% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % % 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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% 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. %% %% %% %% Things to Check: %% %% %% %% Spellcheck: OK %% %% 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 %% %% %% %% Summary of log file: %% %% No errors %% %% One warning from babel package regarding Greek hyphenation %% %% No overfull or underfull hboxes. %% %% %% %% Command block: %% %% 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) %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \listfiles \makeindex \makeatletter \documentclass[12pt,reqno,oneside]{book}[2005/09/16] \usepackage[paperwidth=5.25in,paperheight=8.65in, textwidth=5in, textheight=8in, headsep=0.1in, footskip=0.1in, 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 \providecommand{\ebook}{00000} \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}, pdfpagelayout=SinglePage, pdfdisplaydoctitle, pdfpagelabels=true, bookmarksopen=true, bookmarksopenlevel=2, colorlinks=false, linkcolor=blue]{hyperref}[2008/11/18] % 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 \normalfont\bfseries \interlinepenalty\@M #1.\par\nobreak \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 \newcommand\tocchapter{\@startsection{section}{1}% {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} \renewcommand\headrulewidth{0pt} % Set up pagestyle for first page of each chapter: % page number centered in header \fancypagestyle{first}{\fancyhf{}% \fancyhead[C]{\thepage}} % customise index - using multicols to balance columns \renewenvironment{theindex}{% \small% \setlength\columnseprule{0.5pt}% \setlength\columnsep{1.5em}% \begin{multicols}{2}[\chapter*{\Large INDEX.}]% \setlength\parindent{0pt}% \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{% \renewcommand\arraystretch{0.6}% \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 \newcommand{\circledigit}[1]{% \setlength\unitlength{1pt}% \begin{picture}(10,10)(-5,-3)% \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}% \end{picture}% } % 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 \def\f@@zl@bar{\null} \def\addto@DPbody#1{\global\DP@lignb@dy\@xp{\the\DP@lignb@dy#1}} \def\parseb@dy#1{\ifx\f@@zl@bar#1\f@@zl@bar \addto@DPbody{{}}\let\@next\parseb@dy \else\ifx\end#1 \let\@next\process@DPb@dy \ifDP@cr\else\addto@DPbody{\DPh@@kr&\DP@rint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&}\fi \addto@DPbody{\end} \else\ifx\intertext#1 \def\@next{\eat@command0}% \else\ifx\shortintertext#1 \def\@next{\eat@command1}% \else\ifDP@cr\addto@DPbody{&\DP@lint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&\DPh@@kl} \DP@crfalse\fi \ifx\begin#1\def\begin@stack{b} \let\@next\eat@environment \else\ifx\lintertext#1 \let\@next\linter@text \else\ifx\rintertext#1 \let\@next\rinter@text \else\ifx\\#1 \addto@DPbody{\DPh@@kr&\DP@rint}\@xp\addto@DPbody\@xp{\@xp{\the\DP@lign@no}&\\}\DP@crtrue \global\advance\DP@lign@no\@ne \let\@next\parse@cr \else\check@braces#1!Q!Q!Q!\ifbr@ce\addto@DPbody{{#1}}\else \addto@DPbody{#1}\fi \let\@next\parseb@dy \fi\fi\fi\fi\fi\fi\fi\fi\@next} \def\process@DPb@dy{\let\lintertext\@gobble\let\rintertext\@gobble \@xp\start@align\@xp\tw@\@xp\st@rredtrue\@xp\m@ne\the\DP@lignb@dy} \def\linter@text#1{\@xp\DPlint\@xp{\the\DP@lign@no}{#1}\parseb@dy} \def\rinter@text#1{\@xp\DPrint\@xp{\the\DP@lign@no}{#1}\parseb@dy} \def\DPlint#1#2{\@xp\def\csname DP@lint:#1\endcsname{\text{#2}}} \def\DPrint#1#2{\@xp\def\csname DP@rint:#1\endcsname{\text{#2}}} \def\DP@lint#1{\ifbalancedlrint\@xp\ifx\csname DP@lint:#1\endcsname\relax\phantom {\csname DP@rint:#1\endcsname}\else\csname DP@lint:#1\endcsname\fi \else\csname DP@lint:#1\endcsname\fi} \def\DP@rint#1{\ifbalancedlrint\@xp\ifx\csname DP@rint:#1\endcsname\relax\phantom {\csname DP@lint:#1\endcsname}\else\csname DP@rint:#1\endcsname\fi \else\csname DP@rint:#1\endcsname\fi} \def\eat@command#1#2{\ifcase#1\addto@DPbody{\intertext{#2}}\or \addto@DPbody{\shortintertext{#2}}\fi\DP@crtrue \global\advance\DP@lign@no\@ne\parseb@dy} \def\parse@cr{\new@ifnextchar*{\parse@crst}{\parse@crst{}}} \def\parse@crst#1{\addto@DPbody{#1}\new@ifnextchar[{\parse@crb}{\parseb@dy}} \def\parse@crb[#1]{\addto@DPbody{[#1]}\parseb@dy} \def\check@braces#1#2!Q!Q!Q!{\def\dp@lignt@stm@cro{#2}\ifx \empty\dp@lignt@stm@cro\br@cefalse\else\br@cetrue\fi} \def\eat@environment#1{\addto@DPbody{\begin{#1}}\begingroup \def\@currenvir{#1}\let\@next\digest@env\@next} \def\digest@env#1\end#2{% \edef\begin@stack{\push@begins#1\begin\end \@xp\@gobble\begin@stack}% \ifx\@empty\begin@stack \@checkend{#2} \endgroup\let\@next\parseb@dy\fi \addto@DPbody{#1\end{#2}} \@next} \def\lintertext{lint}\def\rintertext{rint} \newif\ifbalancedlrint \let\DPh@@kl\empty\let\DPh@@kr\empty \def\DPg@therl{&\omit\hfil$\displaystyle} \def\DPg@therr{$\hfil} \newenvironment{DPalign*}[1][a]{% \if m#1\balancedlrintfalse\else\balancedlrinttrue\fi \global\DP@lign@no\z@\DP@crfalse \DP@lignb@dy{&\DP@lint0&}\parseb@dy }{% \endalign } \newenvironment{DPgather*}[1][a]{% \if m#1\balancedlrintfalse\else\balancedlrinttrue\fi \global\DP@lign@no\z@\DP@crfalse \let\DPh@@kl\DPg@therl \let\DPh@@kr\DPg@therr \DP@lignb@dy{&\DP@lint0&\DPh@@kl}\parseb@dy }{% \endalign } \makeatother %[**TN: spellings "show" and "shew" are both used in the original text - retained] \begin{document} \thispagestyle{empty} \pagenumbering{alph} \pdfbookmark[0]{Project Gutenberg Boilerplate}{Project Gutenberg Boilerplate} \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} \vfill {% \setlength{\parindent}{0pt} \setlength{\parskip}{6pt plus 2pt minus 1pt} \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]} } \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{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{$ 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} \begin{verbatim} End of the Project Gutenberg EBook of A Short Account of the History of Mathematics, by W. 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Rouse Ball % % % % *** END OF THIS PROJECT GUTENBERG EBOOK MATHEMATICS *** % % % % ***** This file should be named 31246-t.tex or 31246-t.zip ***** % % This and all associated files of various formats will be found in: % % http://www.gutenberg.org/3/1/2/4/31246/ % % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \end{document} ### @MathEnvironments = ( ['\\begin{DPgather*}','\\end{DPgather*}',''], ['\\begin{DPalign*}','\\end{DPalign*}',''] ); @ControlwordReplace = ( ['\\nbnamespace'," "] ); @ControlwordArguments = ( ['\\nbChapter', 1,1,"\n\nCHAPTER: ","\n", 1,1,'',"\n\n", 1,1,'',"\n\n", 1,0,'','', 1,1," [Footnote: ","]\n"], ['\\nbpart', 1,1,"\n\nPART: ","\n", 1,1,'',"\n\n", 1,0,'',''], ['\\nblabel', 1,0,'',''], ['\\inlinegraphics', 1,0,'','', 1,0,'','', 1,0,'',''], ['\\inlinetabular', 1,0,'','', 1,0,'',''], ['\\figureref', 1,0,'','', 1,1,'',''], ['\\labelincludegraphics', 0,0,'','',1,0,'',''], ['\\tocperiod', 0,0,'','', 1,0,'',''], ['\\rotatebox', 0,0,'','', 1,0,'','', 1,0,'',''], ['\\resizebox', 1,0,'','', 1,0,'','', 1,0,'',''], ['\\footnoteibid', 1,0,'','', 1,0,'','', 1,1," [Footnote: "," ", 1,1,'',"]\n"], ['\\multifootnote', 1,0,'','', 1,1," [Footnote: ","]\n"], ['\\mytextgreek', 1,1,"[Greek: ","]"], ['\\hyperref', 0,0,'',''] ); ### This is pdfTeXk, Version 3.141592-1.40.3 (Web2C 7.5.6) (format=pdflatex 2010.5.6) 28 MAY 2010 10:21 entering extended mode %&-line parsing enabled. **31246-t.tex (./31246-t.tex LaTeX2e <2005/12/01> Babel and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, arabic, farsi, croatian, ukrainian, russian, bulgarian, czech, slov ak, danish, dutch, finnish, basque, french, german, ngerman, ibycus, greek, mon ogreek, ancientgreek, hungarian, italian, latin, mongolian, norsk, icelandic, i nterlingua, turkish, coptic, romanian, welsh, serbian, slovenian, estonian, esp eranto, uppersorbian, indonesian, polish, portuguese, spanish, catalan, galicia n, swedish, ukenglish, pinyin, loaded. \@indexfile=\write3 \openout3 = `31246-t.idx'. 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sequences out of 10000+50000 24323 words of font info for 74 fonts, out of 1200000 for 2000 672 hyphenation exceptions out of 8191 26i,15n,46p,616b,593s stack positions out of 5000i,500n,6000p,200000b,5000s Output written on 31246-t.pdf (466 pages, 2864253 bytes). PDF statistics: 8446 PDF objects out of 8907 (max. 8388607) 1166 named destinations out of 1200 (max. 131072) 472 words of extra memory for PDF output out of 10000 (max. 10000000)