, 4 volumes, Paris, 1893--1901;
by \index{Appell, P. E.}P.~E.~Appell and \index{Lacour on elliptic functions}E.~Lacour, Paris, 1896; by H.~Weber,
Brunswick, 1891; and by G.~H.~Halphen, 3 volumes,
Paris, 1886--1891. To these I may add one by \index{Greenhill, A. G., on elliptic functions}A.~G.~Greenhill
on the \emph{Applications of Elliptic Functions}, London, 1892.

\tbskip
\nblabel{png485b}\nbtopictitle{The Theory of Functions.} I have already mentioned that
the modern theory of functions is largely due to Weierstrass and
\index{Meray@Méray, H. C. R.}H.~C.~R.~Méray. It is a singularly attractive subject, and has
proved an important and far-reaching branch of mathematics.
Historically its modern presentation may be said to have been
initiated by \index{Cauchy!ref.\ to}\emph{A.~Cauchy}, who laid the foundations of the theory of
synectic functions of a complex variable. Work on these lines
was continued by \emph{J.~Liouville}, who wrote chiefly on doubly periodic
functions. These investigations were extended and connected
in the work by \index{Bouquet, Briot and}\index{Briot and Bouquet}\emph{A.~Briot} (1817--1882), and \emph{J.~C.~Bouquet} (1819--1885),
and subsequently were further developed by \index{Hermite!ref.\ to}\emph{C.~Hermite}.

Next I may refer to the researches on the theory of algebraic
functions which have their origin in \index{Puiseux, V. A.}\emph{V.~A.~Puiseux's} memoir of
1851, and \index{Riemann!ref.\ to}\emph{G.~F.~B.~Riemann's} papers of 1850 and 1857; in continuation
of which \emph{H.~A.~Schwarz} of Berlin established accurately
certain theorems of which the proofs given by Riemann were
open to objection. To Riemann also we are indebted for
valuable work on modular %[**TN: 'nodular' in original text]
functions which has been recently
published in his \emph{Nachträge}. Subsequently \index{Klein, F. C.}\emph{F.~C.~Klein} of
Göttingen connected \index{Riemann!ref.\ to}Riemann's theory of functions with the
theory of groups, and wrote on automorphic and modular
functions; \index{Poincaré, H.}\emph{H.~Poincaré} of Paris also wrote on automorphic
functions, and on the \index{Equations, differential!theory of}general theory with special applications to
\index{Differential equations}\index{Equations, differential}differential equations. Quite recently \index{Hensel, K.}\emph{K.~Hensel} of Marburg
has written on algebraic functions; and \emph{W.~Wirtinger} of Vienna
on \index{Abelian functions}Abelian functions.

I have already said that the work of \index{Weierstrass, K.!ref.\ to}Weierstrass shed a new
light on the whole subject. His theory of analytical functions
has been developed by \index{Mittag-Leffler}\emph{G.~Mittag-Leffler} of Stockholm; and
\index{Hermite!ref.\ to}\emph{C.~Hermite}, \index{Appell, P. E.}\emph{P.~E.~Appell}, \index{Picard, C. E.}\emph{C.~E.~Picard}, \index{Goursat, E., on functions}\emph{E.~Goursat}, \index{Laguerre, E. K}\emph{E.~N.~Laguerre},
and \index{Hadamard, J. S.}\emph{J.~S.~Hadamard}, all of Paris, have also written
on special branches of the general theory; while \emph{E.~Borel},
\index{Baire, R. L.}\emph{R.~L.~Baire}, %[**TN: "Baize" in original index entry]
\index{Lebesgue}\emph{H.~L.~Lebesgue}, and \index{Lindelöf, E. L.}\emph{E.~L.~Lindelöf} have produced
a series of tracts on uniform functions which have had a wide
circulation and influence.

As text-books I may mention the \emph{Theory of Functions of
a Complex Variable}, by \index{Forsyth, A. R.}A.~R.~Forsyth, second edition, Cambridge,
1900; \index{Abel's theorem}\emph{Abel's Theorem} by \index{Baker, H. F.}H.~F.~Baker, Cambridge, 1897,
and \emph{Multiple Periodic Functions} by the same writer, Cambridge,
1907; the \emph{Théorie des fonctions algébriques} by P.~E.~Appell
and E.~Goursat, Paris, 1895; parts of C.~E.~Picard's \emph{Traité
d'Analyse}, in 3 volumes, Paris, 1891 to 1896; the \emph{Theory of
Functions} by \index{Harkness, J., on functions}J.~Harkness and \index{Morley, F., on functions}F.~Morley, London, 1893; the
\emph{Theory of Functions of a Real Variable and of Fourier's
Series} by \index{Hobson, E. W.}E.~W.~Hobson, Cambridge, 1907; and \emph{Die Theorie
des Abel'schen Functionen} by \index{Stahl, H. B. L.}H.~B.~L.~Stahl, Leipzig, 1896.

\tbskip
\index{Algebra.!z9@Recent extensions of|(}\index{Algebraa@Algebra,!higher|(}\nblabel{png486a}\nbtopictitle{Higher Algebra.} The \index{Numbers, theory of.!Treatment of, by Pythagoras!q7@by Gauss and other mathematicians of recent times}theory of numbers may be considered
as a higher arithmetic, and the theory of elliptic and Abelian
functions as a higher trigonometry. The theory of higher
algebra (including the theory of equations) has also attracted
considerable attention, and was a favourite subject of study of
the mathematicians whom I propose to mention next, though
the interests of these writers were by no means limited to this
subject.

\index{Cauchy|(}\nblabel{png487a}\nbparagraph{Cauchy.\footnote
  {See \emph{La Vie et les travaux de Cauchy} by \index{Valson, ref.\ to}L.~Valson, two volumes, Paris,
  1868. A complete edition of his works is now being issued by the French
  Government.}}
\emph{Augustin Louis Cauchy}, the leading representative
of the French school of analysis in the nineteenth century,
was born at Paris on Aug.~21, 1789, and died at Sceaux on
May~25, 1857. He was educated at the Polytechnic school, the
nursery of so many French mathematicians of that time, and
adopted the profession of a civil engineer. His earliest mathematical
paper was one on polyhedra in 1811. \index{Legendre!ref.\ to}Legendre thought
so highly of it that he asked Cauchy to attempt the solution of
an analogous problem which had baffled previous investigators,
and his choice was justified by the success of Cauchy in 1812.
Memoirs on analysis and the \index{Numbers, theory of.!Treatment of, by Pythagoras!q7@by Gauss and other mathematicians of recent times}theory of numbers, presented in
1813, 1814, and 1815, showed that his ability was not confined
to geometry alone. In one of these papers he generalised some
results which had been established by \index{Gauss!ref.\ to}Gauss and Legendre; in
another of them he gave a theorem on the number of values
which an algebraical function can assume when the literal
constants it contains are interchanged. It was the latter
theorem that enabled \index{Abel!ref.\ to}Abel to show that in general an algebraic
equation of a degree higher than the fourth cannot be solved by
the use of a finite number of purely algebraical expressions.

To \index{Abel!ref.\ to}Abel, Cauchy, and Gauss we owe the scientific treatment
of \index{Infinite series, difficulties in connection with}series which have an infinite number of terms. \index{Expansions@Expansion in series}In particular,
Cauchy established general rules for investigating the \index{Convergency}convergency
and divergency of such series, rules which were extended
by \index{Bertrand}J.~L.~F.~Bertrand (1822--1900) of Paris, Secretary of the
French Académie des Sciences, \index{Pringsheim}A.~Pringsheim of Munich,
and considerably amplified later by \index{Borel, E.}E.~Borel, by \index{Servant, M. G.}M.~G.~Servant,
both of Paris, and by other writers of the modern
French school. In only a few works of an earlier date
is there any discussion as to the limitations of the series
employed. It is said that \index{Laplace!ref.\ to}Laplace, who was present when
Cauchy read his first paper on the subject, was so impressed
by the illustrations of the danger of employing such
series without a rigorous investigation of their \index{Convergency}convergency,
that he put on one side the work on which he was then
engaged and denied himself to all visitors, in order to see
if any of the demonstrations given in the earlier volumes of the
\emph{Mécanique céleste} were invalid; and he was fortunate enough to
find that no material errors had been thus introduced. The
treatment of series and of the fundamental conceptions of the
calculus in most of the text-books then current was based on
\index{Euler!ref.\ to}Euler's works, and was not free from objection. It is one
of the chief merits of Cauchy that he placed these subjects
on a stricter foundation.\index{Algebraa@Algebra,!higher|)}

On the restoration in 1816 the French Academy was
purged, and, incredible though it may seem, Cauchy accepted
a seat procured for him by the expulsion of \index{Monge!ref.\ to}Monge. He
was also at the same time made professor at the Polytechnic;
and his lectures there on algebraic analysis, the calculus, and
the theory of curves, were published as text-books. On the
revolution in 1830 he went into exile, and was first appointed
professor at Turin, whence he soon moved to Prague to
undertake the education of the \index{Chambord, Comte de, ref.\ to}Comte de Chambord. He
returned to France in 1837; and in 1848, and again in 1851,
by special dispensation of the \index{Napoleon III.}Emperor was allowed to occupy
a chair of mathematics without taking the oath of allegiance.

His activity was prodigious, and from 1830 to 1859 he
published in the \emph{Transactions} of the Academy, or the \emph{Comptes
Rendus}, over 600 original memoirs and about 150 reports.
They cover an extraordinarily wide range of subjects, but are of
very unequal merit.

Among the more important of his other researches are those
on the legitimate use of \index{Imaginary numbers}\index{Imaginary quantities}imaginary quantities; the determination
of the \index{Equations, differential!number of roots}\index{Roots of equations,!number of}number of real and imaginary roots of any algebraic
equation within a given contour; his method of calculating
these roots approximately; his theory of the \index{Roots of equations,!symmetrical functions of}\index{Symmetrical functions of roots of an equation}symmetric functions
of the coefficients of equations of any degree; his \emph{a priori}
valuation of a quantity less than the least difference between the
roots of an equation; his papers on \index{Determinants}determinants in 1841, which
assisted in bringing them into general use; and his investigations
on the \index{Numbers, theory of.!Treatment of, by Pythagoras!q7@by Gauss and other mathematicians of recent times}theory of numbers. Cauchy also did something to
reduce the art of determining definite integrals to a science;
the rule for finding the principal values of integrals was
enunciated by him. The calculus of residues was his invention.
His proof of \index{Taylors@Taylor's theorem}Taylor's theorem seems to have originated from a
discussion of the double periodicity of \index{Elliptic functions}elliptic functions. The
means of showing a connection between different branches of a
subject by giving complex values to independent variables is
largely due to him.

He also gave a direct analytical method for determining
planetary inequalities of long period. To physics he contributed
memoirs on waves and on the quantity of light
reflected from the surfaces of metals, as well as other papers
on optics.\index{Cauchy|)}

\index{Argand, J. R.}\nblabel{png489a}\nbparagraph{Argand.} I may mention here the name of \emph{Jean Robert
Argand}, who was born at Geneva on July~18, 1768, and
died at Paris on August~13, 1822. In his \emph{Essai}, issued
in 1806, he gave a geometrical representation of a \index{Complex numbers}\index{Imaginary numbers}complex
number, and applied it to show that every \index{Roots of equations,!imaginary}algebraic equation
has a root. This was prior to the memoirs of \index{Gauss!ref.\ to}Gauss and
\index{Cauchy!ref.\ to}Cauchy on the same subject, but the essay did not attract
much attention when it was first published. An even
earlier demonstration that $\surd(-1)$ may be interpreted to
indicate perpendicularity in two-dimensional space, and even
the extension of the idea to three-dimensional space by a
method foreshadowing the use of \index{Quaternions}quaternions, had been given
in a memoir by \index{Wessel}C.~Wessel, presented to the Copenhagen
Academy of Sciences in March 1797; other memoirs on the
same subject had been published in the \emph{Philosophical
Transactions} for 1806, and by \index{Kuhn@Kühn}H.~Kühn in the \emph{Transactions}
for 1750 of the Petrograd Academy.\footnote
  {See \index{Beman, W. W.}W.~W.~Beman in the \emph{Proceedings of the American Association for
  the Advancement of Science}, vol.~xlvi, 1897.}

I have already said that the idea of a \index{Number, simple complex}simple \index{Complex numbers}complex number
like $a + bi$ where $i^{2} = -1$ %[**TN: 'i^{2} = 0' in original text]
was extended by \index{Kummer!ref.\ to}Kummer. The general
theory has been discussed by \index{Weierstrass, K.!ref.\ to}K.~Weierstrass, \index{Schwarz, H. A.}H.~A.~Schwarz
of Berlin, \index{Dedekind, J. W. R., ref.\ to}J.~W.~R.~Dedekind, \index{Poincaré, H.}H.~Poincaré, and other writers.

\index{Hamilton, Sir Wm.|(}\nblabel{png490a}\nbparagraph{Hamilton.\footnote
  {See the life of Hamilton (with a bibliography of his writings) by \index{Graves on Hamilton}R.~P.~Graves,
  three volumes, Dublin, 1882--89; the leading facts are given in an
  article in the \emph{North British Review} for 1886.}}
In the opinion of some writers the theory
of \index{Quaternions}quaternions will be ultimately esteemed one of the great
discoveries of the nineteenth century in pure mathematics. That
discovery is due to \emph{Sir William Rowan Hamilton}, who was
born in Dublin on August~4, 1805, and died there on September~2,
1865. His education, which was carried on at home, seems
to have been singularly discursive. Under the influence of an
uncle who was a good linguist, he first devoted himself to
linguistic studies; by the time he was seven he could read
Latin, Greek, French, and German with facility; and when
thirteen he was able to boast that he was familiar with as many
languages as he had lived years. It was about this time that
he came across a copy of \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's \emph{Universal Arithmetic}. This
was his introduction to modern analysis, and he soon mastered
the elements of analytical geometry and the calculus. He next
read the \index{Newtons@Newton's \emph{Principia}!ref.\ to}\emph{Principia} and the four volumes then published of
\index{Laplace!ref.\ to}Laplace's \emph{Mécanique céleste}. In the latter he detected a mistake,
and his paper on the subject, written in 1823, attracted considerable
attention. In the following year he entered at Trinity
College, Dublin. His university career is unique, for the chair of
Astronomy becoming vacant in 1827, while he was yet an undergraduate,
he was asked by the electors to stand for it, and was
elected unanimously, it being understood that he should be left
free to pursue his own line of study.

\index{Optics (geometrical).!Discussed by (among others) Euclid!z9@and Sir William Hamilton}His earliest paper on optics, begun in 1823, was published
in 1828 under the title of a \emph{Theory of Systems of
Rays}, to which two supplements were afterwards added; in
the latter of these the phenomenon of conical \index{Refraction}refraction is predicted.
This was followed by a paper in 1827 on the principle
of \emph{Varying Action}, and in 1834 and 1835 by memoirs on
a \emph{General Method in Dynamics}---the subject of theoretical
dynamics being properly treated as a branch of pure mathematics.
His lectures on \index{Quaternions}\emph{Quaternions} were published in 1852.
Some of his results on this subject would seem to have
been previously discovered by \index{Gauss!ref.\ to}Gauss, but these were unknown
and unpublished until long after Hamilton's death. Amongst
his other papers, I may specially mention one on the
form of the solution of the general algebraic equation of the
fifth degree, which confirmed \index{Abel!ref.\ to}Abel's conclusion that it cannot
be expressed by a finite number of purely algebraical expressions;
one on fluctuating functions; one on the \index{Hodograph}hodograph;
and, lastly, one on the numerical solution of differential
equations. His \emph{Elements of Quaternions} was issued in
1866: of this a competent authority says that the methods
of analysis there given show as great an advance over those of
analytical geometry, as the latter showed over those of Euclidean
geometry. In more recent times the subject has been further
developed by \index{Tait}P.~G.~Tait (1831--1901) of Edinburgh, by \index{Macfarlane, A.}A.~Macfarlane
of America, and by \index{Joly, C. J., on quaternions}C.~J.~Joly in his \emph{Manual of
Quaternions}, London, 1905.

Hamilton was painfully fastidious on what he published, and
he left a large collection of manuscripts which are now in the
library of Trinity College, Dublin, some of which it is to be
hoped will be ultimately printed.\index{Hamilton, Sir Wm.|)}

\index{Grassmann}\nblabel{png491a}\nbparagraph{Grassmann.\footnote
  {Grassmann's collected works in three volumes, edited by \index{Engel, F., on Grassmann}F.~Engel, are
  now in course of issue at Leipzig, 1894, \&c.}}
The idea of non-commutative algebras and of
quaternions seems to have occurred to Grassmann and \index{Boole, G.!ref.\ to}Boole at
about the same time as to \index{Hamilton, Sir Wm.!ref.\ to}Hamilton. \emph{Hermann Günther Grassmann}
was born in Stettin on April~15, 1809, and died there in
1877. He was professor at the gymnasium at Stettin. His
researches on non-commutative algebras are contained in his
\emph{Ausdehnungslehre}, first published in 1844 and enlarged in 1862.
This work has had great influence, especially on the continent,
where Grassmann's methods have generally been followed in
preference to Hamilton's. Grassmann's researches have been
continued and extended, notably by \index{Schlegel S. F. V.}S.~F.~V.~Schlegel and \index{Peano, G.}G.~Peano.

The scientific treatment of the fundamental principles of
algebra initiated by Hamilton and Grassmann was continued by
\index{Demorgana@De~Morgan, A.!ref.\ to}De~Morgan and \index{Boole, G.!ref.\ to}Boole in England, and was further developed
by \index{Hankel, ref.\ to}H.~Hankel (1839--1873) in Germany in his work on complexes,
1867, and, on somewhat different lines, by \index{Cantor, G.}G.~Cantor in
his memoirs on the theory of irrationals, 1871; the discussion
is, however, so technical that I am unable to do more than allude
to it. Of Boole and De~Morgan I say a word or two in passing.

\nblabel{png492a}\index{Boole, G.}\nbparagraph{Boole.} \emph{George Boole}, born at Lincoln on November~2, 1815,
and died at Cork on December~8, 1864, independently invented a
system of non-commutative algebra, and was one of the creators
of \index{Symbolic and mathematical logic}symbolic or mathematical logic.\footnote
  {On the history of mathematical logic, see \index{Jourdain, P. E. B.}P.~E.~B.~Jourdain, \emph{Quarterly
  Journal of Mathematics}, vol.~xliii, 1912, pp.~219--314.}
From his memoirs on
linear transformations part of the theory of invariants has
developed. His \emph{Finite Differences} remains a standard work on
that subject.

\index{Demorgana@De~Morgan, A.}\nbparagraph{De~Morgan.\footnote
  {De~Morgan's life was written by his widow, \index{Demorgans@De~Morgan, S. E.}S.~E.~de~Morgan, London,
  1882.}}
\emph{Augustus de~Morgan}, born in Madura
(Madras) in June 1806, and died in London on March~18,
1871, was educated at Trinity College, Cambridge. In 1828
he became professor at the then newly-established University
of London (University College). There, through his works
and pupils, he exercised a wide influence on English mathematicians.
He was deeply read in the philosophy and
history of mathematics, but the results are given in scattered
articles; of these I have made considerable use in this book.
His memoirs on the foundation of algebra; his treatise on the
differential calculus published in 1842, a work of great ability,
and noticeable for his \index{Infinite series, difficulties in connection with}treatment of \index{Expansions@Expansion in series}infinite series; and his
articles on the calculus of functions and on the \index{Probabilities, theory of}theory of
probabilities, are worthy of special note. The article on the
calculus of functions contains an investigation of the principles
of symbolic reasoning, but the applications deal with the solution
of functional equations rather than with the general theory of
functions.

\index{Galois}\nblabel{png493a}\nbparagraph{Galois.\footnote
  {On Galois's investigations, see the edition of his works with an introduction
  by \index{Picard, E.}E.~Picard, Paris, 1897.}}
A new development of algebra---the \index{Groups, theories of}theory of
groups of substitutions---was suggested by \emph{Evariste Galois}, who
promised to be one of the most original mathematicians of the
nineteenth century, born at Paris on October~26, 1811, and
killed in a duel on May~30, 1832, at the early age of 20.

The theory of groups, and of \index{Invariants}invariant subgroups, has profoundly
modified the treatment of the theory of equations.
An immense literature has grown up on the subject. The
modern theory of groups originated with the treatment by
Galois, \index{Cauchy!ref.\ to}Cauchy, and \index{Serret}J.~A.~Serret (1819--1885), professor at
Paris; their work is mainly concerned with finite discontinuous
substitution groups. This line of investigation has been
pursued by \index{Jordan, C.}M.~E.~C.~Jordan (1838--1922) of Paris and \index{Netto, E.}E.~Netto
of Strassburg. The problem of operations with discontinuous
groups, with applications to the theory of functions, has been
further taken up by (among others) \index{Frobenius}F.~G.~Frobenius of Berlin,
\index{Klein, F. C.}F.~C.~Klein of Göttingen, and \index{Burnside, W.}W.~Burnside formerly of Cambridge
and now of Greenwich.

\index{Cayley}\nblabel{png493b}\nbparagraph{Cayley.\footnote
  {Cayley's collected works in thirteen volumes were issued at Cambridge,
  1889--1898.}}
Another Englishman whom we may reckon
among the great mathematicians of this prolific century was
\emph{Arthur Cayley}. Cayley was born in Surrey, on Aug.~16, 1821,
and after education at Trinity College, Cambridge, was called
to the bar. But his interests centred on mathematics; in 1863
he was elected Sadlerian Professor at Cambridge, and he spent
there the rest of his life. He died on Jan.~26, 1895.

Cayley's writings deal with considerable parts of modern
pure mathematics. I have already mentioned his writings on
the partition of numbers and on \index{Elliptic functions}elliptic functions treated from
\index{Jacobi!ref.\ to}Jacobi's point of view; his later writings on elliptic functions
dealt mainly with the theory of transformation and
the modular equation. It is, however, by his investigations
in analytical geometry and on higher algebra that he will be
best remembered.

In analytical geometry the conception of what is called
(perhaps, not very happily) the \emph{absolute} is due to Cayley. As
stated by himself, the ``theory, in effect, is that the metrical
properties of a figure are not the properties of the figure
considered \emph{per se} \dots but its properties when considered in
connection with another figure, namely, the conic termed the
absolute''; hence metric properties can be subjected to descriptive
treatment. He contributed largely to the general
theory of curves and surfaces, his work resting on the
assumption of the necessarily close connection between algebraical
and geometrical operations.

In higher algebra the theory of \index{Invariants}invariants is due to Cayley;
his ten classical memoirs on binary and ternary forms, and his
researches on matrices and non-commutative algebras, mark an
epoch in the development of the subject.

\index{Sylvester}\nblabel{png494a}\nbparagraph{Sylvester.\footnote
  {Sylvester's collected works, edited by \index{Baker, H. F.!ref.\ to}H.~F.~Baker, are in course of
  publication at Cambridge; 2 volumes are already issued.}}
Another teacher of the same time was \emph{James
Joseph Sylvester}, born in London on Sept.~3, 1814, and died on
March~15, 1897. He too was educated at Cambridge. Like
\index{Cayley!ref.\ to}Cayley, with whom he was on intimate terms of friendship,
he was called to the bar, and yet preserved all his interests
in mathematics. He held professorships successively at
Woolwich, Baltimore, and Oxford. He had a strong personality
and was a stimulating teacher, but it is difficult to describe
his writings, for they are numerous, disconnected, and discursive.

On the \index{Numbers, theory of.!Treatment of, by Pythagoras!q7@by Gauss and other mathematicians of recent times}theory of numbers Sylvester wrote valuable papers
on the \index{Primes!distribution of}distribution of primes and on the partition of numbers.
On analysis he wrote on the calculus and on \index{Equations, differential!theory of}differential
equations. But perhaps his favourite study was higher
algebra, and from his numerous memoirs on this subject I
may in particular single out those on canonical forms, on the
theory of \index{Contravariants}contravariants, on \index{Reciprocants}reciprocants or differential \index{Invariants}invariants,
and on the theory of equations, notably on \index{Newton, Isaac, chapter \textsc{xvi} (see table of contents)!ref.\ to}Newton's
rule. I may also add that he created the language and
\index{Functions, notation for!theory of}notation of considerable parts of those subjects on which he
wrote.

The writings of Cayley and Sylvester stand in marked
contrast: Cayley's are methodical, precise, formal, and complete;
Sylvester's are impetuous, unfinished, but none the
less vigorous and stimulating. Both mathematicians found
the greatest attraction in higher algebra, and to both that
subject in its modern form is deeply indebted.

\index{Lie}\nblabel{png495a}\nbparagraph{Lie.\footnote
  {See the obituary notice by \index{Forsyth, A. R.}A.~R.~Forsyth in the \emph{Year-Book of the
  Royal Society}, London, 1901.}}
Among the great analysts of the nineteenth century
to whom I must allude here, is \emph{Marius Sophus Lie}, born on
Dec.~12, 1842, and died on Feb.~18, 1899. Lie was educated
at Christiania, whence he obtained a travelling scholarship,
and in the course of his journeys made the acquaintance of
\index{Klein, F. C.}Klein, \index{Darboux}Darboux, and \index{Jordan, C.}Jordan, to whose influence his subsequent
career is largely due.

In 1870 he discovered the transformation by which a sphere
can be made to correspond to a straight line, and, by the use
of which theorems on aggregates of lines can be translated into
theorems on aggregates of spheres. This was followed by a
thesis on the theory of tangential transformations for space.

In 1872 he became professor at Christiania. His earliest
researches here were on the relations between \index{Equations, differential!theory of}\index{Differential equations}\index{Equations, differential}differential equations
and infinitesimal transformations. This naturally led him
to the general theory of finite continuous groups of substitutions;
the results of his investigations on this subject are embodied in
his \emph{Theorie der Transformationsgruppen}, Leipzig, three volumes,
1888--1893. He proceeded next to consider the \index{Groups, theories of}theory of
infinite continuous groups, and his conclusions, edited by
G.~Scheffers, were published in 1893. About 1879 Lie
turned his attention to differential geometry; a systematic
exposition of this is in course of issue in his \emph{Geometrie der
Berührungstransformationen}.

Lie seems to have been disappointed and soured by the
absence of any general recognition of the value of his results.
Reputation came, but it came slowly. In 1886 he moved to
Leipzig, and in 1898 back to Christiania, where a post had
been created for him. He brooded, however, over what he
deemed was the undue neglect of the past, and the happiness
of the last decade of his life was much affected by it.

\index{Hermite}\nblabel{png496a}\nbparagraph{Hermite.\footnote
  {Hermite's collected works, edited by \index{Picard, E.}E.~Picard, were issued at Paris
  in four volumes, 1905 to 1917.}}
Another great algebraist of the century was
\emph{Charles Hermite}, born in Lorraine on December~24, 1822, and
died at Paris, January~14, 1901. From 1869 he was professor at
the Sorbonne, and through his pupils exercised a profound influence
on the mathematicians of to-day.

While yet a student he wrote to \index{Jacobi!ref.\ to}Jacobi on \index{Abelian functions}Abelian functions,
and the latter embodied the results in his works. Hermite's
earlier papers were largely on the transformation of these
functions, a problem which he finally effected by the use of
modular functions. He applied \index{Elliptic functions}elliptic functions to find solutions
of the \index{Quintic equation}quintic equation and of \index{Lame@Lamé}Lamé's \index{Equations, differential!theory of}\index{Differential equations}\index{Equations, differential}differential equation.

Later he took up the subject of algebraic continued fractions,
and this led to his celebrated proof, given in 1873, that $e$ cannot
be the root of an algebraic equation, from which it follows that
\index{e@$e$, symbol for 2.71828\ldots}$e$ is a transcendental number. \index{Lindemann}F.~Lindemann showed in a
similar way in 1882 that \index{P1@$\pi$,!z4@transcendental}$\pi$ is transcendental. The proofs have
been subsequently improved and simplified by \index{Weierstrass, K.!ref.\ to}K.~Weierstrass,
\index{Hilbert, D.}D.~Hilbert, and \index{Klein, F. C.}F.~C.~Klein.\footnote
  {The value of $\pi$ was calculated to 707 places of decimals by \index{Shanks, W.}W.~Shanks
  in 1873; see \emph{Proceedings of the Royal Society}, vol.~xxi, p.~318, vol.~xxii, p.~45.
  The value of $e$ was calculated to 225 places of decimals by \index{Tichanek, F.}F.~Tichanek;
  see \index{Studnicka, F. J.}F.~J.~Studnicka, \emph{Vorträge über monoperiodische Functionen}, Prague,
  1892, and \emph{L'Intermédiaire des Mathématiciens}, %[**TN: 'Intermédiare' changed to 'Intermédiaire' for consistency]
  Paris, 1912, vol.~xix, p.~247.}

To the end of his life Hermite maintained his creative
interest in the subjects of the integral calculus and the theory
of functions. He also discussed the theory of associated covariants
in binary quantics and the theory of ternary quantics.

\nblabel{png497a}So many other writers have treated the subject of \emph{Higher
Algebra} (including therein the theory of \index{Forms in algebra}forms and the theory
of equations) that it is difficult to summarise their conclusions.

The \index{Convergency}convergency of series has been discussed by \index{Raabe on convergency}\emph{J.~L.~Raabe}
(1801--1859) of Zürich, \index{Bertrand}\emph{J.~L.~F.~Bertrand}, the secretary of the
French Academy; \index{Kummer!ref.\ to}\emph{E.~E.~Kummer} of Berlin; \index{Dini, U.}\emph{U.~Dini} of Pisa;
\index{Pringsheim}\emph{A.~Pringsheim} of Munich;\footnote
  {On the researches of Raabe, Bertrand, Kummer, Dini, and Pringsheim,
  see the \emph{Bulletin} of the New York (American) Mathematical Society, vol.~ii,
  1892--3, pp.~1--10.}
and \index{Stokes, G. G.}\emph{Sir George Gabriel Stokes}
(1819--1903) of Cambridge,\footnote
  {Stokes's collected mathematical and physical papers in five volumes, and
  his memoir and scientific correspondence in two volumes, were issued at
  Cambridge, 1880 to 1907.}
to whom the well-known theorem on
the critical values of the sums of periodic series is due. The last-named
writer introduced the important conception of non-uniform
convergence; a subject subsequently treated by \index{Seidel, P. L.}P.~L.~Seidel.

Perhaps here, too, I may allude in passing to the work of
\index{Riemann!ref.\ to}\emph{G.~F.~B.\allowbreak~Riemann, G.~G.~Stokes, \index{Hankel, ref.\ to}H.~Hankel}, and \index{Darboux}\emph{G.~Darboux}
on asymptotic expansions; of \index{Poincaré, H.}\emph{H.~Poincaré} on the application
of such expansions to \index{Equations, differential!theory of}\index{Differential equations}\index{Equations, differential}differential equations; and of \index{Borel, E.}\emph{E.~Borel}
and \index{Cesàre, E.}\emph{E.~Cesàro} on divergent series.

On the theory of groups of substitutions I have already
mentioned the work, on the one hand, of \index{Galois}Galois, \index{Cauchy!ref.\ to}Cauchy, \index{Serret}Serret,
\index{Jordan, C.}Jordan, and \index{Netto, E.}Netto, and, on the other hand, of \index{Frobenius}Frobenius, \index{Klein, F. C.}Klein,
and \index{Burnside, W.}Burnside in connection with discontinuous groups, and that
of \index{Lie!ref.\ to}Lie in connection with continuous groups.

I may also mention the following writers: \index{Borchardt}\emph{C.~W.~Borchardt}\footnote
  {A collected edition of Borchardt's works, edited by \index{Hettner on Borchardt}G.~Hettner, was
  issued at Berlin in 1888.}
(1817--1880) of Berlin, who in particular discussed generating
functions in the theory of equations, and arithmetic-geometric
means. \index{Hermite!ref.\ to}\emph{C.~Hermite}, to whose work I have alluded above.
\index{Betti, E.}\emph{Enrico Betti} of Pisa and \index{Brioschi, F.}\emph{F.~Brioschi} of Milan, both of whom
discussed binary \index{Quantics}quantics; the latter applied hyperelliptic functions
to give a general solution of a \index{Sextic Equation}sextic equation. \index{Aronhold, S. H.}\emph{S.~H.~Aronhold}
(1819--1884) of Berlin, who developed symbolic
methods in connection with the \index{Invariants}invariant theory of quantics.
\index{Gordan, P. A.}\emph{P.~A.~Gordan}\footnote
  {An edition of Gordan's work on invariants (\index{Determinants}determinants and binary
  forms), edited by \index{Kerschensteiner on Gordan}G.~Kerschensteiner, was issued at Leipzig in three volumes,
  1885, 1887, 1908.}
of Erlangen, who has written on the theory of
equations, the theories of groups and forms, and shown that there
are only a finite number of concomitants of quantics. \index{Clebsch, R. F. A.}\emph{R.~F.~A.~Clebsch}\footnote
  {An account of Clebsch's life and works is printed in the \emph{Mathematische
  Annalen}, 1873, vol.~vi, pp.~197--202, and 1874, vol.~vii, pp.~1--55.}
(1833--1872) of Göttingen, who independently investigated
the theory of binary forms in some papers collected and
published in 1871; he also wrote on \index{Abelian functions}Abelian functions. \emph{P.~A.~MacMahon},
formerly an officer in the British army, who has
written on the connection of symmetric functions, invariants and
covariants, the concomitants of binary forms, and combinatory
analysis. \index{Klein, F. C.}\emph{F.~C.~Klein} of Göttingen, who, in addition to his
researches, already mentioned, on functions and on finite discontinuous
groups, has written on \index{Differential equations}\index{Equations, differential}differential equations. \index{Forsyth, A. R.}\emph{A.~R.~Forsyth}
of Cambridge, who has developed the theory of invariants
and the general theory of differential equations, ternariants, and
quaternariants. \index{Painlevé, P.}\emph{P.~Painlevé} of Paris, who has written on the
theory of differential equations. And, lastly, \index{Hilbert, D.}\emph{D.~Hilbert} of
Göttingen, who has treated the theory of homogeneous forms.

No account of contemporary writings on higher algebra
would be complete without a reference to the admirable \emph{Higher
Algebra} by \index{Salmon}G.~Salmon (1819--1904), provost of Trinity College,
Dublin, and the \emph{Cours d'algèbre supérieure} by \index{Serret}J.~A.~Serret, in
which the chief discoveries of their respective authors are
embodied. An admirable historical summary of the theory of
the \index{Complex numbers}complex variable is given in the \emph{Vorlesungen über die
complexen Zahlen}, Leipzig, 1867, by \index{Hankel, ref.\ to}H.~Hankel, of Tübingen.

\tbskip
\nblabel{png498a}\nbtopictitle{Analytical Geometry.} It will be convenient next to call
attention to another division of pure mathematics---analytical
geometry---which has been greatly developed in recent years.
It has been studied by a host of modern writers, but I do not
propose to describe their investigations, and I shall content
myself by merely mentioning the names of the following
mathematicians.

\index{Booth, J.}\emph{James Booth}\footnote
  {See Booth's \emph{Treatise on some new Geometrical Methods}, London, 1873.}
(1806--1878) and \index{MacCullagh}\emph{James MacCullagh}\footnote
  {See MacCullagh's collected works edited by \index{Jellett on MacCullagh}Jellett and \index{Haughton on MacCullagh}Haughton,
  Dublin, 1880.}
(1809--1846),
both of Dublin, were two of the earliest British writers
in this century to take up the subject of analytical geometry,
but they worked mainly on lines already studied by others.
Fresh developments were introduced by \index{Plucker@Plücker}\emph{Julius Plücker}\footnote
  {Plücker's collected works in two volumes, edited by \index{Schonflies@Schönflies A.}A.~Schoenflies and
  \index{Pockels on Plücker}F.~Pockels, were published at Leipzig, 1875, 1896.}
(1801--1868)
of Bonn, who devoted himself especially to the study of
algebraic curves, of a geometry in which the line is the element
in space, and to the theory of \index{Congruences}congruences and complexes; his
equations connecting the singularities of curves are well known; in
1847 he exchanged his chair for one of physics, and subsequently
gave up most of his time to researches on spectra and \index{Magnetism}magnetism.

\nblabel{png499a}The majority of the memoirs on analytical geometry by
\index{Cayley!ref.\ to}\emph{A.~Cayley} and by \index{Smith, Henry!ref.\ to}\emph{Henry Smith} deal with the theory of curves
and surfaces; the most remarkable of those of \index{Hesse}\emph{L.~O. Hesse}
(1811--1874) of Munich are on the plane geometry of curves;
of those of \index{Darboux}\emph{J.~G.~Darboux} of Paris are on the geometry of
surfaces; of those of \index{Halphen, G. H.}\emph{G.~H.~Halphen} (1844--1889) of Paris are
on the singularities of surfaces and on \index{Curves, tortuous}\index{Tortuous curves}tortuous curves; and of
those of \index{Bonnet, P. O.}\emph{P.~O.~Bonnet} are on ruled surfaces, curvature, and
torsion. The singularities of curves and surfaces have also been
considered by \index{Zeuthen}\emph{H.~G.~Zeuthen} of Copenhagen, and by \index{Schubert, H. C. H.}\emph{H.~C.~H.~Schubert}\footnote
  {Schubert's lectures were published at Leipzig, 1879.}
of Hamburg. The theory of tortuous curves has
been discussed by \index{Nother@Nöther, M.}\emph{M.~Nöther} of Erlangen; and \index{Clebsch, R. F. A.}\emph{R.~F.~A.~Clebsch}\footnote
  {Clebsch's lectures have been published by \index{Lindemann}F.~Lindemann, two volumes,
  Leipzig, 1875, 1891.}
of Göttingen has applied \index{Abel!ref.\ to}\index{Abel's theorem}Abel's theorem to geometry.

Among more recent text-books on analytical geometry are
\index{Darboux}J.~G.\allowbreak~Darboux's \emph{Théorie générale des surfaces}, and \emph{Les Systèmes
orthogonaux et les coordonnées curvilignes}; R.~F.~A.~Clebsch's
\emph{Vorlesungen über Geometrie}, edited by F.~Lindemann; and
\index{Salmon}G.~Salmon's \index{Conics (Analytical).!q4@recent extensions of}\index{Conic Sections (Geometrical).!Treatment of, by modern synthetic geometry|(}\emph{Conic Sections, Geometry of Three Dimensions}, and
\emph{Higher Plane Curves}; in which the chief discoveries of these
writers are embodied.

\index{Plucker@Plücker}\nblabel{png500a}Plücker suggested in 1846 that the straight line should be
taken as the element of space. This formed the subject of investigations
by \index{Battaglini, G.}\emph{G.~Battaglini} (1826--1892) of Rome, \index{Klein, F. C.}\emph{F.~C.~Klein}, and
\index{Lie!ref.\ to}\emph{S.~Lie}.\footnote
  {On the history of this subject see \index{Loria, ref.\ to}G.~Loria, \emph{Il passato ed il presente delle
  principali teorie geometriche}, Turin, 1st ed.\ 1887; 2nd ed.\ 1896.}
Recent works on it are \index{Sturm, ref.\ to}R.~Sturm's \emph{Die Gebilde ersten und
zweiten Grades der Liniengeometrie}, 3 volumes, Leipzig, 1892,
1893, 1896, and \index{Jessop, C. M.}C.~M.~Jessop's \emph{Treatise on the Line Complex},
Cambridge, 1903.

Finally, I may allude to the extension of the subject-matter
of analytical \index{Geometryz@Geometry, origin of!line}geometry in the writings of \index{Cayley!ref.\ to}\emph{A.~Cayley} in 1844,
\index{Grassmann!ref.\ to}\emph{H.~G.~Grassmann} in 1844 and 1862, \index{Riemann!ref.\ to}\emph{G.~F.~B.~Riemann} in
1854, whose work was continued by \index{Veronese, G.}\emph{G.~Veronese} of Padua,
\index{Schubert, H. C. H.}\emph{H.~C.~H.~Schubert} of Hamburg, \index{Segre, C.}\emph{C.~Segre} of Turin, \index{Castelnuovo, G.}\emph{G.~Castelnuovo}
of Rome, and others, by the introduction of the idea of
space of $n$ dimensions.\index{Algebra.!z9@Recent extensions of|)}

\tbskip
\index{Analysis,!higher}\nblabel{png500b}\nbtopictitle{Analysis.} Among those who have extended the range of
analysis (including the calculus and \index{Differential equations}\index{Equations, differential}differential equations) or
whom it is difficult to place in any of the preceding categories
are the following, whom I mention in alphabetical order.
\index{Appell, P. E.}\emph{P.~E.~Appell}\multifootnote
  {ADPP}{Biographies of Appell, Darboux, Picard, and Poincaré, with bibliographies,
  by \index{Lebon, E.}E.~Lebon, were issued in Paris in 1909, 1910.}
of Paris; \index{Bertrand}\emph{J.~L.~F.~Bertrand} of Paris; \index{Boole, G.!ref.\ to}\emph{G.~Boole}
of Cork; \index{Cauchy!ref.\ to}\emph{A.~L.~Cauchy} of Paris; \index{Darboux}\emph{J.~G.~Darboux}\multifootnote
  {ADPP}
of Paris;
\index{Forsyth, A. R.}\emph{A.~R.~Forsyth} of Cambridge; \index{Frobenius}\emph{F.~G.~Frobenius} of Berlin;
\index{Fuchs}\emph{J.~Lazarus Fuchs} (1833--1902) of Berlin; \index{Halphen, G. H.}\emph{G.~H.~Halphen} of
Paris; \index{Jacobi!ref.\ to}\emph{C.~G.~J.~Jacobi} of Berlin; \index{Jordan, C.}\emph{M.~E.~C.~Jordan} (1838--1922)
of Paris; \index{Konigsberger@Königsberger, L.}\emph{L.~Königsberger} of Heidelberg; \index{Kowalevski, S.}\emph{Sophie Kowalevski}\footnote
  {See the \emph{Bulletin des sciences mathématiques}, vol.~xv, pp.~212--220.}
(1850--1891) of Stockholm; \emph{M.~S.~Lie} of Leipzig; \index{Picard, E.}\emph{E.~Picard}\multifootnote
  {ADPP}
of Paris; \index{Poincaré, H.}\emph{H.~Poincaré}\multifootnote
  {ADPP}
of Paris; \emph{G.~F.~B.~Riemann} of Göttingen;
\index{Schwarz, H. A.}\emph{H.~A.~Schwarz} of Berlin; \emph{J.~J.~Sylvester}; and \index{Weierstrass, K.!ref.\ to}\emph{K.~Weierstrass}
of Berlin, who developed the \index{Calculus of operations!of variations}\index{Variations, calculus of}calculus of variations.

The subject of \index{Differential equations}\index{Equations, differential}\emph{differential equations} should perhaps have been
separated and treated by itself. But it is so vast that it is
difficult---indeed impossible---to describe recent researches in a
single paragraph. It will perhaps suffice to refer to the admirable
series of treatises, seven volumes, on the subject by \index{Forsyth, A. R.}A.~R.~Forsyth,
which give a full presentation of the subjects treated.

A recent development on \index{Equations, differential!integral}\index{Integral equations}integral equations, or the inversion
of a definite integral, has attracted considerable attention. It
originated in a single instance given by \index{Abel!ref.\ to}Abel, and has been
treated by \index{Volterra, V., ref.\ to}V.~Volterra of Rome, \index{Fredholm, J.}J.~Fredholm of Stockholm, \index{Hilbert, D.}D.~Hilbert
of Göttingen, and numerous other recent writers.

\tbskip
\index{Geometry.!z8@Modern synthetic geometry, originated with Desargues!x6@recent development of|(}\nblabel{png501a}\nbtopictitle{Synthetic Geometry.} The writers I have mentioned above
mostly concerned themselves with analysis. I will next describe
some of the more important works produced in this century on
synthetic geometry.\footnote
  {The \emph{Aperçu historique sur l'origine et le développement des méthodes en
  géométrie}, by \index{Chasles, M., ref.\ to}M.~Chasles, Paris, second edition, 1875; and \emph{Die synthetische
  Geometrie im Alterthum und in der Neuzeit}, by \index{Reye on modern geometry}Th.~Reye, Strassburg, 1886,
  contain interesting summaries of the history of geometry, but Chasles's work
  is written from an exclusively French point of view.}

Modern synthetic geometry may be said to have had its
origin in the works of \index{Monge!ref.\ to}Monge in 1800, \index{Carnot, Lazare!ref.\ to}Carnot in 1803, and
\index{Poncelet!ref.\ to}Poncelet in 1822, but these only foreshadowed the great extension
it was to receive in Germany, of which \index{Steiner!ref.\ to}Steiner and \index{Von Staudt!ref.\ to}\index{Staudt, von!ref.\ to}von
Staudt are perhaps the best known exponents.

\index{Steiner}\nblabel{png501b}\nbparagraph{Steiner.\footnote
  {Steiner's collected works, edited by \index{Weierstrass, K.!ref.\ to}Weierstrass, were issued in two
  volumes, Berlin, 1881--82. A sketch of his life is contained in the \emph{Erinnerung
  an Steiner}, by \index{Geiser on Steiner}C.~F.~Geiser, Schaffhausen, 1874.}}
\emph{Jacob Steiner}, ``the greatest geometrician since
the time of \index{Apollonius!ref.\ to}Apollonius,'' was born at Utzensdorf on March~18,
1796, and died at Bern on April~1, 1863. His father was a
peasant, and the boy had no opportunity to learn reading and
writing till the age of fourteen. He subsequently went to
Heidelberg and thence to Berlin, supporting himself by giving
lessons. His \emph{Systematische Entwickelungen} was published in
1832, and at once made his reputation: it contains a full discussion
of the principle of duality, and of the projective and
homographic relations of rows, pencils, \&c., based on metrical
properties. By the influence of \index{Crelle, ref.\ to}Crelle, \index{Jacobi!ref.\ to}Jacobi, and the \index{Von Humboldt}\index{Humboldt}von
Humboldts, who were impressed by the power of this work,
a chair of geometry was created for Steiner at Berlin, and
he continued to occupy it till his death. The most important
of his other researches are contained in papers which appeared
in \emph{Crelle's Journal}: these relate chiefly to properties of algebraic
curves and surfaces, \index{Pedals}pedals and roulettes, and \index{Maxima and minima, determination of}maxima and
minima: the discussion is purely geometrical. Steiner's works
may be considered as the classical authority on recent synthetic
geometry.

\index{Staudt, von}\index{Von Staudt}\nblabel{png502a}\nbparagraph{Von Staudt.} A system of pure geometry, quite distinct
from that expounded by \index{Steiner!ref.\ to}Steiner, was proposed by \emph{Karl Georg
Christian von Staudt}, born at Rothenburg on Jan.~24, 1798,
and died in 1867, who held the chair of mathematics at
Erlangen. In his \emph{Geometrie der Lage}, published in 1847, he
constructed a system of geometry built up without any reference
to number or magnitude, but, in spite of its abstract form, he
succeeded by means of it alone in establishing the non-metrical
projective properties of figures, discussed imaginary points, lines,
and planes, and even obtained a geometrical definition of a
number: these views were further elaborated in his \emph{Beiträge zur
Geometrie der Lage}, 1856--1860. This geometry is curious and
brilliant, and has been used by \index{Culmann on graphics}Culmann as the basis of his
graphical statics.

{\nbstretchyspace \nblabel{png502b}As usual text-books on synthetic geometry I may mention
\index{Chasles, M., ref.\ to}M.~Chasles's \emph{Traité de géométrie supérieure}, 1852; J.~Steiner's
\emph{Vorlesungen über synthetische Geometrie}, 1867; \index{Cremona, L.}L.~Cremona's
\emph{Éléments de géométrie projective}, English translation by
\index{Leudesdorf on Cremona}C.~Leudesdorf, Oxford, second edition, 1893; and \index{Reye on modern geometry}Th.~Reye's
\emph{Geometrie der Lage}, Hanover, 1866--1868, English translation
by \index{Holgate on Reye}T.~F.~Holgate, New York, part i, 1898.} A good presentation
of the modern treatment of pure geometry is contained in
the \emph{Introduzione ad una teoria geometrica delle curve piane},
1862, and its continuation \emph{Preliminari di una teoria geometrica
delle superficie} by \index{Cremona, L.}Luigi Cremona (1830--1903): his collected
works, in three volumes, may be also consulted.

The differences in ideas and methods formerly observed in
analytic and synthetic geometries tend to disappear with their
further development.\index{Conic Sections (Geometrical).!Treatment of, by modern synthetic geometry|)}

\tbskip
\index{Non-Euclidean geometry|(}\nblabel{png503a}\nbtopictitle{Non-Euclidean Geometry.} Here I may fitly add a few words
on recent investigations on the foundations of geometry.

\index{Hyperg3@Hyper-geometry|(}The question of the truth of the assumptions usually
made in our geometry had been considered by \index{Saccheri}J.~Saccheri
as long ago as 1733; and in more recent times had been
discussed by \index{Lobatschewsky}N.~I.~Lobatschewsky (1793--1856) of Kasan,
in 1826 and again in 1840; by \index{Gauss!ref.\ to}Gauss, perhaps as early as
1792, certainly in 1831 and in 1846; and by \index{Bolyai, J.}J.~Bolyai (1802--1860)
in 1832 in the appendix to the first volume of his
father's \emph{Tentamen}, but \index{Riemann!ref.\ to}Riemann's memoir of 1854 attracted
general attention to the subject of non-Euclidean geometry,
and the theory has been since extended and simplified by various
writers, notably by \index{Cayley!ref.\ to}A.~Cayley of Cambridge, \index{Beltrami, E.}E.~Beltrami\footnote
  {Beltrami's collected works are (1908) in course of publication at Milan.
  A list of his writings is given in the \emph{Annali di matematica}, March 1900.}
(1835--1900) of Pavia, by \index{Von Helmholtz}\index{Helmholtz, von, ref.\ to}H.~L.~F. von~Helmholtz (1821--1894)
of Berlin, by S.~P.~Tannery (1843--1904) of Paris, by \index{Klein, F. C.}F.~C.~Klein
of Göttingen, and by \index{Whitehead, A. N.}A.~N.~Whitehead of Cambridge in
his \emph{Universal Algebra}. The subject is so technical that I confine
myself to a bare sketch of the argument\footnote
  {For references see \index{Ball, W. W. R., ref.\ to}my \emph{Mathematical Recreations and Essays}, London,
  ninth edition, 1920, chaps, xv, xxi. A historical summary of the treatment
  of non-Euclidean geometry is given in \emph{Die Theorie der Parallellinien} by
  \index{Engel, F., on Grassmann!ref.\ to}F.~Engel and \index{Stäckel, P.}P.~Stäckel, Leipzig, 1895, 1899;
  see also \index{Frischauf on absolute geometry}J.~Frischauf's \emph{Elemente
  der absoluten Geometrie}, Leipzig, 1876; and a report by \index{Halsted, G. B., on hyper-geometry}G.~B.~Halsted on
  progress in the subject is printed in \emph{Science}, N.S., vol.~x, New York, 1899,
  pp.~545--557.}
from which the idea
is derived.\index{Geometry.!z8@Modern synthetic geometry, originated with Desargues!x6@recent development of|)}

The Euclidean system of geometry, with which alone most
people are acquainted, rests on a number of independent
axioms and postulates. Those which are necessary for Euclid's
geometry have, within recent years, been investigated and
scheduled. They include not only those explicitly given by
him, but some others which he unconsciously used. If these are
varied, or other axioms are assumed, we get a different series
of propositions, and any consistent body of such propositions
constitutes a system of geometry. Hence there is no limit to
the number of possible Non-Euclidean geometries that can be
constructed.

\nblabel{png504a}Among Euclid's axioms and postulates is one on \index{Parallel lines|(}parallel
lines, which is usually stated in the form that if a straight
line meets two straight lines, so as to make the sum of the two
interior angles on the same side of it taken together less than
two right angles, then these straight lines being continually
produced will at length meet upon that side on which
are the angles which are less than two right angles. Expressed
in this form the axiom is far from obvious, and from
early times numerous attempts have been made to prove
it.\footnote
  {Some of the more interesting and plausible attempts have been collected
  by \index{Thompson, T. P.}T.~P.~Thompson in his \emph{Geometry without Axioms}, London, 1833, and later
  by \index{Richard, J.}J.~Richard in his \emph{Philosophie de mathématique}, Paris, 1903.}
All such attempts failed, and it is now known that the
axiom cannot be deduced from the other axioms assumed by
Euclid.

\index{Geometry.!z9@Non-Euclidean geometry, originated with Saccheri, Lobatschewsky, and John Bolyai}\nblabel{png504b}The earliest conception of a body of Non-Euclidean geometry
was due to the discovery, made independently by Saccheri,
Lobatschewsky, and \index{Bolyai, J.}John Bolyai, that a consistent system of
geometry of two dimensions can be produced on the assumption
that the axiom on parallels is not true, and that through
a point a number of straight (that is, geodetic) lines can be
drawn parallel to a given straight line. The resulting geometry
is called \index{Geometryz@Geometry, origin of!hyperbolic}\index{Hyperbolic geometry}\emph{hyperbolic}.

\index{Riemann!ref.\ to}Riemann later distinguished between boundlessness of space
and its infinity, and showed that another consistent system of
geometry of two dimensions can be constructed in which all
straight lines are of a finite length, so that a particle moving
along a straight line will return to its original position. This
leads to a geometry of two dimensions, called \index{Geometryz@Geometry, origin of!elliptic}\index{Elliptic geometry}\emph{elliptic geometry},
analogous to the hyperbolic geometry, but characterised by the
fact that through a point no straight line can be drawn which,
if produced far enough, will not meet any other given straight
line. This can be compared with the geometry of figures drawn
on the surface of a sphere.

Thus according as no straight line, or only one straight line,
or a pencil of straight lines can be drawn through a point
parallel to a given straight line, we have three systems of
geometry of two dimensions known respectively as elliptic,
parabolic or homaloidal or Euclidean, and \index{Geometryz@Geometry, origin of!hyperbolic}\index{Hyperbolic geometry}hyperbolic.\index{Parallel lines|)}

In the parabolic and hyperbolic systems straight lines are
infinitely long. In the \index{Geometryz@Geometry, origin of!elliptic}\index{Elliptic geometry}elliptic they are finite. In the hyperbolic
system there are no similar figures of unequal size; the
area of a triangle can be deduced from the sum of its angles,
which is always less than two right angles; and there is a finite
maximum to the area of a triangle. In the elliptic system all
straight lines are of the same finite length; any two lines intersect;
and the sum of the angles of a triangle is greater than
two right angles.

In spite of these and other peculiarities of hyperbolic and
elliptical geometries, it is impossible to prove by observation
that one of them is not true of the space in which we live.
For in measurements in each of these geometries we must
have a unit of distance; and if we live in a space whose
properties are those of either of these geometries, and such
that the greatest distances with which we are acquainted
(\emph{ex.~gr.}\ the distances of the fixed stars) are immensely smaller
than any unit, natural to the system, then it may be impossible
for us by our observations to detect the discrepancies between the
three geometries. It might indeed be possible by observations
of the parallaxes of stars to prove that the parabolic system and
either the hyperbolic or elliptic system were false, but never
can it be proved by measurements that Euclidean geometry
is true. Similar difficulties might arise in connection with
excessively minute quantities. In short, though the results of
Euclidean geometry are more exact than present experiments
can verify for finite things, such as those with which we have
to deal, yet for much larger things or much smaller things or
for parts of space at present inaccessible to us they may not
be true.

{\nbstretchyspace Other systems of Non-Euclidean geometry might be constructed
by changing other axioms and assumptions made by
Euclid.} Some of these are interesting, but those mentioned
above have a special importance from the somewhat sensational
fact that they lead to no results inconsistent with the properties
of the space in which we live.

\index{Spherical space|(}\nblabel{png506a}We might also approach the subject by remarking that in
order that a space of two dimensions should have the geometrical
properties with which we are familiar, it is necessary that it
should be possible at any place to construct a figure congruent
to a given figure; and this is so only if the product of the
principal radii of curvature at every point of the space or
surface be constant. This product is constant in the case (i)
of spherical surfaces, where it is positive; (ii)~of plane surfaces
(which lead to Euclidean geometry), where it is zero; and (iii)
of \index{Pseudo-spherical space}pseudo-spherical surfaces, where it is negative. A tractroid
is an instance of a pseudo-spherical surface; it is saddle-shaped
at every point. Hence on spheres, planes, and tractroids we
can construct normal systems of geometry. These systems are
respectively examples of \index{Geometryz@Geometry, origin of!elliptic}\index{Elliptic geometry}elliptic, Euclidean, and
\index{Geometryz@Geometry, origin of!hyperbolic}\index{Hyperbolic geometry}hyperbolic %[**TN: original text had 'hyperbolic, Euclidean, and elliptic']
geometries. Moreover, if any surface be bent without dilation
or contraction, the measure of curvature remains unaltered. Thus
these three species of surfaces are types of three kinds on which
congruent figures can be constructed. For instance a plane can
be rolled into a cone, and the system of geometry on a conical
surface is similar to that on a plane.

In the preceding sketch of the foundations of Non-Euclidean
geometry I have assumed tacitly that the measure of a distance
remains the same everywhere.

The above refers only to hyper-space of two dimensions.
Naturally there arises the question whether there are different
kinds of hyper-space of three or more dimensions. Riemann
showed that there are three kinds of hyper-space of three
dimensions having properties analogous to the three kinds of
hyper-space of two dimensions already discussed. These are
differentiated by the test whether at every point no geodetical
surfaces, or one geodetical surface, or a fasciculus of geodetical
surfaces can be drawn parallel to a given surface; a geodetical
surface being defined as such that every geodetic line joining
two points on it lies wholly on the surface.\index{Spherical space|)}\index{Hyperg3@Hyper-geometry|)}

\tbskip
\nblabel{png507a}\nbtopictitle{Foundations of Mathematics. \index{Assumptions}Assumptions made in the
Subject.} The discussion on the Non-Euclidean geometry
brought into prominence the logical foundations of the subject.
The questions of the principles of and underlying assumptions
made in mathematics have been discussed of late by \index{Dedekind, J. W. R., ref.\ to}J.~W.~R.~Dedekind
of Brunswick, \index{Cantor, G.}G.~Cantor of Halle, \index{Frege, G.}G.~Frege of Jena,
\index{Peano, G.}G.~Peano of Turin, the \index{Russell, B. A. W.}Hon.~B.~A.~W.~Russell and \index{Whitehead, A. N.}A.~N.~Whitehead,
both of Cambridge.\index{Non-Euclidean geometry|)}

\tbskip
\index{Kinematics}\nblabel{png507b}\nbtopictitle{Kinematics.} The theory of kinematics, that is, the investigation
of the properties of motion, displacement, and deformation,
considered independently of force, mass, and other physical conceptions,
has been treated by various writers. It is a branch
of pure mathematics, and forms a fitting introduction to the
study of natural philosophy. Here I do no more than allude
to it.

\index{Mechanics.!z5@Recent work on|(}\nblabel{png507c}I shall conclude the chapter with a few notes---more or less
discursive\allowbreak---on branches of mathematics of a less abstract
character and concerned with problems that occur in nature.
I commence by mentioning the subject of \emph{Mechanics}. The
subject may be treated graphically or analytically.

\index{Graphical methods|(}\nbtopictitle{Graphics.} In the science of graphics rules are laid down
for solving various problems by the aid of the drawing-board:
the modes of calculation which are permissible are considered
in modern projective geometry, and the subject is closely
connected with that of modern geometry. This method of
attacking questions has been hitherto applied chiefly to problems
in mechanics, elasticity, and electricity; it is especially useful in
engineering, and in that subject an average draughtsman ought
to be able to obtain approximate solutions of most of the
equations, differential or otherwise, with which he is likely to
be concerned, which will not involve errors greater than would
have to be allowed for in any case in consequence of our imperfect
knowledge of the structure of the materials employed.

The theory may be said to have originated with \index{Poncelet!ref.\ to}Poncelet's
work, but I believe that it is only within the last twenty years
that systematic expositions of it have been published. Among
the best known of such works I may mention the \emph{Graphische
Statik}, by \index{Culmann on graphics}\emph{C.~Culmann}, Zürich, 1875, recently edited by
\index{Ritter on Culmann}W.~Ritter; the \emph{Lezioni di statica grafica}, by \index{Favaro, A., ref.\ to}\emph{A.~Favaro}, Padua,
1877 (French translation annotated by \index{Terrier on graphics}P.~Terrier in 2 volumes,
1879--85); the \emph{Calcolo grafico}, by \index{Cremona, L.}\emph{L.~Cremona}, Milan, 1879
(English translation by \index{Beare, T. H., on graphics}T.~H.~Beare, Oxford, 1889), which is
largely founded on \index{Mobius@Möbius!ref.\ to}Möbius's work; \emph{La statique graphique}, by
\index{Levy@Lévy on graphics}\emph{M.~Levy}, Paris, 4 volumes, 1886--88; and \emph{La statica grafica}, by
\index{Sairotti on graphics}\emph{C.~Sairotti}, Milan, 1888.

The general character of these books will be sufficiently
illustrated by the following note on the contents of Culmann's
work. Culmann commences with a description of the geometrical
representation of the four fundamental processes of
addition, subtraction, multiplication, and division; and proceeds
to evolution and involution, the latter being effected by
the use of \index{Equiangular spiral}\index{Spiral, the equiangular} the equiangular spiral. %[**TN: "the" added]
He next shows how the quantities
considered---such as volumes, moments, and moments of inertia---may
be represented by straight lines; thence deduces the
laws for combining forces, couples, \&c.; and then explains the
construction and use of the ellipse and ellipsoid of inertia,
the neutral axis, and the kern; the remaining and larger part
of the book is devoted to showing how geometrical drawings,
made on these principles, give the solutions of many practical
problems connected with arches, bridges, frameworks, earth
pressure on walls and tunnels, \&c.

The subject has been treated during the last twenty years
by numerous writers, especially in Italy and Germany, and
applied to a large number of problems. But as I stated at the
beginning of this chapter that I should as far as possible avoid
discussion of the works of living authors I content myself
with a bare mention of the subject.\index{Graphical methods|)}\footnote
  {In an English work, I may add here a brief note on \index{Clifford, W. K.}Clifford, who was
  one of the earliest British mathematicians of later times to advocate the use of
  graphical and geometrical methods in preference to analysis. \emph{William
  Kingdon Clifford}, born at Exeter on May~4, 1845, and died at Madeira on
  March~3, 1879, was educated at Trinity College, Cambridge, of which society
  he was a fellow. In 1871 he was appointed professor of applied mathematics
  at University College, London, a post which he retained till his death. His
  remarkable felicity of illustration and power of seizing analogies made him
  one of the most brilliant expounders of mathematical principles. His health
  failed in 1876, when the writer of this book undertook his work for a few
  months; Clifford then went to Algeria and returned at the end of the year,
  but only to break down again in 1878. His most important works are his
  \emph{Theory of Biquaternions, On the Classification of Loci} (unfinished), and \emph{The
  Theory of Graphs} (unfinished). His \emph{Canonical Dissection of a \index{Riemann!ref.\ to}Riemann's
  Surface} and the \emph{Elements of Dynamic} also contain much interesting matter.
  For further details of Clifford's life and work see the authorities quoted in the
  article on him in the \emph{Dictionary of National Biography}, vol.~xi.}

\nblabel{png509a}\nbtopictitle{Analytical Mechanics.} I next turn to the question of
mechanics treated analytically. The knowledge of mathematical
mechanics of solids attained by the great mathematicians of the
last century may be said to be summed up in the admirable
\emph{Mécanique analytique} by \index{Lagrange!ref.\ to}Lagrange and \emph{Traité de mécanique}
by \index{Poisson!ref.\ to}Poisson, and the application of the results to astronomy
forms the subject of \index{Laplace!ref.\ to}Laplace's \emph{Mécanique céleste}. These works
have been already described. The mechanics of fluids is
more difficult than that of solids and the theory is less
advanced.

\emph{Theoretical Statics}, especially the theory of \index{Potential, the}\emph{the potential}
and \index{Attraction, theories of}\emph{attractions}, has received considerable attention from the
mathematicians of this century.

I have previously mentioned that the introduction of the idea
of the potential is due to Lagrange, and it occurs in a memoir
of a date as early as 1773. The idea was at once grasped by
Laplace, who, in his memoir of 1784, used it freely and to
whom the credit of the invention was formerly, somewhat
unjustly, attributed. In the same memoir Laplace also extended
the idea of zonal \index{Harmonic analysis}harmonic analysis which had been
introduced by \index{Legendre!ref.\ to}Legendre in 1783. Of \index{Gauss!ref.\ to}\emph{Gauss's} work on \index{Attraction, theories of}attractions
I have already spoken. The theory of level surfaces and lines
of force is largely due to \index{Chasles, M., ref.\ to}\emph{Chasles}, who also determined the
attraction of an ellipsoid at any external point. I may also here
mention the \emph{Barycentrisches Calcul}, published in 1826 by
\index{Mobius@Möbius}\emph{A.~F.~Möbius}\footnote
  {Möbius's collected works were published at Leipzig in four volumes, 1885--87.}
(1790--1868), who was one of the best known of
Gauss's pupils. Attention must also be called to the important
memoir, published in 1828, on \index{Potential, the}the potential and its properties,
by \index{Green}G.~Green\footnote
  {A collected edition of Green's works was published at Cambridge in
  1871. Other papers of Green which deserve mention here are those in 1832
  and 1833 on the equilibrium of fluids, on attractions in space of $n$ dimensions,
  and on the motion of a fluid agitated by the vibrations of a solid ellipsoid;
  and those in 1837 on the motion of waves in a canal, and on the reflexion and
  \index{Refraction}refraction of sound and light. \index{Light,!physical theories of}\index{Optics (geometrical).!z11@(physical)}In the last of these, the geometrical laws of
  sound and light are deduced by the principle of energy from the undulatory
  theory, the phenomenon of total reflexion is explained physically, and certain
  properties of the vibrating medium are deduced. Green also discussed the
  propagation of light in any crystalline medium.}
(1793--1841) of Cambridge. Similar results were
independently established, in 1839, by Gauss, to whom their
general dissemination was due.

\emph{Theoretical Dynamics}, which was cast into its modern form
by \index{Jacobi!ref.\ to}Jacobi, has been studied by most of the writers above
mentioned. I may also here repeat that the principle of
\index{Action,!varying}``Varying Action'' was elaborated by \index{Hamilton, Sir Wm.!ref.\ to}Sir William Hamilton
in 1827, and the ``Hamiltonian equations'' were given in
1835; and I may further call attention to the dynamical
investigations of \index{Bour, J. E. E.}J.~E.~E.~Bour (1832--1866), of \index{Liouville, J.}Liouville, and
of \index{Bertrand}J.~L.~F.~Bertrand, all of Paris. The use of \index{Coord@Co-ordinates!generalized}\index{Generalized co-ordinates}generalised co-ordinates,
introduced by \index{Lagrange!ref.\ to}Lagrange, has now become the customary
means of attacking dynamical (as well as many physical)
problems.

\nblabel{png510a}As usual text-books I may mention those on particle and
rigid dynamics by \index{Routh on mechanics}E.~J.~Routh, Cambridge; \emph{Leçons sur
l'intégration des équations différentielles de la mécanique} by
\index{Painlevé, P.}P.~Painlevé, Paris, 1895; %[**TN: comma changed to semicolon for consistency]
\emph{Intégration des équations de la
mécanique} by \index{Graindorge, J., ref.\ to}J.~Graindorge, Brussels, 1889; and \index{Appell, C. E.}C.~E.~Appell's
\emph{Traité de mécanique rationnelle}, Paris, 2 vols., 1892,
1896. Allusion to the treatise on Natural Philosophy by Sir
William Thomson (later known as \index{Kelvin, Lord}Lord Kelvin) of Glasgow, and
\index{Tait}P.~G.~Tait of Edinburgh, may be also here made.

On the mechanics of fluids, liquids, and gases, apart from
the physical theories on which they rest, I propose to say
nothing, except to refer to the memoirs of \index{Green}Green, \index{Stokes, G. G.}Sir George
Stokes, Lord Kelvin, and \index{Von Helmholtz}\index{Helmholtz, von, ref.\ to}von Helmholtz. The fascinating but
difficult theory of vortex rings is due to the two writers last
mentioned. One problem in it has been also considered by
\index{Thomson, Sir J. J.}Sir J.~J.~Thomson, of Cambridge, but it is a subject which is
as yet beyond our powers of analysis. The subject of sound
may be treated in connection with hydrodynamics, but on
this I would refer the reader who wishes for further information
to the work first published at Cambridge in 1877 by
\index{Rayleigh, Lord}Lord Rayleigh.\index{Mechanics.!z5@Recent work on|)}

\tbskip
\nblabel{png511a}\emph{Theoretical Astronomy} is included in, or at any rate closely
connected with, theoretical dynamics. Among those who in this
century have devoted themselves to the study of theoretical
astronomy the name of \index{Gauss!ref.\ to}\emph{Gauss} is one of the most prominent; to
his work I have already alluded.

\index{Bessel}\nblabel{png511b}\nbparagraph{Bessel.\footnote
  {See pp.~35--53 of \index{Clerke, A. M.}A.~M.~Clerke's \emph{History of Astronomy}, Edinburgh, 1887.
  Bessel's collected works and correspondence have been edited by \index{Engelmann on Bessel}R.~Engelmann
  and published in four volumes at Leipzig, 1875--82.}}
The best known of Gauss's contemporaries was
\emph{Friedrich Wilhelm Bessel}, who was born at Minden on
July~22, 1784, and died at Königsberg on March~17, 1846.
Bessel commenced his life as a clerk on board ship, but in
1806 he became an assistant in the observatory at Lilienthal,
and was thence in 1810 promoted to be director of the new
Prussian Observatory at Königsberg, where he continued to
live during the remainder of his life. Bessel introduced into
pure mathematics those \index{Bessel's functions}functions which are now called by his
name (this was in 1824, though their use is indicated in a
memoir seven years earlier); but his most notable achievements
were the reduction (given in his \emph{Fundamenta Astronomiae},
Königsberg, 1818) of the Greenwich observations by \index{Bradley!ref.\ to}Bradley
of \index{Stars, lists of}3222 stars, and his determination of the annual parallax
of 61 Cygni. Bradley's observations have been recently reduced
again by \index{Auwers, A.}A.~Auwers of Berlin.

\index{Leverrier}\nblabel{png512a}\nbparagraph{Leverrier.\footnote
  {For further details of his life see \index{Bertrand}Bertrand's \emph{éloge} in vol.~xli of the
  \emph{Mémoires de l'académie}; and for an account of his work see \index{Adams, J. C.!ref.\ to}Adams's
  address in vol.~xxxvi of the \emph{Monthly Notices} of the Royal Astronomical
  Society.}}
Among the astronomical events of this century
the discovery of the \index{Neptune, the planet|(}planet Neptune by Leverrier and Adams is
one of the most striking. \emph{Urbain Jean Joseph Leverrier}, the
son of a petty Government employé in Normandy, was born at
St.~Lô on March~11, 1811, and died at Paris on September~23,
1877. He was educated at the Polytechnic school, and in 1837
was appointed as lecturer on astronomy there. His earliest
researches in astronomy were communicated to the Academy in
1839: in these he calculated, within much narrower limits
than \index{Laplace!ref.\ to}Laplace had done, the extent within which the inclinations
and eccentricities of the \index{Planetary motions|(}planetary orbits vary. The independent
discovery in 1846 by Leverrier and Adams of the planet
Neptune by means of the disturbance it produced on the orbit
of Uranus attracted general attention to physical astronomy,
and strengthened the opinion as to the universality of gravity.
In 1855 Leverrier succeeded \index{Arago!ref.\ to}Arago as director of the Paris
observatory, and reorganised it in accordance with the requirements
of modern astronomy. Leverrier now set himself the
task of discussing the theoretical investigations of the planetary
motions and of revising all tables which involved them. He
lived just long enough to sign the last proof-sheet of this
work.

\index{Adams, J. C.}\nbparagraph{Adams.\footnote
  {Adams's collected papers, with a biography, were issued in two volumes,
  Cambridge, 1896, 1900.}}
The co-discoverer of Neptune was \emph{John Couch
Adams}, who was born in Cornwall on June~5, 1819, educated
at St.~John's College, Cambridge, subsequently appointed
Lowndean professor in the University, and director of the
Observatory, and who died at Cambridge on January~21, 1892.

There are three important problems which are specially
associated with the name of Adams. The first of these is his
discovery of the planet Neptune from the perturbations it
produced on the orbit of Uranus: in point of time this was
slightly earlier than \index{Leverrier!ref.\ to}Leverrier's investigation.\index{Neptune, the planet|)}

The second is his memoir of 1855 on the \index{Moon, secular acceleration of}\index{Secular lunar acceleration}secular acceleration
of the moon's mean motion. \index{Laplace!ref.\ to}Laplace had calculated this
on the hypothesis that it was caused by the eccentricity of
the earth's orbit, and had obtained a result which agreed substantially
with the value deduced from a comparison of the
records of ancient and modern eclipses. Adams shewed that
certain terms in an expression had been neglected, and that
if they were taken into account the result was only about
one-half that found by Laplace. The results agreed with
those obtained later by \index{Delaunay!ref.\ to}Delaunay in France and \index{Cayley!ref.\ to}Cayley in
England, but their correctness has been questioned by \index{Plana!ref.\ to}Plana,
\index{Pontécoulant!ref.\ to}Pontécoulant, and other continental astronomers. The point is
not yet definitely settled.

The third investigation connected with the name of Adams,
is his determination in 1867 of the orbit of the \index{Leonids (shooting stars)}Leonids or
shooting stars which were especially conspicuous in November,
1866, and whose period is about thirty-three years. \index{Newton, H. A., of Yale}H.~A.~Newton
(1830--1896) of Yale, had shewn that there were only
five possible orbits. Adams calculated the disturbance which
would be produced by the planets on the motion of the node
of the orbit of a swarm of meteors in each of these cases, and
found that this disturbance agreed with observation for one of
the possible orbits, but for none of the others. Hence the orbit
was known.

\nblabel{png513a}Other well-known astronomers of this century are \index{Plana}\emph{G.~A.~A.~Plana}
(1781--1864), whose work on the motion of the moon
was published in 1832; \index{Pontécoulant}\emph{Count P.~G.~D.~Pontécoulant} (1795--1871);
\index{Delaunay}\emph{C.~E.~Delaunay} (1816--1872), whose work on the lunar
theory indicates the best method yet suggested for the analytical
investigations of the whole problem, and whose (incomplete)
lunar tables are among the astronomical achievements of this
century; \index{Hansen}\emph{P.~A.~Hansen}\footnote
  {For an account of Hansen's numerous memoirs see the \emph{Transactions
  of the Royal Society of London for} 1876--77.}
(1795--1874), head of the observatory
at Gotha, who compiled the lunar tables published in London
in 1857 which are still used in the preparation of the Nautical
Almanack, and elaborated the methods employed for the
determination of lunar and planetary perturbations; \index{Tisserand}\emph{F.~F.~Tisserand}
(1845--1896) of Paris, whose \emph{Mécanique céleste} is now
a standard authority on dynamical astronomy; and \index{Newcomb, S.}\emph{Simon Newcomb}
(1835--1909), superintendent of the \emph{American Ephemeris},
who re-examined the Greenwich observations from the earliest
times, applied the results to the lunar theory, and revised
Hansen's tables.

Other notable work is associated with the names of \index{Hill, G. W.}Hill,
\index{Darwin, G. H}Darwin, and \index{Poincaré, H.}Poincaré. \emph{G.~W.~Hill},\footnote
  {G.~W.~Hill's collected works have been issued in four volumes,
  Washington, 1905.}
until recently on the
staff of the \emph{American Ephemeris}, determined the inequalities
of the moon's motion due to the non-spherical figure of the
earth---an investigation which completed \index{Delaunay!ref.\ to}Delaunay's lunar
theory.\footnote
  {On recent development of the lunar theory, see the \emph{Transactions of the
  British Association}, vol.~lxv, London, 1895, p.~614.}
Hill also dealt with the secular motion of the moon's
perigee and the motion of a planet's perigee under certain
conditions; and wrote on the analytical theory of the motion
of Jupiter and Saturn, with a view to the preparation of tables
of their positions at any given time. \emph{Sir G.~H.~Darwin} (1845--1912),
of Cambridge, wrote on the effect of tides on viscous
spheroids, the development of planetary systems by means of
\index{Tidal friction}tidal friction, the mechanics of meteoric swarms, and the
possibility of pear-shaped planetary figures. \emph{H.~Poincaré} (1854--1912),
of Paris, discussed the difficult \index{Three bodies, problem of}problem of three bodies,
and the form assumed by a mass of fluid under its own attraction,
and is the author of an admirable treatise, the \emph{Mécanique
céleste}, three volumes. The treatise on the lunar theory by \index{Brown, E. W.}\emph{E.~W.~Brown},
Cambridge, 1896; his memoir on \emph{Inequalities in the
Motion of the Moon} due to Planetary Action, Cambridge,
1908; and a report (printed in the \emph{Report of the British
Association}, London, 1899, vol.~\textsc{lxix}, pp.~121--159) by \index{Whittaker, E. T.}\emph{E.~T.~Whittaker}
on researches connected with the solution of the
problem of three bodies, contain valuable accounts of recent
progress in the lunar and planetary theories.

Within the last half century the results of spectrum analysis
have been applied to determine the \index{Universe, constitution of the}\index{Constitution of the universe}constitution of the heavenly
bodies, and their directions of motions to and from the earth.
The early history of spectrum analysis will be always associated
with the names of \index{Kirchhoff}\emph{G.~R.~Kirchhoff} (1824--1887) of Berlin, of
\index{Angstrom@Ångström}\emph{A.~J.~Ångström} (1814--1874) of Upsala, and of \index{Stokes, G. G.}\emph{George G.~Stokes}
of Cambridge, but it pertains to optics rather than to astronomy.
How unexpected was the application to astronomy is illustrated
by the fact that \index{Comte, A., ref.\ to}A.~Comte in 1842, when discussing the study
of nature, regretted the waste of time due to some astronomers
paying attention to the fixed stars, since, he said, nothing
could possibly be learnt about them; and indeed a century ago
it would have seemed incredible that we could investigate the
chemical constitution of worlds in distant space.

\nblabel{png515a}During the last few years the range of astronomy has
been still further extended by the art of photography. To
what new results this may lead it is as yet impossible to say.
In particular we have been thus enabled to trace the forms of
gigantic spiral nebulae which seem to be the early stages of vast
systems now in process of development.

The constitution of the universe, in which the \index{Solar system}solar system
is but an insignificant atom, has long attracted the attention of
thoughtful astronomers, and noticeably was studied by \index{Herschel, Sir William}William
Herschel. Recently \index{Kapteyn, J. C.}J.~C.~Kapteyn of Groningen has been able
to shew that all the stars whose proper motions can be detected
belong to one or other of two streams moving in different
directions, one with a velocity about three times as great as the
other. The solar system is in the slower stream. These results
have been confirmed by \index{Eddington, A. S.}A.~S.~Eddington and \index{Dyson, F. W.}F.~W.~Dyson.
It would appear likely that we are on the threshold of
wide-reaching discoveries about the constitution of the visible
universe.\index{Planetary motions|)}

\tbskip
\index{Physics, mathematical}\nblabel{png516a}\nbtopictitle{Mathematical Physics.} An account of the history of
mathematics and allied sciences in the last century would be
misleading if there were no reference to the application of
mathematics to numerous problems in \index{Heat, theory of}heat, elasticity, light,
electricity, and other physical subjects. The history of mathematical
physics is, however, so extensive that I could not pretend
to do it justice, even were its consideration properly included in
a history of mathematics. At any rate I consider it outside the
limits I have laid down for myself in this chapter. I abandon
its discussion with regret because the \index{Cambridge, university of}Cambridge school has
played a prominent part in its development, as witness (to
mention only three or four of those concerned) the names
of \index{Stokes, G. G.}Sir George G.~Stokes, professor from 1849 to 1903, \index{Kelvin, Lord}Lord
Kelvin, \index{Maxwell, J. C.}J.~Clerk Maxwell (1831--1879), professor from 1871
to 1879, Lord Rayleigh, professor from 1879 to 1884,
\index{Thomson, Sir J. J.}Sir J.~J.~Thomson, professor from 1884, \index{Larmor, Sir J.}Sir Joseph Larmor,
professor from 1903, and \index{Rutherford, E.}Sir Ernest Rutherford, professor from
1919.\index{Analytical geometry,!x2@on development of, see chapters \textsc{xv-xix}|)}

\backmatter

\index{Abu Djefar|see{Alkarismi}}
\index{Abul-Wafa|see{Albuzjani}}
\index{Alembert, d'|see{D'Alembert}}
\index{Algebraic equations|see{Simple equations, Quadratic equations, \&c.}}
\index{Alkhw@Al-Khw\={a}rizm\={\i}|see{Alkarismi}}
\index{Anharmonic or Cross ratios|see{Geometry (modern synthetic)}}
\index{Arithmetic1@Arithmetic, higher|see{Numbers, theory of}}
\index{Boethian arithmetic|see{Arithmetic}}
\index{Boetius|see{Boethius}}
\index{Bonacci|see{Leonardo of Pisa}}
\index{Calculation|see{Arithmetic}}
\index{Cartes, Des|see{Descartes}}
\index{Ciphers|see{Numerals}}
\index{Circle, quadrature of (or squaring the)!also|see{$\pi$}}
\index{Clerk Maxwell|see{Maxwell}}
\index{Curves, areas of|see{Quadrature}}
\index{Delian problem|see{Cube}}
\index{Descartes, vortices of|see{Cartesian}}
\index{Differential calculus|see{Calculus}}
\index{Double theta functions|see{Elliptic functions}}
\index{Duplication of cube|see{Cube}}
\index{Dynamics|see{Mechanics}}
\index{Elements of Euclid|see{Euclid}}
\index{Equations|see{Simple equations, Quadratic equations, \&c.}}
\index{Fibonacci|see{Leonardo of Pisa}}
\index{Fontana|see{Tartaglia}}
\index{Harmonic ratios|see{Geometry (modern synthetic)}}
\index{Hugens|see{Huygens}}
\index{Huyghens|see{Huygens}}
\index{Hyperelliptic@Hyper-elliptic functions|see{Elliptic functions}}
\index{Infinitesimal calculus|see{Calculus}}
\index{Integral calculus|see{Calculus}}
\index{Involution|see{Geometry (modern synthetic)}}
\index{Lejeune Dirichlet|see{Dirichlet}}
\index{Lucas di Burgo|see{Pacioli}}
\index{Minus|see{Subtraction}}
\index{Mohammed ibn Musa|see{Alkarismi}}
\index{Morgan, A. de|see{De~Morgan}}
\index{Muller1@Müller|see{Regiomontanus}}
\index{Paciolus|see{Pacioli}}
\index{Plus|see{Addition}}
\index{Poles and polars|see{Geometry (modern synthetic)}}
\index{Powers|see{Exponents}}
\index{Quadrature of circle!|see{Circle, also \emph{see} $\pi$}}
\index{Quipus|see{Abacus}}
\index{Quotient!|see{Division}}
\index{Saint-Mesme|see{L'Hospital}}
\index{Series|see{Expansion}}
\index{Slusius|see{Sluze, de}}
\index{Solids|see{Polyhedrons}}
\index{Squaring the circle|see{Circle}}
\index{Statics|see{Mechanics}}
\index{Stiffelius|see{Stifel}}
\index{Swan-pan|see{Abacus}}
\index{Synthetic geometry|see{Geometry}}
\index{Tartalea|see{Tartaglia}}
\index{Ten as radix|see{Decimal}}
\index{Thomson, Sir William|see{Kelvin}}
\index{Trigonometry, higher|see{Elliptic functions}}
\index{Tschotü|see{Abacus}}

% Bookmark the index for ease of navigation, even though it is not in ToC
\pdfbookmark[1]{INDEX}{INDEX}
\nblabel{pngend}

\printindex

\vfill
\begin{center}
\small THE END
\end{center}
\vfill

\cleardoublepage

\pdfbookmark[0]{Project Gutenberg License}{Project Gutenberg License}

\pagestyle{fancy}
\fancyhead[R]{\thepage}%
\fancyhead[C]{\small LICENSING}%
\pagenumbering{Alph}

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